Smoothing properties of the fractional Gauss-Weierstrass semi-group in Morrey smoothness spaces

1.   Introduction

The aim of this paper is to derive a caloric smoothing estimate of the fractional Gauss-Weierstrass semigroup in Morrey smoothness spaces of Besov and Triebel-Lizorkin type. More precisely, we are interested in the inequality

where $0 < t\leq 1, \, d\geq 0$. Here, $\rho\text{-}A^{s}_{p, q}(\mathbb{R}^{n})$ denotes the so-called $\rho\,$-clan of Morrey smoothness spaces where $A = B$ stands for spaces of Besov type and $A = F$ for spaces of Triebel-Lizorkin type. These spaces were introduced in ^[10] and provide a unified approach to several types of Morrey(-Campanato) spaces, global and hybrid spaces; see Subsection 2.2 below. Estimates of type (1.1) play a significant role in the analysis of evolution equations such as (nonlinear) heat, Burgers or Navier-Stokes equations. Due to these applications, we restrict our consideration to Banach spaces, hence, to $s\in \mathbb{R}, \, 1\leq p < \infty, \mbox{ and }1\leq q\leq\infty$. Moreover, we focus on $-n < \rho < 0$; see explanations in Remark 3. Further, $W^\alpha_t$ denotes the fractional Gauss-Weierstrass semigroup, formally defined as

where $\wedge$ and $\vee$ stand for the Fourier transform and its inverse, respectively. Estimate (1.1) extends known results with respect to considered function spaces and gives an alternative proof concerning $\alpha$ compared with former results. Concerning the global spaces $A^s_{p, q}(\mathbb{R}^{n})$, $A\in\{B, F\}$, we refer to ^[1] for $\alpha\in \mathbb {N}$ based on the ideas for $\alpha = 1$ developed in ^[25]. The case $\alpha > 0$ has been considered in ^[11,13]. An alternative proof was presented in ^[4] using characterizations of the spaces in terms of fractional Gauss-Weierstrass semigroups. For local spaces $\mathcal{L}^r A^s_{p, q}(\mathbb{R}^{n})$, we refer again to ^[25] and for hybrid spaces $L^r A^s_{p, q}(\mathbb{R}^{n})$ to ^[26] (both consider $\alpha = 1$). As mentioned above, the definition of $W^\alpha_t$ in terms of a convolution is rather formal due to the fact that the fractional heat kernel is not smooth in $\xi = 0$. More precisely, $e^{-|\xi|^{2\alpha}}$ does not belong to the Schwartz space of rapidly decreasing functions but still has a sufficiently fast polynomial decay; see ^[17] Lemmas 1 and 2. Our proof of (1.1) combines the results presented in ^[1,17] using characterizations by means of wavelets and molecules adapted to Morrey smoothness spaces. The paper is structured as follows. Section 2 provides necessary notation, definitions of global function spaces, a Fourier analytic approach to Morrey smoothness spaces and their characterization in terms of wavelets and molecules. Moreover, we recall useful embeddings and coincidences of related spaces. Our main result is contained in Theorem 1 in Section 3. We start by introducing the fractional Gauss-Weierstrass semigroup and provide estimates of their fractional derivatives which turn out to be our main tool besides decomposition techniques of $\rho\text{-}A^{s}_{p, q}(\mathbb{R}^{n})$-spaces. Finally, Section 4 sketches some application.

