Strong unique continuation for the higher order fractional Laplacian

1.   Introduction

Higher order local and nonlocal elliptic equations arise naturally in problems from conformal geometry and scattering theory ^[1,2], (nonlinear) higher order elliptic PDEs and free boundary value problems ^[3,4], control theory ^[5,6] and inverse problems ^[7,8,9]. Motivated by these applications, in this article we study the strong unique continuation property for higher order fractional Schrödinger equations. More precisely, here we are concerned with equations of the form

where $\gamma \in \mathbb{R}_+ \setminus \mathbb{N}$ with suitable, possibly singular (critical and subcritical) potentials $q$. Here we say that a solution $u$ to (1.1) satisfies the strong unique continuation property if the condition that $u$ vanishes of infinite order at a point $x_0 \in \mathbb{R}^n$, i.e., if for all $m\in \mathbb{N}$

already implies that $u\equiv 0$ in $\mathbb{R}^n$. The strong unique continuation property can hence be viewed as a generalisation of analyticity to rougher equations.

Apart from dealing with the model setting of equation (1.1), we also address the corresponding differential inequalities and the setting of variable coefficient fractional Schrödinger operators with coefficients which might be only of low regularity. This provides new proofs for the results from ^[10] and ^[11], where respectively $C^2$ and $C^{0, 1}$ regular coefficients had been treated in the case $\gamma \in (0, 1)$, and extends appropriately adapted versions of these low regularity results to higher order equations. Further, we discuss possible applications of the unique continuation results to inverse and control theoretic problems.

1.1.   Main results on the strong unique continuation property

Let us outline our main results: As a model situation we deal with the strong unique continuation property (SUCP) for Schrödinger equations of the form (1.1). Without loss of generality, here we normalise our set-up such that $x_0 = 0$.

Theorem 1 (SUCP). Let $\gamma \in \mathbb{R}_+ \setminus \mathbb{N}$ and let $u\in H^{2 \gamma}(\mathbb{R}^n)$ be a solution to (1.1), where the potential $q$ satisfies the following bounds

Here $c_0 > 0$ is a sufficiently small constant, and $C_q > 0$ is an arbitrarily large, finite constant. Assume that $u$ vanishes of infinite order at $x_0 = 0$, i.e., for all $m\in \mathbb{N}$

Then $u\equiv 0$ in $\mathbb{R}^n$.

Remark 1.1. We remark that the limitation of the result to $\gamma > \frac{1}{4}$ arises naturally and was already present in ^[10]: Relying on Carleman estimates with weights which only have a radial dependence, we do not directly obtain positive boundary contributions but have to derive these through boundary-bulk interpolation estimates (see Lemma 2.2). With respect to bulk estimates, $L^2$ Carleman estimates are however subelliptic in the large parameter $\tau$. Through the boundary-bulk estimates this is propagated to the boundary which is then reflected in the loss of a quarter derivative in $\tau$ on the boundary. In the case that one only considers radial Carleman weights this loss seems unavoidable. In order to extend the unique continuation results to the regime $\gamma \in (0, \frac{1}{4})$ in a setting where only radial Carleman weights are used, the loss in $\tau$ has thus to be compensated by regularity of the potential (see ^[10] for corresponding results). In this case the lower order contributions would be included in the main part of the operator in the Carleman estimates (this then allows one to treat exact Hardy type potentials, but any type of perturbation of such potentials will need to obey regularity assumptions).

One could hope to avoid this loss of derivatives by considering Carleman weights which are not only of a radial structure but also depend on the normal directions. However, due to the weighted form of the inequalities, the exact Lopatinskii type conditions necessary for this are not immediate. We do not pursue this further in this article but postpone this to future work.

Remark 1.2. It would also have been possible to treat additional nonlinear terms in the the equations. As we are mainly interested in the associated differential inequalities, we do not consider them here.

Motivated by the work on the strong unique continuation properties on higher order elliptic equations (see ^[12] and the references therein), it is natural to wonder whether it is possible to extend the unique continuation property to (Hardy type higher) gradient potentials. Using iterative applications of our main Carleman estimate, we note that this is indeed the case:

Theorem 2 (SUCP with gradient potentials). Let $\gamma \in \mathbb{R}_+ \setminus \mathbb{N}$ and let $u\in H^{2 \gamma}(\mathbb{R}^n)$ be a weak solution to the differential inequality

where the potentials $q_j$ satisfy the following bounds

Here $c_0 > 0$ is a sufficiently small constant, and $C_{q_j} > 0$ are arbitrarily large, finite constants. Assume that $u$ vanishes of infinite order at $x_0 = 0$. Then $u\equiv 0$ in $\mathbb{R}^n$.

Remark 1.3. As in ^[12] it would have been possible to extend this result even further to (slightly subcritical) gradient potentials involving also contributions $|q_j(x)||\nabla^j u(x)|$ with $j \in (\lfloor{\gamma}\rfloor, \frac{3}{2}\lfloor{\gamma}\rfloor)$. As this requires some extra care, we do not present the details of this here but refer to the ideas in ^[12].

Further, relying on the methods from ^[10,13] as well as the spliting argument from ^[8], we also address the case with variable coefficient metrics and study the unique continuation properties at a point $x_0 \in \mathbb{R}^n$. In the sequel, without loss of generality, we will normalise our set-up such that $x_0 = 0$. Under this assumption, we consider the operator

for Lipschitz metrics with the following structural conditions:

$({\rm A}1)$ $\tilde{a}: \mathbb{R}^n \rightarrow \mathbb{R}^{n\times n}$ is symmetric, (strictly) positive definite, bounded.

$({\rm A}2)$ $\tilde{a}\in C^{\mu, 1}_{loc}(\mathbb{R}^n, \mathbb{R}^{n\times n}_{sym})$ with $[\tilde{a}^{ij}]_{\dot{C}^{\mu, 1}(B_4')} + [\tilde{a}^{ij}]_{\dot{C}^{0, 1}(B_4')} \ll \delta$, where ${B_4': = \{x\in \mathbb{R}^n: \ |x|\leq 4\}}$, for some small parameter $\delta > 0$. The constant $\mu > 0$ is specified below.

$({\rm A}3)$ We assume that $\tilde{a}^{ij}(0) = \delta^{ij}$.

For this class of coefficients, we can prove the analogue of Theorem 2:

Theorem 3 (SUCP with variable coefficients). Let $\gamma \in \mathbb{R}_+ \setminus \mathbb{N}$ and let $u\in \text{Dom}(L^{\gamma})$ be a solution to

where

● the metric $\tilde{a}$ satisfies the conditions (A1)–(A3) with $\mu = 2\lfloor{\gamma}\rfloor$,

● the potentials $q_j$ satisfy the following bounds

where $c_0 > 0$ is a sufficiently small constant, and $C_{q_j} > 0$ are arbitrarily large, finite constants.

Then the strong unique continuation property holds at $x_0 = 0$, i.e., if $u$ vanishes of infinite order at $x_0 = 0$, then $u\equiv 0$ in $\mathbb{R}^n$.

Remark 1.4. As explained in the Appendix (Section A.2), we interpret the variable coefficient fractional Laplacian through its spectral decomposition as directly related to a generalised Caffarelli-Silvestre extension. Relying on spectral theory in order to establish this, we restrict to functions ${u\in \text{Dom}(L^{\gamma})}$. Using ideas as outlined in ^[14] and ^[15], it would also have been possible to lower the required regularity of $u$ in this discussion.

Remark 1.5. Based on counterexamples to the weak unique continuation property with metrics of any $C^{0, \alpha}$ Hölder regularity with $\alpha \in (0, 1)$ due to Miller ^[16] and Mandache ^[17], it is expected that the coefficient regularity stated in condition (A2) in our variable coefficient strong unique continuation result is optimal in the case $\lfloor{\gamma}\rfloor = 0$. This strengthens the results from [10,Section 7] and ^[11].

The condition (A3) is to be read as a normalisation condition which can be assumed without loss of generality. We remark that condition (A2) together with interpolation estimates implies that $[a^{ij}]_{\dot{C}^{\ell, 1}(B_4')}\leq \tilde{C} \delta$ for any $\ell \in \{1, \dots, \mu\}$.

In both the settings of Theorems 1 and 3 also the unique continuation property from measurable sets (MUCP) holds:

Theorem 4 (MUCP). Let $\gamma \in \mathbb{R}_+\setminus \mathbb{N}$ and let $u \in \text{Dom}(L^{\gamma})$ be a solution to

where $\tilde{a}^{ij}$ satisfies the conditions (A1)–(A3) with $\mu = 2\lfloor{\gamma}\rfloor$ and $q\in L^\infty(\mathbb{R}^n)$. If there exists a measurable set $E\subset \mathbb{R}^n$ with $|E| > 0$ and density one at $x_0 = 0$ such that $u|_{E} = 0$, then $u\equiv 0$ in $\mathbb{R}^n$.

Let us comment on the results of Theorems 1–4 in the context of the literature on fractional Schrödinger equations: The weak unique continuation property, i.e., the question whether for solutions $u$ of (1.1) the condition that $u = 0$ on an open set in $\mathbb{R}^n$ already implies that ${u\equiv 0}$ in the whole of $\mathbb{R}^n$ is rather well understood for constant coefficient fractional Schrödinger equations, even for potentials in very rough, non-$L^2$-based function spaces (see ^[18]). In contrast, the understanding of the strong unique continuation properties of solutions to (higher order) fractional Schrödinger equations is still much less developed (for non-fractional higher order Schrödinger operators we refer to ^[12] and the references therein). The main known results are here given in the regime $\gamma \in(0, 1)$ and can be summarised as the following statements:

● Strong unique continuation for constant coefficient fractional Schrödinger equations with essentially $L^{\infty}$ potentials. In the regime $\gamma\in(0, 1)$ the articles ^[10,19] deal with the strong unique continuation property for $L^{\infty}$ as well as "Hardy type" critical and subcritical potentials. Both results crucially exploit the possibility of rephrasing the fractional Schrödinger operator in terms of the Caffarelli-Silvestre extension (see ^[20]), i.e., in terms of a (degenerate) Dirichlet-to-Neumann map associated with a (degenerate) elliptic, local equation in the upper half-plane. Technically, this allowed the authors of ^[19] to rely on frequency function methods for local equations, while, similarly, the key tool in ^[10] consisted of several Carleman inequalities for the Caffarelli-Silvestre extension.

● Unique continuation property from measurable sets. Relying on the arguments from ^[10,19] also unique continuation results from measureable sets can be proved in the regime $\gamma\in(0, 1)$. Indeed, in ^[19] this is formulated as one of the main results. For rougher equations this is deduced in ^[8] based on variants of the Carleman estimates from ^[10].

● Variable coefficient operators. Using more refined frequency function or Carleman estimates, also the case of variable coefficient fractional Schrödinger equations has been treated in [10,Section 7] ($C^2$ regular coefficients) and in ^[11] (Lipschitz regular coefficients).

In contrast, the situation for higher order fractional Schrödinger operators is much less studied. Here the main known properties are:

● Representation of the equation through a Caffarelli-Silvestre type extension problem. In ^[15] and in ^[1] and later also in ^[21] it was observed that the higher order fractional Laplacian can be realised as a Dirichlet-to-Neumann map of a Caffarelli-Silvestre type extension problem. This can either take the form of a scalar equation (however with a weight which is no longer in the Muckenhoupt class) or a system of Caffarelli-Silvestre extensions.

● Strong unique continuation for fractional harmonic functions. Exploiting the systems characterisation from ^[15], Yang also sketches the proof of the strong unique continuation property for fractional harmonic functions based on frequency function methods for systems of equations. In the regime $\gamma \in(1, 2)$ this was further detailed in ^[22], where the authors also obtained precise asymptotics of the solutions under consideration.

● Strong unique continuation for fractional Schrödinger equations. In the case $\gamma = \frac{3}{2}$ the strong unique continuation property for Schrödinger equations with Hardy type potentials was recently proved in ^[23]. In this context, the regime $\gamma = \frac{3}{2}$ is special, since the degeneracy of the problem disappears and the result can be viewed as a boundary unique continuation result for the Bilaplacian. The authors again rely on frequency function methods.

In contrast, in this article we study Carleman estimates to deduce the desired unique continuation property. Also relying on the systems Caffarelli-Silvestre extension of the higher order fractional Laplacian (which is recalled in the Appendix), we view the various unique continuation properties from above as boundary unique continuation results. In order to deduce these, we hence derive Carleman estimates for the associated systems. This is inspired by the work in ^[12] in which unique continuation in the interior is discussed for higher order equations (or equivalently systems). As in ^[10] we combine these Carleman estimates with careful compactness and blow-up arguments.

We emphasise that the results from Theorems 1–4 improve significantly on the known strong unique continuation results for the fractional Laplacian by for instance including (Hardy type) potentials and variable coefficients of low regularity. The strength of these aspects are even novel for the regime $\gamma\in(0, 1)$.

Remark 1.6. Using a characterisation of the higher order fractional Laplacian through a system and a bootstrap argument, for $a^{ij} = \delta^{ij}$ we here impose $H^{2\gamma}$ regularity on the solutions to the equations at hand. We remark that even in the setting of fractional Schrödinger equations in bounded domains, it would be possible to apply our arguments: Indeed, starting from $H^{\gamma}$ solutions, it would be possible to bootstrap the regularity properties of the solutions by means of the estimates in ^[24] (away from the boundary).

We however emphasise that by pseudolocality of the fractional Laplacian our results could also be formulated under only local regularity assumptions: Assuming that we had a weak notion of a generalised Caffarelli-Silvestre extension (which is the case for any $H^{r}(\mathbb{R}^n)$, $r\in \mathbb{R}$, boundary datum, see Section A.1) as well as only local $H^{2\gamma}(B_1')$ regularity for $u$, it would have been possible to invoke our unique continuation arguments. We refer to Proposition 1.9 and Remark 1.10 for more on this.

1.2.   Main ideas

Approaching the problem by means of the generalised systems Caffarelli-Silvestre extension, our arguments for unique continuation rely on several Carleman estimates for the localised equation in combination with a careful blow-up analysis. To this end, we rely on similar ideas as in ^[8,10].

A crucial technical tool thus is the derivation of higher order Carleman estimates, which we address by iteration of second order estimates. As a consequence, we obtain the following bounds:

Proposition 1.7. Let $m\in \mathbb{N}\cup \{0\}$ and $b\in (-1, 1)$. Let $a\in C^{2m, 1}(B_4^+, \mathbb{R}^{(n+1)\times(n+1)})\cap C^{0, 1}(B_4^+, \mathbb{R}^{(n+1)\times (n+1)})$ be of a block form

where the metric $\tilde{a}$ satisfies the conditions (A1)–(A3) from above with $\mu = 2m$. Assume that $\tilde{u}_0, \dots, \tilde{u}_m \in H^{1}(B_4^+, x_{n+1}^b)$ with $\mathit{\text{supp}}(\tilde{u}_j) \subset B_{4}^+\setminus\{0\}$, $j\in \{0, \dots, m\}$, are solutions to the bulk system

in $B_4^+$ with $f_{0}, \dots, f_m \in H^{1}(B_4^+, x_{n+1}^b)$. Further suppose that on $B_4'$ we have

where all limits are considered with respect to the $L^2$ topology. Then, there exists $\tau_0 > 1$ such that for all $\tau > \tau_0$ there is a weight $h$ such that

Here $\overline{h}(x): = h''(t)|_{t = -\ln(|x|)}$.

This estimate improves previous results even in the case $m = 0$ by allowing for only Lipschitz continuous metrics $a$. It includes strengthened bounds which exploit the spectral gap of the fractional Laplacian on the sphere with Neumann (or Dirichlet) conditions (see the Appendix A, Section 8.3 in ^[25] for these spectral gap properties). A relevant ingredient in the derivation of this Carleman estimate involves the use of a splitting technique in a similar way as in ^[8].

Estimating the commutators, the bounds from Proposition 1.7 can be further improved to include higher order tangential derivatives on the left hand side of (1.7):

Proposition 1.8. Let $m\in \mathbb{N}\cup \{0\}$, $b\in (-1, 1)$ and $a: \mathbb{R}^{n+1}_+ \rightarrow \mathbb{R}^{(n+1)\times (n+1)}$ as in Proposition 1.7. Assume that $\tilde{u}_0, \dots, \tilde{u}_m \in H^{1}(B_4^+, x_{n+1}^b)$ with $\mathit{\text{supp}}(\tilde{u}_j) \subset B_{4}^+\setminus\{0\}$, $j\in \{0, \dots, m\}$, are solutions to the bulk system (1.5) in $B_4^+$ with $f_{j}\in{H^{m-j}(B_4^+, x_{n+1}^b)}$ for $j\in \{0, \dots, m\}$. Further suppose that on $B_4'$ the boundary conditions (1.6) hold, where all limits are considered with respect to the $L^2$ topology. Then, there exists $\tau_0 > 1$ such that for all $\tau > \tau_0$ there is a weight $h$ such that

Here $\overline{h}(x): = h''(t)|_{t = -\ln(|x|)}$.

It is in this form that we exploit the Carleman estimates to infer our main results.

1.3.   Applications of the unique continuation results

Motivated by the recent introduction of the fractional Calderón problem ^[7,8,9], we here discuss applications of our unique continuation results in inverse problems. As a first main property, we deduce the antilocality of the higher order fractional Laplacian:

Proposition 1.9 (Antilocality). Let $\gamma \in \mathbb{R}_+\setminus \mathbb{N}$ and let $L = -\nabla \cdot \tilde{a} \nabla$, where $\tilde{a}$ satisfies the conditions (A1)–(A3) from above with $\mu = 2\lfloor{\gamma}\rfloor$. Let $u\in \text{Dom}(L^{\gamma})$. Assume that for some open set $W\subset \mathbb{R}^n$ containing the ($n$-dimensional) unit ball $B_1 \subset \mathbb{R}^n$ we have

Then, $u\equiv 0$ in $\mathbb{R}^n$.

We emphasise that in this result the function $u$ is not assumed to satisfy any equation globally. This result thus provides a strong global rigidity property in which the nonlocality of the equation under consideration plays a major role. Originally, the notion of antilocality appeared in the context of quantum field theory as the Reeh-Schlieder theorem ^[26], but has recently found various applications in inverse problems ^{[7,8,9,27,28]} and control theory ^[5,6].

Remark 1.10. We remark that a result of the form stated in Proposition 1.9 also holds in a large range of less regular spaces. The only ingredient needed is the presence of a Caffarelli-Silvestre extension at the given regularity (but this holds under very weak assumptions, see Proposition A.6 in the constant coefficient setting). The pseudolocality of the associated operators then allows one to locally deduce the vanishing of $u$ from which it is possible to propagate the deduced information to an arbitrary point through the upper half plane (in which the extension problem is formulated).

As a property dual to the antilocality of the higher order fractional Laplacian, we further obtain approximation properties for these operators:

Proposition 1.11 (Runge approximation). Let $\Omega \subset \mathbb{R}^n$ be open, bounded and non-empty and let $\widetilde{\Omega} \subset \mathbb{R}^n$ be open with $\Omega \subset \tilde\Omega$ and $B'_1\subset\tilde\Omega\setminus\Omega$. Let $v\in H^{\gamma}(\Omega)$ with $\gamma \in \mathbb{R}_+ \setminus \mathbb{N}$. Assume that $L = -\nabla \cdot \tilde{a} \nabla$, where $\tilde{a}$ satisfies the conditions (A1)–(A3) from above with $\mu = 2\lfloor{\gamma}\rfloor$. Suppose that $q\in L^{\infty}(\Omega)$ is such that zero is not a Dirichlet eigenvalue of the operator $L^{\gamma} + q$. Then, for any $\epsilon > 0$ there exists a solution to

such that $\|v-u\|_{L^2(\Omega)}\leq \epsilon$.

Remark 1.12. The assumptions on the sets $W$ and $\tilde\Omega\setminus\Omega$ with respect to the unit ball $B'_1 = \{x\in \mathbb{R}^n: |x| < 1\}$ are taken without loss of generality after normalising the set-up in such a way that the assumptions (A2), (A3) on $\tilde a$ are satisfied.

These type of approximation properties again crucially exploit the nonlocality of the operator. They were first observed in ^[29] and generalised to larger classes of equations in ^{[28,30,31,32,33,34]}. As first highlighted in ^[7] in the context of nonlocal inverse problems they play an important role in deducing injectivity. In ^[9] these properties were strengthened to quantitative estimates which were applied in proving stability of the associated inverse problem.

In addition to the rigidity and flexibility properties from Propositions 1.9 and 1.11, it is possible to make use of our unique continuation results in many further contexts. As in ^[22,23,35] one could for instance study the associated problems more quantitatively and derive vanishing order or nodal domain estimates. Using Carleman estimates, one could here proceed similarly as in ^[36]. Also control theoretic questions similar to for instance ^[6] could be addressed with our results. We postpone such a discussion to future work.

1.4.   Organisation of the article

The remainder of the article is organised as follows: After first recalling a number of auxiliary results (including the generalised Caffarelli-Silvestre extension) in Section 2, in Section 3 we then deduce the Carleman estimates which form the basis of our unique continuation results. In Section 4 we derive compactness results for the systems which are associated with the SUCP for fractional Schrödinger operators. Here we reduce the SUCP to the weak unique continuation property (WUCP) for the associated systems. This is complemented by a bootstrap argument to derive the WUCP in Section 5. In Section 6 we combine all the previous results and deduce the statements of Theorems 1–4 and Propositions 1.9 and 1.11. Finally, in the Appendix, we present a sketch of the derivation of the generalised Caffarelli-Silvestre extension for the higher order fractional Laplacian which had been introduced in ^[15] and which we here discuss at low regularity.

2.   Auxiliary results

In this section we recall several auxiliary results that will be used frequently throughout the text: On the one hand, we recall a higher order Caffarelli-Silvestre extension result. On the other hand, we discuss appropriate boundary-bulk estimates.

2.1.   Notation

We summarise the notation that we will use in the sequel:

2.1.1.   Sets

Working in $\mathbb{R}^{n+1}_+: = \{x\in \mathbb{R}^{n+1}: \ x_{n+1}\geq 0\}$, we will always use the convention that $x = (x', x_{n+1})$ with $x'\in \mathbb{R}^n$ and $x_{n+1}\geq 0$. For $x_0 \in \mathbb{R}^n \times \{0\}$ we will denote (half) balls in $\mathbb{R}^{n+1}_+$ and $\mathbb{R}^n \times \{0\}$ by

If $x_0 = 0$, we will simply write $B_r^+$ and $B_{r}'$.

In the sequel and in particular in the proofs of the Carleman inequalities, we will often use conformal polar coordinates in the upper half space: $x = e^{-t}\theta$, where $t = -\ln(|x|)$ and $\theta = \frac{x}{|x|}$. Here $\theta \in S^{n}_+$, where $S^n_+$ denotes the upper half (unit) sphere in $\mathbb{R}^{n+1}_+$.

2.1.2.   Operators

We will frequently use the operator

where $L = -\nabla'\cdot\tilde a\nabla'$ and $\tilde a$ satisfies the conditions from (A1)–(A3) with $\mu = 2\lfloor{\gamma}\rfloor$. Usually, we will be thinking of $b = 1-2\lfloor{\gamma}\rfloor+2\gamma$, where $\lfloor{t} = {\max}\{k\in \mathbb{N}: \ k\leq t\}$.

We define the variable coefficient fractional Laplacian through functional calculus, see Section A.2. The set $\text{Dom}(L^{\gamma})$ is then defined as the space in which this functional calculus can be directly applied.

In analogy to our convention for the space variables, we often use the notation

to denote the corresponding tangential gradient and partial derivatives.

2.1.3.   Function spaces

Studying operators of the form (2.1), in the sequel, we will often work in function spaces of the form

where $\Omega \subset \mathbb{R}^{n+1}_+$ is a relatively open set. Similarly, we also deal with the function spaces

We denote the homogeneous Hölder norms by $[\cdot]_{\dot{C}^{k, \alpha}(\Omega)}$ for $k\in \mathbb{N}\cup\{0\}$, $\alpha \in (0, 1)$ and $\Omega \subset \mathbb{R}^n$ or $\Omega \subset \mathbb{R}^{n+1}_+$.

For a sequence $\{a_{k}\}_{k\in \mathbb{N}} \subset \ell^1$, we set

2.2.   The higher order fractional Laplacian

We first recall that the higher order fractional Laplacian can be interpreted by means of a Caffarelli-Silvestre-type extension ^[1,15,20,21]:

Proposition 2.1. Let $\gamma > 0$, $f\in \text{Dom}(L^{\gamma})$ and $L: = -\nabla' \cdot \tilde{a} \nabla'$, where $\tilde{a}$ satisfies the conditions (A1)–(A3) with $\mu = 2\lfloor{\gamma}\rfloor$. Then, there exists an extension operator

such that $u = E_{\gamma}(f)$ is a weak solution to the scalar higher order problem

Here $L_b: = x_{n+1}^{-b}(\partial_{x_{n+1}} x_{n+1}^b \partial_{x_{n+1}} + x_{n+1}^{b} \nabla' \cdot \tilde{a}\nabla')$ and $b = 1-2\lfloor{\gamma}\rfloor+2\gamma$. All boundary conditions in (2.2) are attained as $L^2(\mathbb{R}^n)$ limits.

Moreover, setting $u_0: = u$ and $u_{j+1} = L_b u_{j}$ for $j\in\{0, \dots, \lfloor{\gamma}\rfloor-1\}$, the functions $u_j$ are in $C^{2, 1}(\mathbb{R}^n \times (0, 1))\cap H^{1}_{loc}(\mathbb{R}^{n+1}_+, x_{n+1}^b)$. They are weak solutions of the following system of second order equations

where $m = \lfloor{\gamma}\rfloor$. All boundary conditions hold in an $L^2(\mathbb{R}^n)$ sense.

In order to keep our presentation self-contained, we provide a short sketch of the proof of this statement in the Appendix.

Compared to the original nonlocal Eqs (1.1) and (1.2), the Eqs (2.2) and (2.3) arising from the generalised Cafferelli-Silvestre extension have the advantage that they can be approached with tools from the analysis of unique continuation properties of local elliptic equations. As in ^[10], ^[37] the price to pay for this localisation is the introduction of the additional dimension in which the solutions $u$ have to be controlled through the corresponding equations. This gives our problem and our argument the flavour of boundary unique continuation results (see for instance ^[38] and the references therein).

2.3.   Boundary-bulk interpolation estimates

We recall the boundary-bulk interpolation inequality from ^[10] for fractional Sobolev spaces on the sphere:

Lemma 2.2. Let $u:S^{n}_+ \rightarrow \mathbb{R}$ and let $b\in (-1, 1)$. Then, identifying $\partial S^{n}_+$ with $S^{n-1}$, there exists a constant $C > 0$ such that for any $\tau > 1$

Proof. The proof follows as in ^[37] by using the trace inequality in the associated weighted Sobolev spaces. We discuss the details:

First, by the trace inequality, any function $w\in H^{1}(\mathbb{R}^{n+1}_+, x_{n+1}^{b})$ with $b\in (-1, 1)$ satisfies

Indeed, for $w \in C^{\infty}(\mathbb{R}^{n+1}_+)$ with compact support in the tangential directions this follows from the fundamental theorem of calculus: For $t\in (0, 1)$

where we used that $b\in (-1, 1)$. Thus, applying the triangle inequality, integrating in $t\in [0, 1]$ and applying the Cauchy-Schwarz inequality, we infer

Taking squares and integrating in $x'\in \mathbb{R}^n$ then yields

By density considerations, this concludes the proof of the trace estimate (2.4).