2.   Preliminaries

2.1.   Notation and basic definitions

Let $\mathbb{R}^{n}$ be the Euclidean $n\,$- space with $n\in \mathbb {N}$ where $\mathbb {N}$ indicates the collection of all natural numbers, $\mathbb {N}_0 = \mathbb {N}\cup\{0\}$. Put $\mathbb{R} = \mathbb{R}^1$, whereas $\mathbb{C}$ is the complex plane. $S(\mathbb{R}^{n})$ denotes the Schwartz space of all complex-valued infinitely differentiable rapidly decreasing functions on $\mathbb{R}^{n}$ and $S'(\mathbb{R}^{n})$ its dual, the space of all tempered distributions. Furthermore, let $L_{p}(\mathbb{R}^{n})$ with $0 < p < \infty$ be the standard complex quasi-Banach space with respect to the Lebesgue measure in $\mathbb{R}^{n}$, quasi-normed by

with the natural modification if $p = \infty$. Similarly, we define $L_p^{\text{loc}}(\mathbb{R}^{n})$ which consists of all $f$ whose restriction to bounded Lebesgue measurable sets $M\subset \mathbb{R}^{n}$ belongs to $L_p(M)$. As usual, $\mathbb{Z}$ is the collection of all integers and $\mathbb{Z}^n$ defines the set of all lattice points $m = (m_1, \dots, m_n)$, $m_i\in \mathbb{Z}$. If $\phi\in S(\mathbb{R}^{n})$, then

denotes the Fourier transform of $\phi$. $\mathcal{F}^{-1}\phi$ and $\phi^{\vee}$ stand for the inverse Fourier transform, given by the righthand side of (2.1) with $i$ in place of $-i$, and $x\xi$ stands for the scalar product in $\mathbb{R}^{n}$. $\mathcal{F}$ and $\mathcal{F}^{-1}$ are extended in the usual way to $S'(\mathbb{R}^{n})$. Let $\phi_0\in S(\mathbb{R}^{n})$ with

and let

Then

i.e., $\{\phi_j\}$ forms a dyadic resolution of unity. Recall that $(\phi_j\widehat{f})^\vee$ is an entire analytic function and makes sense point-wise for any $f\in S'(\mathbb{R}^{n})$. All function spaces, which we consider here, are subspaces of $S'(\mathbb{R}^{n})$. Let

be the dyadic cube in $\mathbb{R}^{n}$ (where $(0, 1)^n$ denotes the open cube with side length 1) with side length $2^{-J}$ parallel to the coordinate axes and $2^{-J}M$ as the lower left corner. For a cube $Q$, we denote by $dQ$ with $d > 0$ the cube concentric with $Q$ and its side length multiplied by $d$. $|\Omega|$ denotes the Lebesgue measure of the Lebesgue measurable set $\Omega\subset \mathbb{R}^{n}$. We write $a_+: = \max(0, \, a)$ for $a\in \mathbb{R}$.

Finally, we write

if there exist two positive constants $c_1, c_2 > 0$ such that $c_1\, a\leq b\leq c_2\, a$. Sometimes we use the symbol "$\lesssim$" instead of "$\leq$". The meaning of $a\lesssim b$ is given by: There exists a positive constant $C$ such that $a\leq C b$.

Definition 1. Let $\phi = \{\phi_j\}_{j = 0}^{\infty}$ be the above dyadic resolution of unity.

(ⅰ) Let $0 < p, q\leq\infty$, $s\in \mathbb{R}$. We define the Besov spaces $B_{p, q}^s(\mathbb{R}^{n})$ as the collection of all $f\in S' (\mathbb{R}^{n})$ such that

is finite (with the usual modification if $q = \infty$).

(ⅱ) Let $0 < p < \infty$, $0 < q\leq\infty$, $s\in \mathbb{R}$. We define the Triebel-Lizorkin spaces $F_{p, q}^s(\mathbb{R}^{n})$ as the collection of all $f\in S' (\mathbb{R}^{n})$ such that

is finite (with the usual modification if $q = \infty$).

(ⅲ) Let $0 < q < \infty$ and $s\in \mathbb{R}$. We define the spaces $F^s_{\infty, q} (\mathbb{R}^{n})$ as the collection of all $f\in S' (\mathbb{R}^{n})$ such that

is finite.