Rescaling (2.4), we then infer

Finally, we apply this to $w(x) = \eta(|x|)u(\frac{x}{|x|})$, where $\eta$ is a smooth, positive cut-off function supported on $B_{2}^+ \setminus B_{1/2}^+$ and which is identically one on $B_{3/2}^+ \setminus B_{3/4}^+$. This yields the desired claim.

2.4.   Caccioppoli inequality

We derive a Caccioppoli type inequality for tangential derivatives of a solution to a variable coefficient equation associated with the operator $L_b$ from above.

Lemma 2.3. Let $b\in(-1, 1)$ and $a:B_4^+ \rightarrow \mathbb{R}^{(n+1)\times(n+1)}$ be of the block form (1.4) with $\tilde a$ satisfying the conditions (A1)–(A3) with $\mu = 2m$. Let $u\in H^1(B^+_4, x_{n+1}^b)$ be a solution to

with $f, |\nabla' f|, \cdots, |(\nabla')^k f| \in L^2(B_4^+, x_{n+1}^b)$ and $g\in H^k(B_4')$ for some $k\in \mathbb{N} \cup \{0\}$. Then there exists a constant $C > 0$ such that for any $J\in \mathbb{N}\cup \{0\}$ with $J\leq\min\{2m, k\}$ and for any $r\in (0, 2)$

Proof. First of all we note that by scaling it suffices to prove the estimate for $r = 1$. Next, we observe that by the block structure of the metric $a$ for any $j\in\{1, \dots, 2m\}$

provided that these expressions are $j$-times differentiable in the tangential directions. Hence, formally $(\nabla')^j u$ is a weak solution to

Using difference quotient arguments together with the regularity of $f$ and $g$, these formal differentiations in the tangential directions can be justified. As a consequence, elliptic regularity implies that $(\nabla')^j u \in H^{1}(B_3^+, x_{n+1}^b)$. In particular, for any test function $\varphi$ it holds

Let $\eta$ be a radial cut-off function equal to one on $B^+_{1/2}$ which vanishes outside of $B^+_{1}$ and satisfies $|\nabla \eta|\leq C$. We remark that the function $\varphi = \eta^2 (\nabla')^j u$ is an admissible test function {since by the above considerations as an $H^{1}(B_1^+, x_{n+1}^b)$ function it is sufficiently regular}. Using that by condition (A2) it holds $|(\nabla')^\alpha\tilde a(x)| \leq C_{\alpha}$ for $x\in B_{4}^+$ and $|\alpha|\leq 2m$, we obtain the following estimate

By virtue of Young's inequality and absorbing terms into the left hand side we infer

Lastly, we use the bounds for $\eta$ to obtain

The estimate for $j = 0$ is straightforward. Rescaling and summing over $j\in \{0, \dots, J\}$ with the corresponding factors, yields the desired estimate.

3.   Carleman estimates for systems in the upper half-plane

In order to deduce the Carleman estimates from Propositions 1.7 and 1.8, we first prove Carleman estimates for second order (degenerate elliptic) equations. Here we proceed in two steps: First, we discuss the situation for constant coefficient metrics but in the presence of divergence form right hand side contributions, and then, in a second step, we deduce the estimates for variable coefficient metrics.

3.1.   Constant coefficient Carleman estimates

As a main ingredient in our argument we make use of the following (constant coefficient) Carleman estimate:

Proposition 3.1. Let $b \in (-1, 1)$ and let $u\in H^{1}(B_4^+, x_{n+1}^b)$ with $\mathit{\text{supp}}(u)\subset B_4^+ \setminus \{0\}$ be a solution to

where $f \in L^{2}(B_4^+, x_{n+1}^{-b})$, $g\in L^2(B_4')$ and $F = (F^1, \dots, F^n) \in L^2(B_4^+, x_{n+1}^b)^n$ with $\mathit{\text{supp}}(f)$, $\mathit{\text{supp}}(F) \subset B_4^+ \setminus \{0\}$ and $\mathit{\text{supp}}(g)\subset B_4' \setminus \{0\}$. Then, for each $\tau > \tau_0\gg 1$ there is a weight function $h(-\ln(|x|))$ such that there exists a constant $C > 0$ which is independent of $\tau$ such that it holds

Here $\overline{h}(x): = h''(t)|_{t = -\ln(|x|)}$.

The proof of Proposition 3.1 relies on a splitting strategy, in which all inhomogeneities (be they bulk or boundary contributions) are dealt with in an elliptic estimate. As a consequence, the subelliptic part, i.e., the actual Carleman estimate, becomes rather clean. In particular, as shown in the following section, the estimates are strong enough to deal with only Lipschitz regular metrics in a perturbative way.

Proof. We proceed in three main steps: First, we construct an appropriate Carleman weight. Then, we deduce the desired Carleman estimate by a splitting argument in conformal polar coordinates. In a final step, we concatenate the obtained information.

Step 1: Construction of the weight. We begin by constructing a family of Carleman weights $h(t): \mathbb{R} \rightarrow \mathbb{R}$. Anticipating the use of polar conformal coordinates, we require it to satisfy

where $\varepsilon > 0$ is a constant which is to be determined later (see Step 1 in the proof of Proposition 1.7), and $C > 1$ is a constant independent of $\tau$. We recall that by the results of ^[25] the operator $\nabla_{S^n} \cdot \theta_n^{{b}} \nabla_{S^n}$ with $\theta_n: = \frac{x_{n+1}}{|x|}$ has a spectral gap if it is considered with Neumann (or Dirichlet) data (see Section 8.3 in the Appendix A in ^[25]). We follow the argument from ^[13] and ^[12] to obtain the desired properties for the Carleman weight. To this end, we consider a sequence $\{c_j \}_{j\in \mathbb{N}} \in \ell^1$, $\|c_j\|_{\ell^1} < \delta$ of non-negative numbers and define the sequence $\{a_j\}_{j\in \mathbb{N}}$ as the convolution of $c_j$ with $2^{-\nu j}$ for some $\nu > 0$ small. Then, the sequence $a_j$ is slowly varying (i.e., $2^{-\nu} a_{j+1}\leq a_j \leq 2^{\nu} a_{j+1}$) and obeys the bound $c_j \leq a_j$. With this preparation, we define $h(0) = 0$, $h'(-\infty) = \lfloor{\tau}\rfloor + c_1$, where $c_1 > 0$ is a suitable constant independent of $\tau$, and

with

In order to obtain the desired higher order regularity properties for $h$, we regularise this by convolution (on the unit scale). Thus, the last estimate in (3.2) is fulfilled for any given, small $\varepsilon > 0$ provided that the difference of consecutive values of $a_j$ is small enough, i.e., if $\delta$ and $\nu$ are taken sufficiently small.

Step 2: Conformal polar coordinates and splitting argument. We proceed by a splitting argument. In order to obtain more transparent expressions, we pass to conformal coordinates by setting $t = -\ln(|x|)$, $\theta = \frac{x}{|x|}$. We further pass from the function $u$ to the function $\tilde{u}(t, \theta) = e^{\frac{1-b-n}{2}t} u(e^{-t}\theta)$. In these coordinates we consider (the weak form) of the equation

where

Here $\text{div}_{S^n_+}$ denotes the divergence with respect to the standard metric on the sphere and the choice of the sign in the expression for $\tilde{F}^t(t, \theta)$ depends on the specific chart.

We split the problem into two parts $\tilde{u} = u_1+u_2$, where $u_1$ is a solution to the following elliptic problem

Here $K\gg 1$ is a sufficiently large parameter (to be specified later). The function $u_2 = \tilde{u}-u_1$ thus solves a corresponding problem. In order to derive the desired estimate, we discuss the bounds for $u_1$ and $u_2$ separately.

Step 2a: Bounds for $u_1$. By virtue of the positivity of $K\gg 1$, the estimates for $u_1$ are elliptic energy estimates. Indeed, by the Lax-Milgram theorem, we obtain that a solution to (3.5) exists in the energy space $H^{1}(S^n_+ \times \mathbb{R}, \theta_n^{b})$. We test the weak form of (3.5) with the test function $\tau^2 e^{2h_{M, \delta}(t)} u_1$, where

for $M \in \mathbb{N}$ and with $\eta_{\delta}$ denoting a standard mollifier.

This leads to the following identity

Here $(\cdot, \cdot): = (\cdot, \cdot)_{L^2(S^n_+ \times \mathbb{R})}$ and $(\cdot, \cdot)_0: = (\cdot, \cdot)_{L^2(\partial S^n_+ \times \mathbb{R})}$. Recalling that by definition (see (3.2))

an application of Young's inequality leads to

Absorbing the contributions involving $u_1$ as well as the other non-positive terms from (3.6) into either the positive derivative contributions or (for $K^2\gg 1$ sufficiently large) into the coercive term involving $K^2$ in (3.6), then yields

As above, here and in the sequel, we use the notation $\|\cdot \|: = \|\cdot \|_{L^2(S^n_+ \times \mathbb{R})}$ and $\|\cdot \|_0 : = \|\cdot \|_{L^2(\partial S^n_+ \times \mathbb{R})}$. Applying the boundary-bulk interpolation estimate from Lemma 2.2, allows us to absorb the first boundary contribution into the left hand side, which then results in

Using the compact supports of $\tilde{f}$, $\tilde{F}$ and $\tilde{g}$, by dominated convergence, we may pass to the limits $M \rightarrow \infty$ and $\delta \rightarrow 0$ which leads to

We remark that this estimate not only contains the right weighted bounds for $u_1$ but also implies that $u_1$ has (quantitative) fast decay as ${|t|} \rightarrow \infty$ (which corresponds to $|x|\rightarrow 0$ and $|x| \rightarrow \infty$). By the compact support assumption on $\tilde{u}$ a similarly fast decay then also holds for $u_2$.

Step 2b: Bounds for $u_2$. The estimates for $u_2$ will be sub-elliptic (in $\tau$) Carleman estimates. We recall that by construction, $u_2$ is a weak solution of

i.e., it satisfies

We test this with a Neumann eigenfunction to the spherical operator, i.e., with a function $\psi_{\lambda}$ which satisfies

We recall that the set $\{\psi_{\lambda}\}_{\lambda}$ forms an orthogonal, complete system in $L^2(\theta_n^{b}, S^n_+)$. More precisely, we insert $\varphi(t, \theta) = \psi_{\lambda}(\theta)\gamma(t)$ with $\gamma(t)$ a compactly supported smooth function into (3.11). Since the set $\{\psi_{\lambda}\}$ forms an orthonormal set in both $H^{1}(S^n_+, \theta_n^b)$ and $L^2(S^n_+, \theta_n^b)$, we infer that $\alpha_{\lambda}(t): = (u_2, \theta_n^{b} \psi_{\lambda})$ is a distributional (but then by uniqueness also a classical) solution to the ODE

where $\beta_{\lambda}(t) = (u_1, \theta_n^{b}\psi_{\lambda})$.

Conjugating this modewise equation with the weight $e^{h(t)}$ yields the equation

where $\tilde{\alpha}_{\lambda} = e^{h(t)} \alpha_{\lambda}$ and $\tilde{\beta}_{\lambda} = e^{h(t)} \beta_{\lambda}$. Noting that the symmetric and antisymmetric parts of the conjugated operator turn into

we expand the conjugated operator to infer

where

We observe that the first two contributions in the expansion of the commutator are non-negative. The last term which does not necessarily carry a sign can be absorbed into these positive contributions (recall the last condition in (3.2) for that) and can hence be neglected for $\tau \geq \tau_0 > 1$ sufficiently large, whence

if $\tau \geq \tau_0 > 1$ is chosen sufficiently large.

Next we exploit the spectral gap of the Neumann data version of the operator $\nabla_{S^n}\cdot \theta_n^{b} \nabla_{S^n}$ (see [25,Appendix A, Section 8.3]) in connection with the symmetric and antisymmetric parts of the operator. To this end, we consider two cases: If $\lambda^2 \in [0, 2C^2 \tau^2)$ with $C > 1$ as in (3.2), we have that $|h'|\geq\frac{C^{-2}}{\sqrt{2}}\lambda$. Using the antisymmetric part of the operator, we estimate

If $\lambda^2 \geq 2C^2 \tau^2$, we estimate the symmetric part of the operator and integrate by parts

We note that non-positive or not necessarily positive contributions are given by the third and the fifth term on the right hand side of (3.16), which we hence seek to bound. Since $\lambda^2 \geq 2C^2 \tau^2\geq \frac{|h'|^2}{2}$, taking $\tau > \tau_0$ large enough, we have

Thus the fifth term on the right hand side of (3.16) can be absorbed into the fourth term on the right hand side of (3.16). We estimate the remaining possibly non-positive contribution which is the third term (as $\lambda^2 \geq 2C^2 \tau^2$): To this end, we again integrate by parts and obtain that for sufficiently large $\tau \geq \tau_0 > 1$

Thus, if $\lambda^2 \geq 2C^2 \tau^2$, we infer

Hence, combining (3.15) and (3.17), we deduce that for some constant $c_0 > 0$ which is independent of $\tau > 1$

Further using the antisymmetric part to bound (for $\tau \geq \tau_0 > 1$ sufficiently large)

and combining this with (3.14) and (3.18) we arrive at

We remark that we have given up a factor $\tau^2$ in the antisymmetric estimate. This is due to the fact, that in undoing the conjugation with the weight $e^{h(t)}$, we obtain a term originating from the $t$ derivative falling onto the weight. Without the loss of the factor $\tau^2$ this would carry a weight $\tau^4$. We would not be able to absorb this into the $L^2$ contributions on the left hand side of the estimates.