Remark 1. These are the classical (global) spaces $A^s_{p, q}(\mathbb{R}^{n})$, where $A$ stands either for $B$ or $F$. In what follows, we will write $A^s_{p, q}(\mathbb{R}^{n})$ if an assertion applies both to $B$- and $F$- spaces. The above definition coincides with [28, Definition 1.1]. A detailed study of these spaces, including their history and properties, can be found in ^[21,22,23]. They are independent (in the sense of equivalent quasi-norms) of the chosen resolution of unity. Therefore, we will omit the subscript $\phi$ in the sequel.

2.2.   Morrey smoothness spaces

Next, we introduce Morrey smoothness spaces following closely the new approach presented in ^[10]. We present two different possibilities to characterize these types of spaces. We start with the Fourier analytic approach followed by characterizations of the spaces by means of wavelets and molecules. This is one of the main tools for proving estimate (1.1). We recall the necessary definitions and assertions as far as we need them for our considerations. Standard references with respect to wavelets are, e.g., ^[6,14,30]. For molecules, we refer, e.g., to ^[20], [8, Section 12], [9, Section 5].

2.2.1.   Fourier analytic approach

Recall that $L_p^{\text{loc}}(\mathbb{R}^{n})$ consists of all $f$ whose restriction to bounded Lebesgue measurable sets $M\subset \mathbb{R}^{n}$ belongs to $L_{p}(M)$, $1\leq p\leq\infty$.

Definition 2. Let $n\in \mathbb {N}$, $0 < p < \infty$ and $-n\leq \rho\leq 0$. Then, $\Lambda^\rho_p(\mathbb{R}^{n})$ collects all $f\in L_p^{\text{loc}}(\mathbb{R}^{n})$ such that

is finite.

Remark 2. Note that we have the coincidence

in the sense of equivalent (quasi-)norms.

Let $0 < p\leq u < \infty$. The Morrey space $\mathcal{M}^u_p(\mathbb{R}^{n})$ collects all $f\in L_p^{\text{loc}}(\mathbb{R}^{n})$ such that

is finite. Compared with Definition 2, one has

Definition 3. Let $n\in \mathbb {N}$, $s\in \mathbb{R}$, $0 < p < \infty$ and $0 < q\leq\infty$.

(ⅰ) Let $-n\leq \rho\leq 0$. Then, $\Lambda^{\rho}B^{s}_{p, q}(\mathbb{R}^{n})$ is the collection of all $f\in S'(\mathbb{R}^{n})$ such that

is finite and $\Lambda^{\rho}F^{s}_{p, q}(\mathbb{R}^{n})$ is the collection of all $f\in S'(\mathbb{R}^{n})$ such that

is finite (usual modification if $q = \infty$).

(ⅱ) Let $-n\leq \rho < 0$. Then, $\Lambda_{\rho}B^{s}_{p, q}(\mathbb{R})$ is the collection of all $f\in S'(\mathbb{R}^{n})$ such that

is finite (usual modification if $q = \infty$). The space $\Lambda_{\rho}F^{s}_{p, q}(\mathbb{R}^{n})$ is the collection of all $f\in S'(\mathbb{R}^{n})$ such that

is finite (usual modification if $q = \infty$).

Remark 3. Let $s\in \mathbb{R}$ and $A\in\{B, F\}.$

(ⅰ) The spaces introduced in Definition 3 coincide with the well-known global spaces $A^s_{p, q}(\mathbb{R}^{n})$ when $\rho = -n$. Thus,

see [10, Remarks 2.7 and 2.9].

Moreover, if $\rho = 0$, we have the coincidence

(see [10, Proposition 2.12 (ⅲ)]).

(ⅱ) The Besov-Morrey spaces $\mathcal{N}^s_{u, p, q}(\mathbb{R}^{n})$, $s\in \mathbb{R}$, $0 < q\leq\infty$ and $0 < p\leq u < \infty$ were introduced by Kozono and Yamazaki in ^[12]. The Triebel-Lizorkin Morrey spaces $\mathcal{E}^s_{u, p, q}(\mathbb{R}^{n})$ go back to Tang and Xu in ^[29]. They coincide with the above scales $\Lambda_{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$ as follows:

For more details; see [10, Remark 2.7] and the references given there.

(ⅲ) The spaces in Definition 3, part (i), are reformulations of corresponding Morrey smoothness spaces $A^{s, \tau}_{p, q}(\mathbb{R}^{n})$ (see [10, Remark 2.9]). It holds

where

(ⅳ) Note that the spaces defined in Definition 3, part (ⅰ), coincide for all admitted parameters with the hybrid spaces as introduced by Triebel in ^[26]. We have

and $0 < q\leq\infty$ (see [10, Remark 2.9]).

(ⅴ) Moreover, it holds

We refer to [10, formula (2.45), p. 1309]) and the references given there to [19, Theorem 1.1 (ii)], and [18, Theorem 6.35, p. 794]. The situation is different if $A = B$. We have the strict embedding

if $0 < q < \infty$ and

(see [10, Theorem 2.21, (iv)]).

In other words, one has only one $\Lambda^\rho F = \Lambda_\rho F$ scale but two $\Lambda^\rho B$ and $\Lambda_\rho B$ scales.

(ⅵ) Furthermore, spaces $\Lambda_{\rho}B^{s}_{p, q}(\mathbb{R})$ can be obtained by means of real interpolation. More precisely, let $n < \rho < 0, \ 0 < p < \infty$ and $0 < q_1, q_2\leq\infty$. Then it holds

if

(see [10, Proposition 3.3 and Remark 3.4]).

Following [10, Definition 2.15], for $-n\leq\rho < 0$ the so-called $\rho$-clan $\rho\text{-} A^s_{p, q}(\mathbb{R}^{n})$ stands for the three families

with

2.2.2.   Lifts and embeddings

We summarize shortly some properties of $\rho\text{-}A^{s}_{p, q}(\mathbb{R}^{n})$ - spaces. Here, we adapt parameters $p, \; q,$ and $\rho$ to our later needs.

Proposition 1. (see [10, Theorem 5.3])

Let $s\in \mathbb{R}$, $1\leq p < \infty$, $1\leq q\leq\infty$ and $-n < \rho < 0$. Then

$A\in\{B, F\}$, and

Let $\sigma$ be a real number. The operator $I_{\sigma}$ given by

is a one-to-one map onto itself both in $S(\mathbb{R}^{n})$ and $S'(\mathbb{R}^{n})$. Furthermore, $I_{\sigma}$ is a lift for the spaces $A^s_{p, q}(\mathbb{R}^{n})$ with $s\in \mathbb{R}$, $0 < p, q\leq\infty$. Thus, we have

in the sense of equivalent quasi-norms. We recall the corresponding results for Morrey smoothness spaces.

Proposition 2. (see [10, Theorem 3.8])

Let $s, \sigma\in \mathbb{R}$, $1\leq p < \infty$, $1\leq q\leq\infty$ and $-n < \rho < 0$. Then $I_{\sigma}$ maps $\Lambda^{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$ isomorphically onto $\Lambda^{\rho}A^{s+\sigma}_{p, q}(\mathbb{R}^{n})$ and $\Lambda_{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$ isomorphically onto $\Lambda_{\rho}A^{s+\sigma}_{p, q}(\mathbb{R}^{n})$, where $A\in\{B, F\}$. Further,

Remark 4. We have the following properties.