We further complement the estimate (3.19) by a bound on the spherical part of the gradient. To this end, we make use of the symmetric part of the operator. Indeed, we have

As a consequence, if $c_0 > 0$ is sufficiently small,

Here the last estimate follows from the previously deduced bounds from (3.19). Hence, we conclude

By orthogonality, summing the estimate (3.20) over $\lambda$, integrating over $S^n_+$, using the properties of $h$ and undoing the conjugation, we thus obtain

Step 3: Conclusion. Last but not least, we combine the estimates from Steps 1 and 2 and deduce the Carleman estimate from Proposition 3.1 from this. We obtain

Using the bulk-boundary interpolation estimate from Lemma 2.2, this can further be strengthened by a boundary contribution on the left hand side:

Transforming back into Cartesian coordinates yields the desired estimate.

Analogous arguments as in the proof of Proposition 3.1 allow us to derive a slight variation of the estimate from Proposition 3.1 which we will exploit in the sequel.

Corollary 3.2. Let $K$ be a finite set of positive integers. Under the same assumptions as in Proposition 3.1, for each $\tau > \tau_0\gg 1$ there is a weight function $h(-\ln(|x|))$ such that there exists a constant $C > 0$ which is independent of $\tau$ such that for any $k\in K$ it holds

Proof. Let $h$ be the family of Carleman weights constructed in the proof of Proposition 3.1 and consider for $k\in K$ the following new family

It also satisfies (3.2) for $\tau > \tau_0$ and $\tau_0$ sufficiently large: Notice that $\tilde h'$ differs from $h'$ by a constant and hence all the higher order derivatives coincide. In particular, the first and the last estimates in (3.2) hold directly. Adding a suitable extra term of the form $\alpha t$ to the original $h$, we can ensure to keep a positive distance between $\tilde h'$ and the spectrum of $\nabla_{S^n}\cdot \theta_n^b\nabla_{S^n}$ for finitely many $k$.

Therefore, estimate (3.1) holds with $\tilde h$ playing the role of $h$. Finally, observe that

so the claimed estimate is obtained.

3.2.   Variable coefficient metrics

Considering second order equations of the form

in the sequel, we seek to introduce variable coefficients in the Carleman estimate of Proposition 3.1. Throughout this section, the metric $a$ is assumed to be of a block form as in (1.4) where the metric $\tilde{a}$ satisfies the conditions (A1)–(A3) from the introduction with $\mu = 0$.

We first note that the estimate in Proposition 3.1 remains valid for a constant coefficient metric in the block form (1.4). This follows immediately from a change of coordinates (only involving the tangential variables). In order to extend Proposition 3.1 to variable coefficient problems, we exploit the presence of the divergence contribution and in conformal coordinates localise the problem to scales of the size $C(a_j \tau)^{-\frac{1}{2}}$ or of size one, respectively (depending on the size of the metric). Here $\{a_j\}_{j\in \mathbb{N}}$ denotes the sequence that was used in the definition of the Carleman weight $h$ (see Step 1 in the proof of Proposition 3.1).

We follow the argument in ^[13] and argue in two steps: First, in the regime in which $h$ is convex, we localise to very small scales (Lemma 3.3). In the regime in which no convexity is present anymore, we localise to scales of order one in conformal coordinates (Lemma 3.4). Finally, we patch these estimates together to derive the desired global bound of Proposition 1.7.

Lemma 3.3. Let $\tau \geq 1$ and $\epsilon > 0$. Assume that $h$ is convex and

Assume that

for some sufficiently small constant $\delta > 0$. Then, for all $u$ with $\mathit{\text{supp}} (u) \subset I_\ell$ and all $\tau \geq \tau_0\geq 1$ we have

Proof. Step 1: Restricted support. We first assume that $u$ is supported on a ball $B_{C_0|x_0|(\epsilon \tau)^{-\frac{1}{2}}}(x_0)$ for some $x_0 \in I_\ell$ or in some half ball $B_{C_0|x_0|(\epsilon \tau)^{-\frac{1}{2}}}^+(x_0)$ for some $x_0 \in I_\ell\cap (\mathbb{R}^n \times \{0\})$ and where the constant $C_0 > 0$ is still to be determined (see Step 2). As the arguments are similar in both cases, we only discuss the case of the full ball in detail in the sequel. We note that for $x\in B_{C_0|x_0|(\epsilon \tau)^{-\frac{1}{2}}}(x_0)\cap I_\ell$ we have

Next, we apply the Carleman estimate from Proposition 3.1 to the equation

where $f = \partial_i x_{n+1}^b a^{ij} \partial_j u$ and where we recall the block structure (1.4) of $a^{ij}$. By virtue of Proposition 3.1 we obtain

In order to deduce the desired estimate under the support constraint, it suffices to bound the second bulk term on the right hand side of (3.24). To this end, we invoke (3.23) and estimate

For $\delta > 0$ sufficiently small (but independent of $u$), it is possible to absorb this contribution into the left hand side of (3.24).

Step 2: Localisation. We seek to apply the previous argument by localising a general solution $u$ with $\text{supp} (u) \subset I_\ell$ by a partition of unity. Here commutator estimates play a crucial role and provide a natural limitation to the possible localisation scale.

We consider a partition of unity $\{\psi_k\}_{k\in \{1, \dots, K\}}$ associated with the half annulus $I_\ell$ and a finite collection $\{x_{k}\}_{k\in \{1, \dots, K\}}$ of points in $I_\ell$ such that $\text{supp}(\psi_k) \subset B_{C_0|x_k|(\epsilon \tau)^{-\frac{1}{2}}}(x_k)$ or $\text{supp}(\psi_m) \subset B_{C_0|x_{m}|(\epsilon \tau)^{-\frac{1}{2}}}^+(x_{m})$ (where in the latter case $x_m \in I_{\ell}\cap (\mathbb{R}^n \times \{0\})$) and such that the balls and half-balls $B_{C_0|x_k|(\epsilon \tau)^{-\frac{1}{2}}}(x_k)$ and $B_{C_0|x_{m}|(\epsilon \tau)^{-\frac{1}{2}}}^+(x_{m})$ cover the interval $I_\ell$ (with controlled overlap). Without loss of generality, we choose the partition of unity and the points $x_k$ such that the following estimate hold:

and $C_1 = C_1(C_0) > 0$. Further, by for instance choosing $\psi_k$ to have a dependence on $|x-x_k|$ only, we may assume that

in $I_{\ell}$. We then write $u = \sum\limits_{k = 1}^K \psi_k u$. With this in hand, we apply the triangle inequality as well as the Carleman estimate from Step 1:

We first consider the bulk contribution on the right hand side for which

Using the bounds for $\psi_k$ from (3.25) as well as the estimate for $|\nabla a^{ij}|$, we obtain

Choosing $C_1 = C_1(C_0) > 1$ sufficiently large (by choosing $C_0 > 0$ appropriately), then allows us to absorb the second and third contribution from the right hand side of (3.27) into the left hand side of (3.26).

For the boundary term in (3.26), we use the fact that by construction of the partition of unity $\lim\limits_{x_{n+1}\rightarrow 0} x_{n+1}^{b} \partial_{n+1} (\psi_k u) = \psi_k \lim\limits_{x_{n+1}\rightarrow 0} x_{n+1}^{b} \partial_{x_{n+1}} u$.

Using this and the finite overlap of the supports of the functions $\psi_k$, allows us to turn (3.26) into

which concludes the proof of the argument.

Similarly as in Lemma 3.3, it is also possible to deal with the situation of even smaller perturbations of the metric without invoking the convexity of the weight:

Lemma 3.4. Let $\tau \geq 1$. Assume that

for some sufficiently small constant $\delta > 0$. Then, for all $u$ with $\mathit{\text{supp}} (u) \subset I_\ell$ and all $\tau \geq \tau_0\geq 1$ we have

Proof. The argument follows along the same lines as the proof of Lemma 3.4, however now we directly localise to scales of the order $C_0|x_k|$ around a finite number of points $x_k \in I_\ell$. Using an associated partition of unity then yields the desired result.

Relying on the previous result, we obtain global Carleman estimates:

Proposition 3.5. Let the metric $a: \mathbb{R}^{n+1}_+ \rightarrow \mathbb{R}^{(n+1)\times (n+1)}$ be of a block form as in (1.4) where the metric $\tilde{a}$ is assumed to satisfy the conditions (A1)–(A3) with $\mu = 0$. Let $u\in H^1(B^+_4, x_{n+1}^b)$ with ${ \mathit{\text{supp}}(u)\subset B^+_4\backslash\{0\}}$. Then, for each $\tau > \tau_0\geq 1$ there exist a weight function $h$ and a constant $C > 0$ independent of $\tau$ such that it holds

Here $\overline{h}(x): = h''(t)|_{t = -\ln(|x|)}$.

Proof. In order to deduce this, we use the properties of the weight $h$. By relying on a partition of unity, we localise the set-up to dyadic intervals. Then, with the constants $a_j$ as in Step 1 in the proof of Proposition 3.1, if $a_j \tau > 1$, we apply Lemma 3.3, while if $a_j \tau < 1$, we invoke Lemma 3.4.

As before, by the same arguments as in the proof of Proposition 3.5, it is possible to obtain a slight variation of this, which we will use in the sequel:

Corollary 3.6. Let $K$ be a finite set of positive integers. Under the same assumptions as in Proposition 3.5, for each $\tau > \tau_0\gg 1$ there is a weight function $h(-\ln(|x|))$ such that there exists a constant $C > 0$ which is independent of $\tau$ such that for any $k\in K$ it holds

Proof. The proof follows in the same way as Proposition 3.5 but instead of using Proposition 3.1 we use Corollary 3.2.

3.3.   Proof of Propositions 1.7 and 1.8

Proposition 1.7 arises as an iteration of the Carleman estimate from Proposition 3.5 (or directly from Proposition 3.1 if $a^{ij} = \delta^{ij}$). In the sequel, we present the variable and constant coefficient proofs simultaneously.

Proof of Proposition 1.7. We seek to iterate the second order Carleman estimates in order to obtain an estimate for the full system. To this end, we apply Corollary 3.6 with $K = \{k_j\}_{j = 0}^m$, $k_j = 2m-2j$ to $\tilde{w}_j = (1+\overline{h})^{\frac{m-j}{2}} \tilde{u}_j$. We argue in two steps.

Step 1: Building block estimate. For $j\in\{0, \dots, m\}$ we have

Hence, as consequence of Corollary 3.6, we deduce the estimate

By virtue of the triangle inequality, the second term on the left hand side of (3.29) can be estimated as

Using the last condition in (3.2), we obtain

Hence, the negative term on the right hand side of (3.30) can be absorbed into the first left hand side contribution in (3.29) if $\tau\geq \tau_0 > 1$ is sufficiently large. Further by the regularity of the metric $a$, and by choosing $\varepsilon$ sufficiently small, we may absorb the last two contributions on the right hand side of (3.29) into the left hand side of (3.29).

As a consequence, we obtain

For later use, we remark that an analogous argument for $\tilde{w}_{j, \ell}: = (1+\overline{h})^{\frac{\ell}{2}} \tilde{u}_j$ for $\ell \in \mathbb{Z}$ and $k\in \{k_j\}_{j = 0}^m$ yields

Step 2: Iteration. Using the $C^{2m, 1}$ coefficient regularity, we iterate the building block estimate:

Here we used that $g_j = 0$ except for $g_m = g$ and applied the triangle inequality to separate the bulk terms on the right hand side. Summing these estimates from $j = 0$ to $m$, then yields the desired bulk estimate in (1.7).

Step 3: Boundary-bulk interpolation. In order to deduce the boundary estimate, we apply Lemma 2.2 to the function

on each sphere $|x| = r$. {Recall that $h$ and $\overline{h}$ are independent of the spherical variables so the integration with respect to the radial directions implies

Combining this with Step 2 then concludes the proof.

Proof of Proposition 1.8. We split the proof into two steps: First we deal with commutation relations arising in iterations of the second order estimates of Proposition 3.5 and then we iterate the resulting bounds.

Step 1: Commutators. We observe that

where $\partial'_{k}$ denotes derivatives in tangential directions.

Due to the assumptions in condition (A2), the contributions in the Carleman estimate arising from $\nabla'L_b u_{j-1}$ for $j\in\{1, \dots, m\}$ can be bounded from below by terms which are controlled by $L_b \nabla'u_{j-1}$ if $\delta > 0$ is chosen sufficiently small: Indeed, by using the commutator (3.33), we first obtain

Invoking the bounds from condition (A2) for the metric and interpolating, we obtain $|x||\nabla a (x)|$, $|x|^2|\nabla^2 a (x)|\leq \tilde{C}\delta$ for $x\in B^+_4$ and thus

Applying the Carleman estimate from Step 1 in combination with equation (3.32) in the proof of Proposition 1.7 with $k_j = 2m+1-2j$, $\ell = m+1-j$, it is possible to bound

whence, in combination with the previous estimates,

Choosing $\delta > 0$ so small that $C\tilde{C}\delta\leq \frac 1 2$, it is possible to absorb the last contribution into the left hand side, which leads to the estimate

This allows us to apply the estimate (3.32) from Step 1 in the proof of Proposition 1.7 with $k_j = 2m+2-2j$, $\ell = m+1-j$ and thus to deduce that for $j\in\{1, \dots, m\}$ the following bounds hold

Step 2: Iteration. With the additional bounds from (3.34) in hand, we can iterate the Carleman estimate. For $j\geq 1$ and $\ell\in\{1, \dots, j\}$ we infer

A similar estimate holds for the gradient term. Adding these estimates to the bounds from Proposition 1.7 and summing over all $j \in \{1, \dots, m\}$ implies the desired bulk estimate.