(ⅰ) It holds

(ⅱ) Let $\sigma_1\ \ \text{and}\ \sigma_2$ be real numbers. Then

2.2.3.   Representation by means of wavelets

We recall the necessary definitions and assertions as far as we need them for our considerations. Standard references with respect to wavelets are, e.g., ^[6,14,30]. Throughout the following sections, we restrict parameters $p, q$ and $\rho$ as indicated above to $1\leq p < \infty$, $1\leq q\leq\infty$ and $-n < \rho < 0$. Let $C^u(\mathbb{R})$, $u\in \mathbb {N}$ denote the space of all complex-valued $u\,$- times continuously differentiable functions with bounded derivatives in $\mathbb{R}$. Let

be real compactly supported Daubechies wavelets with $\widehat{\psi_F}(0) = (2\pi)^{-1/2}$ and

$\psi_F$ is called the scaling function (father wavelet) and $\psi_M$ is the so-called associated wavelet (mother wavelet). We extend these wavelets from $\mathbb{R}$ to $\mathbb{R}^{n}$ by the usual multi-resolution procedure. Let either

which means that the components $G_r$ of $G$ where $r\in\{1, \dots, n\}$ are either $F$ or $M$ or let

Here, $*$ indicates that at least one of the components of $G$ must be an $M$. In the sequel, we denote such a set $G^j$ with $G^*$. Let

(where $m_r$ denote the components of $m$), $x\in \mathbb{R}^{n}$, now with $j\in \mathbb {N}_0$. We always assume that $\psi_F$ and $\psi_M$ have an $L_2$-norm 1. Then, for any $u\in \mathbb {N}$,

is an orthonormal basis in $L_2(\mathbb{R}^{n})$ and

with

is the corresponding expansion, where $2^{-jn/2} \Psi^j_{G, m}$ are uniformly bounded functions with respect to $j$ and $m$. For more detailed explanations, cf. ^[22,23] and [24, Subsection 1.2.1].

Let $\chi_{j, m}$, $j\in \mathbb {N}_0$, $m\in \mathbb{Z}^{n}$ be the characteristic function of the usual dyadic cubes $Q_{j, m}$ as defined in (2.2).

Definition 4. Let $1\leq p < \infty$, $1\leq q\leq\infty$, $s\in \mathbb{R}$ and $-n < \rho < 0$. Let

(ⅰ) Then

with

(ⅱ) Let $1\leq p < \infty$, $1\leq q\leq\infty$, $s\in \mathbb{R}$ and $-n < \rho < 0$. Let

Then

where

(usual modification if $q = \infty$).

Remark 5. Part (ⅰ) is covered by [28, Definition 1.13] with a reference to [26, 3.26]. Concerning part (ⅱ), we refer to [26, Definition 3.24] which is already adapted to our notation.

We shall use the notation $a^s_{p, q}$ with $a = b$ or $a = f$. Based on [26, Theorem 3.26] and Remark 3 part (ⅳ), we have the following wavelet characterization of $\Lambda^\rho A^s_{p, q}(\mathbb{R}^{n})$.

Proposition 3. Let $s\in \mathbb{R}$, $1\leq p < \infty, \ 1\leq, q\leq\infty$, and $-n < \rho < 0$. Let $\Psi$ be the wavelet system (2.30) based on (2.25)–(2.29), where $u > \max(s, -s)$.

Let $f\in S'(\mathbb{R}^{n}).$ Then $f\in \Lambda^{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$ if, and only if, it can be represented as

where $a = b$ if $A = B$ and $a = f$ if $A = F$. The series converges unconditionally in $S'(\mathbb{R}^{n})$. The representation (2.33) is unique,

and

is an isomorphic map of $\rho\mathit{{-}}A^{s}_{p, q}(\mathbb{R}^{n})$ onto $\Lambda^{\rho}a^{s}_{p, q}$. Hence,

Remark 6. For a detailed discussion of how to understand the dual pairing $\langle\, {f}, { \Psi^j_{G, m}}\rangle$ in (2.34) we refer to [26, Thm 3.26] and the references given there.

2.2.4.   Representation by means of molecules

Next, we introduce molecular decompositions of the spaces $\Lambda^{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$. We recall first the definition of molecules related to $Q_{j, m}$ according to [25, Section 2.4.2]. Molecular decompositions have been considered, for instance, in ^[8,9,20]. We refer also to [32, Chapter 3] where one finds, in particular, corresponding representations for the spaces $A^{s, \tau}_{p, q}(\mathbb{R}^{n})$ briefly mentioned in Remark 3 part (ⅲ).