Step 3: Boundary-bulk interpolation. The boundary estimates can be deduced as in the proof of Proposition 1.7. Applying Lemma 2.2 to the functions

on each sphere $|x| = r$ and integrating with respect to the radial directions we infer

Combining the previous steps concludes the proof.

4.   On the strong unique continuation property for the extension problem

In this section we study the strong unique continuation property for the system (2.3) and seek to reduce it to the weak unique continuation property. As in ^[8,10,19] we achieve this by careful compactness and blow-up arguments.

From a technical point of view, the main challenge is to control solutions to our system also in the normal direction in which we can only obtain information through the equation itself. Here two cases arise:

● If we had vanishing of infinite order in the tangential and normal directions, an immediate application of the Carleman estimate (1.8) would allow us to prove the strong unique continuation property.

● If we are however dealing with solutions which a priori do not vanish of infinite order in the normal {direction}, we have to argue more carefully, exploiting properties of our equations.

In this section, we consider solutions $u_j \in H^1_{loc}(\mathbb{R}^{n+1}_+, x_{n+1}^b)$ with $b\in (-1, 1)$ of the system (1.5), (1.6) with $f_0, \dots, f_m = 0$ satisfying the following conditions:

$({\rm C})$ $a$ is of a block form (1.4) and such that $\tilde a$ satisfies (A1)–(A3) with $\mu = 2m$ and

with $|q_j(x')|\leq C_{q_j}|x|^{-2m+j+b-1}$ for $j = \{0, \dots, m-1\}$ and

Here $c_0 > 0$ is a sufficiently small constant (which is specified below), and $C_{q_j} > 0$ are arbitrarily large, finite constants.

As the vanishing of infinite order in the normal directions is not a direct consequence of our assumptions on the infinite order vanishing in the tangential directions, we split the argument into two parts:

● In the case of infinite order vanishing in all directions, i.e., for all $j\in\{0, \dots, m\}$

we directly apply the Carleman estimate from Proposition 1.8 (see Section 4.2).

● If this is (a priori) not the case, i.e., there exist some $j\in \{0, \dots, m\}$, a subsequence $r_{\ell}\rightarrow 0$ and a constant $k_0\in \mathbb{N}$ such that

we deduce doubling properties and then exploit these in a compactness argument to reduce the strong unique continuation property to the weak unique continuation property (see Section 4.1).

4.1.   Reduction to the weak unique continuation property

In the sequel, we seek to reduce the strong unique continuation property to a weak unique continuation result by a blow-up argument under the assumption that the solution vanishes to infinite order just in the tangential directions (but not in the normal directions, see (4.1)).

In order to deduce sufficient compactness for a blow-up argument, we first prove a doubling estimate for the functions $u_j$. Here we exploit elliptic estimates and deal with the resulting boundary contributions by absorbing these into the bulk terms with finite order of vanishing (for sufficiently small radii).

Proposition 4.1 (Doubling). Let $u_j\in H^1_{loc}(\mathbb{R}^{n+1}_+, x_{n+1}^b)$ for $j\in\{0, \dots, m\}$ be weak solutions of the system (1.5), (1.6) with $f_0, \dots, f_m = 0$ satisfying the conditions from (C). Assume also that the tangential restrictions $u_j(x', 0)$ vanish of infinite order at $x_0 = 0$ and that there exist some $j\in \{0, \dots, m\}$, a subsequence $r_{\ell}\rightarrow 0$ and a constant $k_0\in \mathbb{N}$ such that (4.1) holds. Then, there exist a universal constant $C > 1$, a positive integer $\ell_0$ and a radius $r_{u_0} > 0$ (the latter depending on $u_0$) such that for any $\ell > \ell_0$ and all $r\in (r_{\ell}, 2^{m{+3}}r_{\ell})\cap(0, r_{u_0})$ we have

Here $r_{\ell} > 0$ denotes the radii from (4.1).

Remark 4.2. In the case of bounded potentials, the same result holds without assuming that the functions $u_j(x', 0)$ vanish of infinite order in the tangential directions. Moreover, in the setting of bounded potentials, the statement holds for all $r\in(0, r_0)$ (there is no intersection with the interval around $r_{\ell}$ here), where $r_0$ is sufficiently small but independent of $u_0$. We refer to the proof of Proposition 4.1 for further details on this.

Remark 4.3. Instead of restricting our doubling results to radii around $r_{\ell}$, we could also have argued as in Section 3 in ^[36]. This would have allowed us to conclude that the vanishing order is defined not only through a subsequence of radii but is independent of such a sequence. As a consequence, we would have obtained the statement of Proposition 4.1 for any choice of radius less that $r_{u_0}$. As our unique continuation argument does not rely on quantitative order of vanishing estimates, we do not further pursue this approach here.

Proof of Proposition 4.1. Consider $\tilde{u}_j : = \eta u_j$, where $\eta$ is a radial cut-off function which is equal to one in $B_{3}^+ \setminus B_{r/4}^+$, vanishes outside of $B_4^+ \setminus B_{r/8}^+$ and satisfies the following bounds

Here $r\in (0, r_0)$ where $r_0 \ll 1$ is chosen sufficiently small (with a choice that is explained later). We note that the functions $\tilde{u}_j$ are solutions to the system from Proposition 1.7 with $f_j : = 2\nabla \eta \cdot(a\nabla u_j) + u_j L_b \eta$ and $|g(x')|\leq|\eta| \sum_{j = 0}^m|q_j(x')||(\nabla')^ju_0(x', 0)|$. Hence the Carleman estimates (1.7) and (1.8) hold.

Step 1: Boundary contributions. Let us assume that for all $j\in \{0, \dots, m\}$ it holds $|q_j(x)|\leq C_{q_j} |x|^{-2m+j+b-1+\epsilon}$, where $\epsilon\geq 0$ is a constant which will be specified below. We seek to absorb the boundary contributions from the right hand side of the estimate (1.8) in Proposition 1.8 into the ones on the left hand side. To this end, we compare the relevant contributions: The left hand side of the Carleman estimate controls contributions of the form

while the boundary terms on the right hand side can be estimated from above by

In order to carry out the absorption argument, we distinguish three cases:

● If $b < m-j$, for any $\epsilon \geq 0$, it suffices to choose $\tau$ sufficiently large, in order to absorb the contributions from (4.3) into (4.4). Note that this always holds for $j\in\{0, \dots, m-1\}$ and also for $j = m$ if $b < 0$.

● If $j = m$ and $b = 0$, it is still possible carry out this absorption argument in the case that $\epsilon = 0$ provided the constant $c_0$ is small enough. More precisely, after plugging the estimate (4.3) into (1.8), the relevant boundary contribution will carry the prefactor $C {c_0}$. Requiring that $c_0\leq\frac{1}{2C}$ then allows us to implement an absorption argument.

● Lastly, in the case of subcritical potentials, i.e., if $\epsilon > 0$, a wider range of values of $b$ is admissible by using the properties of $\overline h$. Indeed, by the construction of $h''$

Analogous estimates hold for the gradient contributions. Hence, by a sufficiently small choice of $\nu > 0$ (in the construction of the Carleman weight in the proof of Proposition 3.1 which in particular is compatible with the other requirements there) it is possible to absorb all boundary contributions as long as $b < \frac{3m+1}{2}-j$ for any finite constant $C_{q_j}$ by choosing $\tau$ sufficiently large. This enlarges the range of $b$ for $m = 0$ to $b < \frac{1}{2}$ and for $j = m\geq 1$ to $b < 1$.

Step 2: Bulk contributions. In discussing the bulk contributions, we first deal with the bulk terms of the right hand side of the Carleman estimate which are localised on the unit scale. We will absorb these into the left hand side of the Carleman estimate. Secondly, we treat the contributions on the small scale $r > 0$ for which we deduce the desired doubling estimate.

In the sequel, we use the following abbreviations for the respective half annuli

For the convenience of the reader, we split the proof of the bulk estimates into two steps: In Step 2a, we deal with the case without gradient contributions. This allows us to introduce the ideas without resorting to too many technicalities. Then, in Step 2b, we deal with the full case including gradient terms.

Step 2a: Lowest order potentials. After having dealt with the boundary terms in Step 1, the Carleman estimate (1.7) turns into

with $|L_b\eta|\leq C (|\nabla^2\eta|+|x|^{-1}|\nabla\eta|)$.

As a first simplification step, we deal with the contributions on the unit scale: Using the monotonicity of $h$, we infer

where $\tilde I_4 = B^+_{4}\backslash B^+_{5/2}$. Estimating the terms on the left hand side of (4.5) from below by

and relying on the monotonicity of $h$, by choosing $\tau > \tau_0$ sufficiently large, we can absorb the contribution (4.6) into the left hand side of (4.5) (this yields a dependence of $\tau$ on $u$, but only on the unit scale).

As a consequence, we are left with the estimate

where $\tau$ has been fixed in the previous step. Using the monotonicity of $h$, the bound of $(1+\overline h)$ and the estimates on the derivatives of $\eta$ in $I_1$, we obtain

We observe that the difference $|h(-\ln \frac r 8)-h(-\ln 2r)|$ is bounded independently of $r > 0$, since

where $\xi\in(\frac 1 8, 2)$, and $h'\in (C^{-1}\tau, C\tau)$.

Thus, dividing (4.8) by $e^{h(-\ln(2r))}$ and adding to both sides

we obtain

Step 2b: Gradient potentials. The proof is similar to the one in Step 2a but instead of the Carleman estimate from Proposition 1.7, we here use the one from Proposition 1.8. After Step 1, estimate (1.8) becomes

Considering the bounds for derivatives of $\eta$ and the metric $a$ (i.e., $|\nabla^\ell a|\leq \tilde{C}\delta |x|^{-\ell}$), and repeating the same arguments as in Step 2a, we arrive at

Step 3: Caccioppoli's inequality. It remains to control the gradient terms of the right hand side in (4.10) and in (4.9). We can apply Lemma 2.3 to $u_k$ with $f = u_{k+1}$, $g = 0$ if $k\in\{0, \dots, m-1\}$ and $f = 0$, $|g|\leq\sum_{j = 0}^m |q_j||(\nabla')^ju_0|$ if $k = m$.

If we just consider the lowest order potentials (i.e., where in the bounds for $|g|$ only $q_0$ is needed), tangential derivatives are not necessary and after summing over $k$ with suitable factors we arrive at

We seek to absorb the boundary terms which occur in (4.11) into the bulk term on the right hand side of (4.11). Before explaining the general case, we discuss the case $q_0 \in L^{\infty}$. If $q_0\in L^\infty$, then the boundary term can be absorbed into the bulk terms for $r\in(0, r_0)$ with some $r_0 > 0$ independent of $u_0$: By scaling, it is enough to prove this for $r = 1$. If $m > 0$, we do not directly start from (4.11) but return to the proof of the Caccioppoli estimate which yields

with $\eta$ as in the proof of Lemma 2.3. By Young's inequality we can then estimate the boundary term as

for some constant $\delta > 0$. Using the Poincaré type inequality $\|\eta u_j\|_{L^2(B_1')}\leq C\|x_{n+1}^{\frac{b}{2}}\nabla(\eta u_j)\|_{L^2(B^+_1)}$ for $j\in\{0, m\}$, we obtain the bound

In order to control the gradient term involving $\eta \nabla u_m$ we note that it is possible to absorb it into the left hand side of (4.12) by choosing $\delta$ small enough. In order to bound the gradient contribution $\eta \nabla u_0$, we use again the Caccioppoli estimate (and observe that in this case no boundary terms appear). Therefore (4.2) is valid for $r\in(0, r_0)$ and $m > 0$. For the case $m = 0$, we refer to Lemma 5.1 in ^[37].

Let us now return to the general case: If we also consider gradient potentials (i.e., where the full bound $|g|\leq\sum_{j = 0}^m |q_j||(\nabla')^ju_0|$ is needed), a similar estimates holds after considering in (2.5) tangential derivatives up to the order $m-k$:

Now the boundary terms in (4.11) and (4.13) (which in general are not in $L^{\infty}$ but might become singular) can be controlled as follows: We first notice that

and

with $\chi$ a suitable cut-off function and $\gamma = \frac{1+2m-b}{2}$. Since $\lim\limits_{r\rightarrow 0} r^{-\ell} \|u_0\|_{L^2(B_{4r}')} = 0$ for any $\ell \in \mathbb{N}$, given any $\delta > 0$ there is a radius $r_{u_0} > 0$ such that if $r\in(0, r_{u_0})$

Therefore

On the other hand, $\sum\limits_{j = 0}^{m}r^{-2m+2j-1}\|x_{n+1}^{\frac b 2}u_j\|_{L^2({B^+_r)}}$ only vanishes of finite order, so choosing $\delta$ sufficiently small, the boundary term can be absorbed into the bulk terms {for suitable ranges of $r$: Indeed, from (4.1) we know that for some $j\in\{0, \dots, m\}$ and some $k_0 > 0$ there exist $C_0 > 0$ and $\ell_0\in \mathbb{N}$ such that if $\ell > \ell_0$

and therefore if $r\in (r_\ell, Rr_\ell)$, with $R > 1$ some constant (which we will fix later),

(where $\tilde C_0 = R^{-k_0}C_0$). So the bulk term can absorb the boundary contribution in the right hand side of (4.11) and (4.13) if $\delta$ is small enough.

Hence, (4.11) becomes

and (4.13) turns into

In order to deal with the remaining derivatives on the left hand side in (4.14), we notice that

In order to estimate the remaining (lower order) gradient terms we seek to iterate the Caccioppoli estimate up to $m$ times. This leads to an estimate of the type (4.15) where the norms on the right hand side are evaluated in increasingly larger balls. In particular, in order to obtain radii of the size $r$ in the end, the beginning of the iteration should be carried out with radii of the size $\tilde{r} = 2^{-m}r$. Notice that in the Caccioppoli estimates the boundary terms only appear in the estimate for $u_m$, so they will only appear in the first iteration, where $\tilde r = 2^{-m}r$. In order to carry out the iteration argument of the Caccioppoli estimate as outlined above, we thus have to ensure that $\tilde r\in(r_\ell, R r_\ell)$ for some $R > 1$. This is satisfied if $r\in(r_\ell, C r_\ell)$ with $C > 2^{m}$. For technical reasons appearing in the proof of Proposition 4.4, we choose $C = 2^{m+3}$.

With the doubling property in hand, we apply a blow-up argument reducing the strong unique continuation property to the weak unique continuation property. To this end, we introduce the following rescaled functions:

We exploit the previous compactness arguments to pass to the blow-up limit $\sigma \rightarrow 0$ which leads to a boundary weak unique continuation problem of the blown-up system:

Proposition 4.4. Let $u_j\in H^1_{loc}(\mathbb{R}^{n+1}_+, x_{n+1}^b)$ for $j\in\{0, \dots, m\}$ be weak solutions of the system (1.5), (1.6) with $f_0, \dots, f_m = 0$ satisfying the conditions from (C). Assume also that the tangential restrictions $u_j(x', 0)$ vanish of infinite order at $x_0 = 0$ and that there exist some $j\in \{0, \dots, m\}$, a subsequence $r_{\ell}\rightarrow 0$ and a constant $k_0\in \mathbb{N}$ such that

Let $u_{\sigma, j}$ be the rescaled functions defined by (4.16) and let $\{r_{\ell}\}$ denote the sequence of radii from (4.1). Then, along a subsequence $\{\sigma_{\ell}\}_{\ell \in \mathbb{N}} \subset \{2r_{\ell}\}_{\ell{ > \ell_0}}$, for some $\ell_0\in \mathbb{N}$, with $\sigma_{{\ell}} \rightarrow 0$ we have $u_{\sigma_l, j} \rightarrow u_{0, j}$ strongly in $L^2(B_4^+, x_{n+1}^b)$, where the functions $u_{0, j}$ are weak solutions to the following elliptic system:

with $\Delta_b = x_{n+1}^{-b}\nabla \cdot x_{n+1}^b\nabla$. Moreover, for all $j\in\{0, \dots, m\}$ we have

and

Proof. We first note that the functions $u_{\sigma, j}$ are constructed in such a way that

Applying Proposition 4.1 twice with $\sigma\in\{2r_{\ell}\}_{\ell > \ell_0}\cap (0, \frac{r_{u_0}}{2})$ (so both $\sigma$ and $2\sigma\in\{(r_\ell, 2^{m+3}r_\ell)\}_{\ell > \ell_0}\cap(0, r_{u_0})$ for any given $m$), we obtain

After rescaling, this turns into

which holds holds uniformly in $\sigma$ for the whole family $u_{\sigma, j}$. By Rellich's compactness theorem, there exist a subsequence $\{\sigma_\ell\}_{\ell \in \mathbb{N}} \subset \{2r_{\ell}\}_{\ell { > \ell_0}}$ with $\sigma_\ell\to 0$ and functions $u_{0, j}\in H^1(x_{n+1}^b, B^+_4)$ such that $u_{\sigma_\ell, j}\to u_{0, j}$ strongly in $L^2(B_4^+, x_{n+1}^b)$ and weakly in $H^1(B_4^+, x_{n+1}^b)$ and the normalisation (4.18) holds. In addition, since the embedding $H^1(B^+_4, x_{n+1}^b)\hookrightarrow H^{\frac{1-b}{2}}(B'_3)$ is continuous and $H^{\frac{1-b}{2}}(B'_3)$ is compactly embedded in $L^2(B'_3)$, up to a redefinition of the subsequence, $u_{\sigma_\ell, j}\to u_{0, j}$ strongly in $L^2(B'_{3})$ and weakly in $H^{\frac{1-b}{2}}(B'_3)$.

The functions $u_{\sigma, j}$ satisfy weakly the same system (1.5), (1.6) as the original functions $u_j$ (again with $f_0, \dots, f_m = 0$) however with a rescaled metric and potentials $a_\sigma(x) = a(\sigma x)$ and

Hence,

for any $\varphi\in C^1_c(B^+_3)\cap C_c(B'_3)$. As a result, in the limit $\sigma_\ell\to 0$

Here we have used that by the normalising condition (A3) the metric satisfies $a^{ij}(\sigma x)\to \delta^{ij}$ as $\sigma \rightarrow 0$ and that the boundary integrals vanish of infinite order as we proved in Step 3 of the proof of Proposition 4.1. This shows that the functions $u_{0, j}$ are indeed weak solutions to the system in the statement.

Finally, we prove that the functions $u_{0, j}$, $j\in\{0, \dots, m\}$, vanish on $B'_1$. Indeed,

and whereas the numerator vanishes of infinite order, by our assumption (4.1), the denominator vanishes of only finite order.

4.2.   Strong unique continuation

Here we deduce the strong unique continuation property for solutions which vanish of infinite order in tangential and normal directions. The proof relies on the associated Carleman estimates from Proposition 1.8.

Proposition 4.5. Let $u_j\in H^1_{loc}(\mathbb{R}^{n+1}, x_{n+1}^b)$ for $j\in\{0, \dots, m\}$ be weak solutions of the system (1.5), (1.6) with $f_0, \dots, f_m = 0$ satisfying the conditions from (C). Assume further that for all $j\in \{0, \dots, m\}$ and all $k\in \mathbb{N}$

Then, $u\equiv 0$ in $B_1^+$.

Proof. Consider the functions $\tilde{u}_j: = \eta_{\epsilon} u_j$, where $\eta_{\epsilon}$ is a smooth, radial cut-off function which satisfies the following bounds:

where $0 < \epsilon\ll1$. The functions $\tilde u_j$ are solutions to the system from Proposition 1.7 with $f_j = 2\nabla \eta_\epsilon\cdot a\nabla u_j+u_jL_b\eta_\epsilon$.

We insert the functions $\tilde{u}_j$ into the Carleman estimate (1.8). Notice that we can pass to the limit $\epsilon \rightarrow 0$ by virtue of the infinite rate of vanishing of $u_j$. Arguing as in Step 1 of the proof of Proposition 4.1, we can drop the boundary contributions. Therefore (1.8) turns into

Finally, passing to the limit $\tau \rightarrow \infty$, we obtain $u_j = 0$ in $B^+_1$ for all $j\in\{0, \dots, m\}$.

5.   Weak unique continuation

In this section we consider the weak unique continuation property for solutions to the system (4.17). In spite of weak unique continuation results for the fractional Laplacian already existing in the literature (see in particular ^[18]), both our argument and our result contain novel aspects: In contrast to the weak unique continuation results from Seo ^[18] our result is a localised unique continuation result (as we do not need the validity of the equation $L^{\gamma} u = q u$ in $\mathbb{R}^n$), and hence in particular it is formulated for a local equation (instead of working with the global fractional Laplacian).

Proposition 5.1. Let $u_j\in H^1_{loc}(B_1^+, x_{n+1}^b)$ for $j\in\{0, \dots, m\}$ be weak solutions of the system (1.5), (1.6) in $B^+_1$ with $f_0, \dots, f_m = 0$, $g = 0$ and the metric $a$ of a block form (1.4) where $\tilde a$ satisfies the conditions (A1)–(A3). Assume also that for all $j\in\{0, \dots, m\}$ the tangential restrictions $u_j(x', 0)$ vanish on $B'_1$. Then, $u_j \equiv 0$ in $B_1^+$.

Proof. We bootstrap the system by applying the weak unique continuation property for scalar equations: Indeed, by the weak unique continuation property of solutions of the fractional Laplacian (see ^[10] and ^[19]) and regularity results from ^[25], we first infer that $u_m \equiv 0$ in $B_1^+$. Iteratively, this then also entails that $u_{j} \equiv 0$ in $B_1^+$ since, once $u_{j+1} \equiv 0$ in $B_1^+$, then $u_j$ satisfies the Caffarelli-Silvestre equation with zero Dirichlet and zero (weighted) Neumann data. We iterate this until we reach $u_0$.

Remark 5.2. We remark that an argument for the WUCP had already been given by Riesz ^[39] (relying on certain regularity conditions, see the discussion in Remark 4.2 in ^[7]). Using a Kelvin transform he reduced it to the situation with data vanishing in the exterior of a domain. An argument of a related flavour for a much larger class of pseudodifferential operators was also used in ^[9] (see also ^[40]).

Remark 5.3. We remark that the (weak) unique continuation property requires the Lopatinskii condition to hold (see ^[41] and the references therein for a precise definition of the Lopatinskii condition). If this is violated even if "formally" there are sufficiently many boundary conditions prescribed, one will in general not be able to infer the vanishing of $u$. This is for instance the case for problem

By simply invoking counting arguments these boundary conditions should yield an overdetermined system. They however do not (the function $w(x, y) = y^2 x$ is a non-trivial solution), as the Lopatinskii condition is not satisfied.

As a consequence of the localised formulation of our weak unique continuation property, it for instance applies to settings which arise in inverse problems ^[7,8,9]. This allows us to prove the antilocality of the fractional Laplacian for any order $\gamma > 0$ with $\gamma \notin \mathbb{N}$, i.e., it allows us to prove Proposition 1.9, which we postpone to the next section.

6.   Proofs of the unique continuation results for the fractional Laplacian

In this section, we rely on the connection between the systems representations for the higher order fractional Laplacian (see Proposition 2.1 as well as Propositions A.6 and A.7 in the Appendix) and–building on the previous compactness results-present the proofs of Theorems 1–4 and of Propositions 1.9 and 1.11.

6.1.   Proofs of strong unique continuation properties for the fractional Laplacian

We begin by proving Theorems 1–3:

Proofs of Theorems 1, 2 and 3. We seek to reduce the strong unique continuation properties for the fractional Laplacian to the previously deduced results on the systems case. We invoke Proposition 2.1 and rewrite the problem as a system of the form (2.3), where $f = u$, $m = \lfloor{\gamma}\rfloor$ and $b = 1-2\gamma+2\lfloor{\gamma}\rfloor$. We seek to apply a combination of Propositions 4.4, 4.5 and 5.1. To this end, we have to show that the functions $u_j(x', 0) = L^j u(x')$ with $j\in\{1, \dots, \lfloor{\gamma}\rfloor\}$ vanish of infinite order in the tangential directions on the boundary. By assumption, we have that the function $u_0(x', 0) = u(x')$ vanishes of infinite order at $x'_0 = 0$ in the tangential directions. In order to obtain the desired infinite order of vanishing of $u_j$ in the tangential directions on the boundary, we use an interpolation argument: Let $\eta$ be a smooth cut-off function which is equal to one on $B_{r}'$ and which is supported in $B_{4r}'$. Then,

Since $\lim\limits_{r\rightarrow 0} r^{-\ell} \|u\|_{L^2(B_{4r}')} = 0$ for any $\ell \in \mathbb{N}$ and as $u\in H^{2\gamma}(B_1')$, this implies that the same holds for $\|L^j u\|_{L^2(B_{r}')}$ and thus for $\|u_j(\cdot, 0)\|_{L^2(B_{r}')}$ with $j\in\{0, \dots, \lfloor{\gamma}\rfloor\}$. Moreover, due to the previous identification of $b$ and $m$ in terms of $\gamma$ and $\lfloor{\gamma}\rfloor$, the conditions from (C) are satisfied.

Hence, if we assume that (4.1) holds for the functions $u_j$ with $j\in\{0, \dots, m\}$, the blow-up argument from Proposition 4.4 is applicable and yields blow-up solutions $u_{0, j}$ with $u_{0, j} = 0$ on $B_1'$ for $j\in \{0, \dots, m\}$ which simultaneously satisfy the normalisation condition (4.18). By the weak unique continuation result from Proposition 5.1, the tangential and normal vanishing of $u_{0, j}$ on $B_1'$ however entails that $u_{0, j} = 0$ in $B^+_1$ for $j\in\{0, \dots, m\}$, which contradicts (4.18). Therefore, (4.1) cannot hold and the functions $u_j$ for $j\in \{0, \dots, m\}$ must vanish of infinite order in tangential and normal directions. Thus we invoke Proposition 4.5 to obtain $u_j = 0$ in $B_1^+$ for $j = 0, \dots, m$. Using that the equation for the generalised Caffarelli-Silvestre extension holds globally, the vanishing of $u$ on $B_1^+$ propagates through the upper half plane $\mathbb{R}^{n+1}_+$: Indeed, by the weak unique continuation property for uniformly elliptic equation and by (2.3) we infer $u_m\equiv 0$ in $\mathbb{R}^{n+1}_+$. Plugging this into the equation for $u_{m-1}$ and again using the weak unique continuation property for solutions to uniformly elliptic equations in the upper half plane implies also $u_{m-1} \equiv 0$ in $\mathbb{R}^{n+1}_+$. Iterating this further leads to $u_j \equiv 0$ in $\mathbb{R}^{n+1}_+$, whence $u \equiv 0$ in $\mathbb{R}^n$. This concludes the argument.

6.2.   Proof of unique continuation from measurable sets

In this section we prove Theorem 4 by reducing it to the weak unique continuation property for the generalised Caffarelli-Silvestre extension.

By the representations from Proposition 2.1 (see also Propositions A.6 and A.7) we rewrite the problem of Theorem 4 as a system of the form (2.3), where $f = u$, $m = \lfloor{\gamma}\rfloor$ and $b = 1-2\gamma+2\lfloor{\gamma}\rfloor$. Notice that by Proposition 2.1 and the assumptions in Theorem 4 we have that $|L^\gamma f(x')|\leq |q(x')||u_0(x', 0)|$ with $q\in L^\infty(\mathbb{R}^n)$ and $u_0(\cdot, 0)\in H^{2\gamma}(\mathbb{R}^n)$. By assumption, $u_0(x', 0) = u(x')$ vanishes on a measurable set $E\subset \mathbb{R}^n\times\{0\}$ of density one at $x_0 = 0$.

Under these assumptions and supposing that (4.1) holds, we prove an analogous blow-up result as in Proposition 4.4:

Proposition 6.1. Let $u_j$ with $j\in \{0, \dots, m\}$ be the functions from above and let $u_{\sigma, j}$ with $j\in \{0, \dots, m\}$ be the associated rescaled functions defined in (4.16). Suppose further that (4.1) holds. Then, along a subsequence $\{\sigma_{\ell } \}_{\ell \in \mathbb{N}}\subset\{2r_{\ell}\}_{\ell \in \mathbb{N}, \ \ell \geq \ell_0}$ with $\sigma_{\ell} \rightarrow 0$ we have $u_{\sigma_{\ell, j}} \rightarrow u_{0, j}$ strongly in $L^2(B_4^+, x_{n+1}^b)$, where $u_{0, j}$ is a weak solution to the following elliptic system

Moreover, for all $j\in\{0, \dots, m\}$ we have

and

In order to obtain the properties of the blow-up limit, we deduce a smallness condition for the (not yet blown-up) function $u_0$ in tangential directions on the boundary. By virtue of an interpolation inequality, this will be inherited to all the (not yet blown-up) functions $u_j$ with $j\in\{0, \dots, m\}$ on the boundary.

Lemma 6.2. Let $u_j$ with $j\in\{0, \dots, m\}$ be as in Proposition 6.1. For any $\epsilon > 0$, there exists a radius $r_0 > 0$ such that if $r\in (0, r_0)$

Proof. Since $x_0 = 0$ is a point of density one in $E\cap B'_4$, given $\delta > 0$, there exists a radius $r_\delta > 0$ such that if $r\in(0, r_\delta)$

On the other hand, using Hölder's inequality ($n > 1-b$)

By Sobolev and trace inequalities

Now we use the estimate from Proposition 4.1, where according with Remark 4.2 no assumptions on the vanishing order of $u_j$ are necessary and it holds for $r\in(0, r_0)$ with $r_0$ independent of $u_0$:

Therefore, by combining this with (6.1) and recalling the definition of $\delta > 0$ from above, we obtain

Choosing $\delta$ such that $C\delta^{\frac{1-b}{2n}} = \epsilon$, the result holds.

Proof of Proposition 6.1. The proof of Proposition 6.1 follows along the same lines as the one of Proposition 4.4 until the moment of proving $u_{0, j}|_{B'_1} = 0$. Here we use Lemma 6.2 to obtain the same result: Indeed, after rescaling it implies

Therefore, in the limit $\sigma_\ell\to 0$,

Since this holds for any $\epsilon > 0$, in particular for any sequence $\epsilon_k\to 0$, we infer $u_{0, 0}|_{B'_1} = 0$.

The proof of $u_{0, j}|_{B'_1} = 0$ for $j = 1, \dots, m$ relies on an interpolation result together with the previous bound: Considering a smooth cut-off function $\eta$ with $\eta = 1$ in $B_{r}'$ and $\text{supp}(\eta) \subset B_{4r}'$, we obtain

By rescaling, we then also infer

Thus, the smallness of $u_0$ and $u_{\sigma, 0}$ also entails the smallness of $u_j$ and $u_{\sigma, j}$ on the boundary. The remainder of the argument follows analogously as in the proof of Proposition 4.4.

Proof of Theorem 4. We again have a dichotomy: Rewriting the equation as a solution to a system of the form (2.3), we either have that all functions $u_j$ with $j\in\{0, \dots, m\}$ vanish of infinite order at some point in $E$ or that (4.1) holds for the points in $E$.

If (4.1) holds, we may apply Proposition 6.1 which leads to normalised blow-up solutions $u_{0, j}$ with $j\in\{0, \dots, m\}$ which are non-trivial in $B_1^+$. This however contradicts the weak unique continuation statement from Proposition 5.1 which implies {that $u_{0, j} = 0$ in $B^+_1$ for $j\in\{0, \dots, m\}$. Hence, (4.1) cannot hold. Therefore, all the functions $u_j$ with $j\in\{0, \dots, m\}$ must vanish of infinite order in tangential and normal directions at $x_0 = 0 \in E$. This allows us to} directly apply Proposition 4.5, whence we conclude that $u_j = 0$ in $\mathbb{R}^{n+1}_+$ for $j\in \{0, \dots, m\}$ and thus also $u \equiv 0$ in $\mathbb{R}^n$.

6.3.   Applications of the unique continuation results

We turn to the proof of the antilocality result. As above we emphasise that in this case, we do not assume the validity of an equation on the whole space $\mathbb{R}^n$. Nevertheless the antilocality of the fractional Laplacian entails the claimed strong rigidity property.

Proof of Proposition 1.9. By Proposition 2.1 we can consider the extension $u$ and the functions $u_j = L_b^j u$ for $j\in\{0, \dots, \lfloor{\gamma}\rfloor\}$, which solve a system of the form (2.3). Thus, if $f = 0$ and $L^\gamma f = 0$ on $B_1'$, (2.3) reduces to the setting in Section 5, whence we conclude that $u_j = 0$ on $B_1^+$. Since the nonlocal equation is assumed to hold in $\mathbb{R}^n$, the vanishing of $u_j$ can be propagated to the whole upper half space $\mathbb{R}^{n+1}_+$, whence we conclude that $u\equiv 0$ in $\mathbb{R}^n$.

With the global unique continuation result of Proposition 1.9 in hand, the proof of Proposition 1.11 follows by a duality argument and the Fredholm property of the fractional Schrödinger equation (see ^[24]). In particular, this is of similar flavour as a number of approximation properties which had been used to treat inverse problems for nonlocal equation in ^[7,8,27].

We consider the fractional Schrödinger equation

where $L$ is as in Proposition 1.11.

Considering the bilinear form

it is possible to show the well-posedness of the problem, provided zero is not a Dirichlet eigenvalue of the problem. This follows similarly as explained for instance in ^[7]. In this setting, we define the associated Poisson operator as

where $u_f$ is a weak solution to (6.2). With this preparation, we address the proof of Proposition 1.11:

Proof of Proposition 1.11. It suffices to prove that the set

is dense in $L^2(\Omega)$, where $W\subset\tilde{\Omega}\setminus \Omega$ is any open subset. We can suppose without loss of generality that $B_1'\subset W$ by assupmtion. As in ^[7] we rely on the Hahn-Banach theorem: Assuming that $v\in L^2(\Omega)$ is such that $(P_q f, v)_{\Omega} = 0$ for all $f \in C_0^{\infty}(W)$, it suffices to show that $v = 0$. In order to infer this, we note that by the well-posedness results for the inhomogeneous fractional Schrödinger equation, we may define $w$ to be a solution to

Then, as in ^[7],

As a consequence, $L^{\gamma} w = 0 = w$ in $W$. Thus, by Proposition 1.9, the function $w$ vanishes identically in $\mathbb{R}^n$, whence also $v\equiv 0$. This concludes the argument.

Conflict of interest

The authors declare no conflict of interest.

A.   Appendix: The higher order fractional Laplacian and degenerate elliptic systems

In this section, in order to keep our presentation self-contained, we connect the previous discussion on systems with certain boundary conditions to the properties of the higher order fractional Laplacian. Here we mainly recall several known results from the literature and rely heavily on the observations from ^[1,15,21] but also refer to ^[42,43,44] and the references therein.

We split the section into two parts: First, we derive the representation of the constant coefficient higher order fractional Laplacian operators through a generalised Caffarelli-Silvestre extension. Next, we deduce analogous results for operators with non-constant coefficients.

A.1.   The constant coefficient operator – characterisation through a system of degenerate elliptic equations

The starting point of our discussion is the definition of the fractional Laplacian as a Fourier multiplier:

where $\mathcal{F}$ denotes the Fourier transform. Since we seek to study the unique continuation property of the higher order fractional Laplacian by techniques which are available for local (possibly weighted) equations, we are particularly interested in Caffarelli-Silvestre type extension properties for the higher order fractional Laplacian. These exist in different generalities, we only recall two of these and refer to the literature for more general results. As we aim at applying these characterisations of the (higher order) fractional Laplacian for our study of the unique continuation property, we limit ourselves here to showing that starting from the fractional Laplacian of a function $f$, it is possible to find a suitable and sufficiently regular extension $u$ of $f$ which obeys a corresponding equation/ a corresponding system of equations. We however do not address the full equivalence (in that we do not show that any solution to the system at hand is related to the fractional Laplacian of a suitable function). For this we refer to the literature cited above.

We begin by recalling that also the higher order fractional Laplacian can be realised as the solution to a degenerate elliptic, second order boundary value problem ^[1,15]:

For $\gamma \in \mathbb{R}_+$ we consider the equation

Here we are interested in solutions

(ⅰ) which (by elliptic regularity) are classical solutions in $\mathbb{R}^n \times (0, \infty)$,

(ⅱ) which attain the boundary data $f\in H^{\mu}(\mathbb{R}^n)$ with $\mu \in \mathbb{R}$ in an $H^{\mu}(\mathbb{R}^n)$ sense as $x_{n+1}\to 0$,

(ⅲ) and which have some decay at infinity in the sense that $u \in \dot{H}^{1}(\mathbb{R}^{n+1}_+, x_{n+1}^{1-2(\gamma-\lfloor{\gamma}\rfloor)})$, where for $t\in \mathbb{R}$ we define $\lfloor{t}\rfloor: = {\max}\{k\in \mathbb{N}: \ k\leq t\}$.

Solutions to degenerate elliptic equations of this form have been investigated in the literature, even in the context of fully nonlinear equations ^[45]. Working with extensions of a problem from $\mathbb{R}^n$ into $\mathbb{R}^{n+1}_+$, in this section, we use the notational convention that

As we view the strong unique continuation property for the fractional Laplacian as a strong boundary unique continuation property of the associated degenerate extension problem, it is one of our main goals to identify the associated boundary values for the generalised Caffarelli-Silvestre extension. In particular, we aim at showing that, as in the original Caffarelli-Silvestre characterisation of the fractional Laplacian, the formulation (A.1) also allows one to compute the fractional Laplacian $(- \Delta')^{\gamma} f(x')$ as an "iterated Neumann" map from the knowledge of its generalised Caffarelli-Silvestre extension $u(x', x_{n+1})$.

Lemma A.1. Let $\gamma > 0$, $\mu \in \mathbb{R}$ and assume that $f\in H^{\mu}(\mathbb{R}^n)$. Let $\mathcal{F}_{x'}$ denote the tangential Fourier transform. Then there exists an extension operator

such that $E_{\gamma}f$ is a solution to

and

Here $\phi_{\gamma}(t) = t^{\gamma}K_{\gamma}(t)$ and $K_{\gamma}(t)$ denotes a modified Bessel function of the second kind.

Also, there exists a constant $\tilde{c}_{\gamma, n} \neq 0$ such that

Proof. We first derive the desired representation of the extension operator: To this end, we solve (A.1) by means of a tangential Fourier transform. Fourier transforming in the tangential directions, one obtains for the partial Fourier transform $\hat{u}(\xi, x_{n+1}): = \mathcal{F}_{x'}u(\xi, x_{n+1})$ the following ODE

We rewrite $\hat{u}(\xi, x_{n+1}) = v(\xi, |\xi|x_{n+1})$ and deduce a similar ODE for this function, but where we can scale out the $|\xi|$ contribution:

Further, setting $v(\xi, x_{n+1}) = x_{n+1}^{\gamma} g(\xi, x_{n+1})$ for some function $g$, we are lead to a modified Bessel equation for $g$ (as a function of $x_{n+1}$):

with corresponding initial conditions. Since we are looking for a function with decay at infinity, by the asymptotics of modified Bessel functions (see ^[47]) we infer that $g(x_{n+1}, \xi) = C(\xi) K_{\gamma}(x_{n+1})$, where $K_{\gamma}(t)$ denotes a modified Bessel function of the second kind.

Returning to our original variables and using the asymptotics of $K_{\gamma}(t)$ as $t \rightarrow 0$, we thus obtain that $\hat{u}(\xi, x_{n+1}) = c_{\gamma, n} \hat{f}(\xi) \phi_{\gamma}(|\xi| x_{n+1})$, where $\phi_{\gamma}(t) = t^{\gamma} K_{\gamma}(t)$ and $c_{\gamma, n}\neq 0$.

By the regularity of $\phi_{\gamma}(t)$ for $t > 0$ the function $u(x): = \mathcal{F}^{-1}_{x'}(c_{\gamma, n} \hat{f}(\xi) \phi_{\gamma}(|\xi| x_{n+1}))$ is $C^{\infty}_{loc}(\mathbb{R}^n \times (0, \infty))$.

In order to observe the $H^{\mu}(\mathbb{R}^n)$ convergence from (A.2) we note that

Using that $c_{n, \gamma}\phi_{\gamma}(\epsilon) \rightarrow 1$ as $\epsilon \rightarrow 0$, the boundedness of $\phi_{\gamma}(t)$ as $t\rightarrow \infty$ and the fact that for $f\in H^{\mu}(\mathbb{R}^n)$, $\|(1+|\xi|^2)^{\frac{\mu}{2}} \mathcal{F}(f)(\xi)\|_{L^2(\{|\xi|\geq \epsilon x_{n+1}^{-1}\})} \rightarrow 0$ as $x_{n+1}\rightarrow 0$, this implies the desired limiting behaviour.

Finally, in order to obtain (A.3), we now use the asymptotics and recurrence relations of the modified Bessel functions ^[47]. We have

Recalling the expression for $u$ (or rather $\hat{u}$), we thus obtain

Abbreviating $N_{\gamma}(f)(x', x_{n+1}): = x_{n+1}^{1-2\gamma + 2\lfloor{\gamma}\rfloor} \partial_{x_{n+1}} ((x_{n+1}^{-1} \partial_{x_{n+1}})^{\lfloor{\gamma}\rfloor} u(x', x_{n+1}))$, we thus infer

As a consequence, for $\overline{C}_{\gamma, n}: = \tilde c_{\gamma, n} C_{\gamma, n}$,

Due to the asymptotics of the modified Bessel functions (see (A.4)) and the regularity of $f$ in the limit $x_{n+1}\rightarrow 0$ this implies the desired result for a proper choice of $\tilde c_{\gamma, n}$.

Corollary A.2. Let $f\in H^{\gamma}(\mathbb{R}^n)$ and let $u = E_{\gamma}f$ be the extension from Lemma A.1. Then we also have the following bulk estimates:

where $\mathcal{F}_{x'}$ denotes the tangential Fourier transform.

Proof. In order to deduce the bulk estimates from (A.5) we note that

where we used the change of coordinates $z = |\xi| x_{n+1}$. Using an analogous change of coordinates, we also obtain

and

Here, in the passage from $\phi'_{\gamma}(t)$ to $t \phi_{\gamma-1}(t)$, we used the recurrence relations (A.4). This concludes the discussion of the mapping properties of $E_{\gamma}$ and provides the estimates from (A.5).

While the formulation (A.3) already provides a convenient alternative local characterisation of the fractional Laplacian as an iterated and weighted Dirichlet-to-Neumann map for a second order equation in the upper half-plane, if $\gamma \notin (0, 1)$ it is not immediately associated in a natural way with a finite energy (the quantity $\|x_{n+1}^{\frac{1-2\gamma}{2}} \nabla u\|_{L^2(\mathbb{R}^{n+1}_+)}$ diverges in general).

In order to remedy this, in the sequel, we recall that the fractional Laplacian is also related to a Dirichlet-to-Neumann map for a system (or, equivalently, a higher order equation) which can naturally be associated with a finite energy ^[15]. This provides the natural functional analytic framework for our discussion of the unique continuation properties of the higher order fractional Laplacian and explains our focus on unique continuation properties for systems with Muckenhoupt weights in the earlier sections.

In order to derive the desired higher order equation for $u$, we begin by discussing the bulk equation:

Lemma A.3. (Lemma 4.2 in ^[15]). Let $\gamma > 0$ and let $u$ be a solution to the bulk equation in (A.1). Then, for $k\in\{0, \dots, \lfloor{\gamma}\rfloor\}$ the function $w_k = (\Delta_b)^k u$ with $\Delta_b : = x_{n+1}^{-b} \nabla \cdot x_{n+1}^{b} \nabla$ and $b = 1-2\gamma + 2\lfloor{\gamma}\rfloor \in (-1, 1)$ satisfies

In particular,

The equation (A.7) provides us with the bulk equation which we are working with in the sequel. For self-containedness, we recall the argument for Lemma A.3 from ^[15].

Proof. We show that if a function $w$ solves

with $a\in \mathbb{R}$, then $w_1: = \Delta_b w$ solves

To this end, we observe that

This concludes the proof.

Next we seek to complement (A.7) with suitable boundary conditions. To this end, we use the explicit form of $u$ which was deduced in the proof of Lemma A.1. It entails the validity of certain weighted Neumann conditions and provides tangential limits for the associated higher order Dirichlet data:

Lemma A.4. Let $\gamma > 0$, $\mu \in \mathbb{R}$, let $f\in H^{\mu}(\mathbb{R}^n)$ and let $u = E_{\gamma}f = G_{\gamma}\ast f$ be the solution to (A.1) from Lemma A.1. For $x_{n+1} > 0$ and $k\in\{0, \dots, \lfloor{\gamma}\rfloor\}$, define $w_k: = (\Delta_b)^k u$, where $\Delta_b : = x_{n+1}^{-b}\nabla \cdot x_{n+1}^b \nabla$ and $b = 1-2\gamma+2\lfloor{\gamma}\rfloor\in(-1, 1)$. Then we have that for all $k\in\{0, \dots, \lfloor{\gamma}\rfloor\}$ it holds

for some constant $c_{n, \gamma, k}\neq 0$. The functions $w_k$ are in $C^{\infty}_{loc}(\mathbb{R}^n \times (0, \infty))$ and for any $\nu \in \mathbb{R}$ they obey the bounds

Moreover, for any $k\in\{0, \dots, \lfloor{\gamma}\rfloor-1\}$

and for some constant $c_{n, \gamma} \neq 0$

Proof. First, by induction, we show that

For $k = 0$ this is true by Lemma A.1. It thus suffices to prove the induction step. By Lemma A.3 and the equation for $w_k$ we have

Using the claimed representation for $w_k$, i.e., $\hat{w}_k(\xi, x_{n+1}) = c_{n, \gamma, k} |\xi|^{2k} \hat{f}(\xi)\phi_{\gamma-k}(|\xi| x_{n+1})$ and the asymptotic recurrence relations for modified Bessel functions (A.4), this directly implies

The representation from (A.13) together with the asymptotics from (A.4) then directly entails (A.9), (A.11) and (A.12). Here as in the proof of Lemma A.1 all limits should be understood in the corresponding Sobolev spaces. Indeed, for instance, if $k\in\{0, \dots, \lfloor{\gamma}\rfloor-1\}$, by invoking (A.4), we have

Thus, similarly as in the proof of Lemma A.1, we infer

Relying on the asymptotics of the Bessel functions and passing to the limit $x_{n+1}\rightarrow 0$ then leads to (A.11). The remaining limits are obtained analogously.

Finally, the bounds from (A.10) are deduced as those from (A.5) in Corollary A.2.

In analogy to the notation from Lemma A.1, we introduce the notation $E_{\gamma, k}f: = w_k$.

We next show that it is possible to obtain localised regularity estimates for the functions $w_k$ from Lemma A.3. This is helpful in the discussion of the global unique continuation properties for the fractional Laplacian. These are particularly relevant in the analysis of associated fractional inverse problems.

Lemma A.5. Let $\nu \in \mathbb{R}$ and assume that $f\in H^{2k+\gamma-\lfloor{\gamma}\rfloor+2N}(B_1')$ for some $N\in \mathbb{N}$ and ${k\in\{0, \dots, \lfloor{\gamma}\rfloor\}}$. Then for all $r\in (0, 1)$ we have that

The argument for this relies on the pseudolocality of the operator at hand.

Proof. We split

where $\eta$ is a cut-off function that is one on $B_{r}'$ for some $r\in(0, 1)$ and vanishes outside of $B'_1$. For $E_{\gamma, k}(f\eta)$ the claim is a direct consequence of Lemma A.4. It hence suffices to study the regularity of $E_{\gamma, k}(f(1-\eta))$. To this end, we argue as in ^[9]. For convenience of notation we only prove the argument for $k = 0$; the argument for $k\in \{1, \dots, \lfloor{\gamma}\rfloor\}$ is analogous. Let $\psi$ be a second smooth cut-off function which is equal to one on the support of $\eta$ and vanishes outside of $B_{1}'$. Then,

By an explicit computation, we obtain that $P_{\epsilon}^{\gamma}(x'): = \mathcal{F}^{-1}_{x'} (\phi_{\gamma}(\epsilon |\cdot|))(x') = c_{\gamma, n} \frac{\epsilon^{2\gamma}}{(|x'|^2 + \epsilon^2)^{\frac{n+2\gamma}{2}}}$ (this exploits formula 9.6.25 in ^[46]). Using the explicit form of $P_{\epsilon}^{\gamma}(x')$ or heat kernel estimates (as outlined in the next section and in ^[14]), we obtain that for any $a > 0$

As the convolution in the expression for $T_{\epsilon}$ is only active in regions in which $|x'-y'| > a$ for some suitable $a > 0$, by virtue of Schur's lemma and an integration by parts, we then deduce that

whence

Integrating in $x_{n+1}$, then implies that

which is the desired statement.

For the estimate of the normal derivative $x_{n+1}^{1-2(\gamma-\lfloor{\gamma}\rfloor)} \partial_{n+1} u$ we notice that a short computation shows that

Arguing as above then concludes the proof.

We summarise the results from this section for $f\in H^{2\gamma}(\mathbb{R}^n)$:

Proposition A.6. Let $\gamma > 0$ and let $f\in H^{2\gamma}(\mathbb{R}^n)$. Then the function $u: = E_{\gamma}(f) \in C^{\infty}_{loc}(\mathbb{R}^{n+1}_+)\cap H^{1}(\mathbb{R}^{n+1}_+, x_{n+1}^{1+2\gamma - 2 \lfloor{\gamma}\rfloor})$ is a solution to the scalar higher order problem

All limits $x_{n+1}\rightarrow 0$ are understood in an $L^{2}(\mathbb{R}^n)$ sense.

Setting $u_0: = u$ and defining the functions $u_{j+1} = \Delta_b u_{j}$ for $j\in\{0, \dots, \lfloor{\gamma}\rfloor-1\}$, this can also be rewritten as the following system of second order equations

where $m = \lfloor{\gamma}\rfloor$. Again, all limits $x_{n+1}\rightarrow 0$ are understood in an $L^{2}(\mathbb{R}^n)$ sense.

A.2.   The variable coefficient setting – characterisation through a system of degenerate elliptic equations

In this section, we derive analogous results to Proposition A.6 in the presence of variable coefficients, i.e., we are now concerned with the operator $L^\gamma$, where $L = -\nabla \cdot \tilde{a}^{ij} \nabla$ and the coefficients $\tilde{a}^{ij}$ satisfy the conditions stated in (A1)–(A3) with $\mu = 2\lfloor{\gamma}\rfloor$. In contrast to the previous argument in which the Fourier transform diagonalised the tangential operator, we here rely on a spectral decomposition. We argue analogously as in ^[21] and thus only present the arguments formally. For convenience of notation we set

where $L$ is as above and $b = 1-2\gamma+2\lfloor{\gamma}\rfloor$.

To this end, we recall that for the self-adjoint, positive operator $L$ we can carry out a spectral decomposition and obtain a unique associated resolution of the identity which is supported on the spectrum of $L$ with

Based on this we can define the action of the heat semigroup and of the fractional powers of $L$:

where $\text{Dom}(L^{\gamma}): = \{f\in L^2(\Omega): \ \int\limits_{0}^{\infty} \lambda^{2\gamma} dE_{f, f}(\lambda) < \infty\}$.

Our main result in the context of variable coefficients mirrors the statement of the constant coefficient case:

Proposition A.7. Let $\gamma > 0$ and let $f\in \text{Dom}(L^{\gamma})$. Then the function

is a solution to the scalar higher order problem (2.2). The function $u(x', x_{n+1})$ can also be represented as

Setting $u_0(x', x_{n+1}): = u(x', x_{n+1})$ and defining the functions $u_{j+1}(x', x_{n+1}) = L_b u_{j}(x', x_{n+1})$ for $j\in\{1, \dots, \lfloor{\gamma}\rfloor\}$, allows one to rewrite (2.2) as the the system (2.3), where $m = \lfloor{\gamma}\rfloor$. All boundary conditions hold in an $L^2(\mathbb{R}^n)$ sense.

Remark A.8. With the systems representation being established, we also obtain that $u_k \in L^{2}_{loc}(\mathbb{R}^{n+1}_+, x_{n+1}^b)$. Then, direct energy estimates also yield that $u_k\in H^{1}_{loc}(\mathbb{R}^{n+1}_+, x_{n+1}^b)$ for all $k\in\{1, \dots, \lfloor{\gamma}\rfloor\}$, if $f\in \text{Dom}(L^{\gamma})$.

Proof. The fact that $u(x', x_{n+1}) = c_{\gamma} x_{n+1}^{2\gamma} \int\limits_{0}^{\infty} e^{-tL}f(x') e^{-\frac{x_{n+1}^{2}}{4t}} \frac{dt}{t^{1+\gamma}}$ solves the equation

follows as in ^[21,42]. By interior elliptic regularity estimates and the assumed coefficient regularity, this also implies the claimed regularity result.

We discuss the attainment of the Dirichlet boundary conditions for the function $u$: To this end, we observe that

Here we used the change of coordinates $z = \frac{x_{n+1}^2}{t}$. Thus, passing to the limit $x_{n+1}\rightarrow 0$ dominated convergence yields the claimed result.

Next, we seek to show that the function $u$ also satisfies the higher order equation (2.2) and the system (2.3). To this end, we set $w_k(x', x_{n+1}): = L_b^k u(x', x_{n+1})$ for $k\in \{0, \dots, \lfloor{\gamma}\rfloor\}$. We claim that these functions solve the system (2.3), where all boundary conditions hold in an $L^2$ sense.

In order to observe this, analogously as in Lemma A.3, we infer that the functions $w_k$ solve the bulk equation

whence

We claim that

The latter in particular also shows the equivalent representation for $u$. We obtain the first representation for $w_k$ by induction and the following computation

Together with the representation for $u$ this yields the first identity in (A.16). Arguing along the same lines as above (where the argument is detailed for $u$), this also immediately implies the claim on the Dirichlet data.

We next deduce the alternative characterisation of $w_{k}(x', x_{n+1})$: We have

Here we used the change of coordinates $r = \frac{x_{n+1}^2}{4t\lambda}$.

It remains to discuss the Neumann data:

If $\gamma-k > 1$, dominated convergence allows us to infer

as the $r$ powers are still integrable in zero.

For $\gamma-\lfloor{\gamma}\rfloor < 1$, we argue similarly as in ^[42]:

Here we used the change of coordinates $z = \frac{x_{n+1}^2}{r}$. By dominated convergence we may again pass to the limit and obtain

This concludes the argument.