Definition 5. Let $K\in \mathbb {N}_0, \ N\in \mathbb {N}\ \text{and}\ L > N+n-1$. Let $j$ be a natural number and $m\in \mathbb{Z}^{n}$. The $L_{\infty}-$ functions $b_{j, m}: \mathbb{R}^{n}\mapsto\mathbb{C}$ are called $(K, N, L)-$ molecules, related to $Q_{j, m}$, if

and

We introduce corresponding sequence spaces.

Definition 6. Let $1\leq p < \infty$, $1\leq q\leq\infty$, $s\in \mathbb{R}$ and $-n < \rho < 0$. Let

Then

with

and

with

(usual modification if $q = \infty$).

The following molecular characterization of $\Lambda^{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$ is already adapted to our needs based on [25, Proposition 2.35].

Proposition 4. $1\leq p < \infty$, $1\leq q\leq\infty$, $s\geq 0$ and $-n < \rho < 0$. Let $K\in \mathbb {N}_0, \ N = 1$ and $L\in \mathbb{R}$ with

Let $f\in S'(\mathbb{R}^{n})$. Then $f\in \Lambda^{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$ if, and only if, it can be represented as

where $b_{j, m}$ are $(K, N, L)$-molecules, unconditional convergence being in $S'(\mathbb{R}^{n}).$ Additionally,

where the infimum is taken over all admissible representations (2.43).

3.   Caloric smoothing

In this section we prove the smoothing property

where

Note that in view of Remark 3, part (ⅰ), the case $\rho = -n$ is already covered by the papers ^[4,13]. At first, we prove (3.1) for the $\Lambda^{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$-spaces. The corresponding result for $\Lambda_\rho A^s_{p, q}(\mathbb{R}^{n})$ follows then from Remark 3 parts (ⅴ) and (ⅵ).

3.1.   Fractional Gauss-Weierstrass semigroup

We start with some observations concerning fractional heat kernels. Consider the function

Clearly, the function $\varphi$ is not smooth in $\xi = 0$ if $\alpha\notin \mathbb {N}$. To define the Gauss-Weierstrass semigroup in a proper way, consider first

Moreover, we need the fractional Laplacian, formally given by

for $\omega$ in an appropriate function space. We define

The following two estimates can be found in [17, Lemmas 1 and 2].

Lemma 1. The kernel function $G^{\alpha}$ satisfies the point-wise estimate

for $\alpha > 0$. Consequently, one has

Lemma 2. The kernel function $G^{\alpha, \sigma}$ has the point-wise estimate

for $\alpha, \, \sigma > 0$. Consequently, one has

These lemmas show that $G^\alpha$ and $G^{\alpha, \sigma}$ provide a sufficiently fast polynomial decay which will be of great use later in the proof of Proposition 5. Now, we consider the fractional heat kernel given by

Obviously, it holds

Note that $G^{\alpha, \sigma}_t$ has the same scaling properties as $G^\alpha_t$, namely,

see [17, p. 6].

Based on Propositions 1 and 2 and Lemma 1, we define the fractional Gauss-Weierstrass semigroup $W^\alpha_t$ as follows.

Definition 7. Let $1\leq p < \infty$, $1\leq q\leq \infty$ and $-n < \rho < 0.$ Let $\omega\in \rho\text{-}A^{s}_{p, q}(\mathbb{R}^{n})$ and $G^\alpha_t$ be as above.

(1) We define

(2) Let $\sigma\in \mathbb{R}$ such that $s+\sigma > \frac{|\rho|}{p}$. Then we define

Note that the definition in part (ⅱ) is independent of $\sigma$. According to Proposition 2, $I_{\sigma}\omega$ is smooth enough to justify the application of $W^\alpha_t$ in the sense of part (ⅰ).

3.2.   Preparations

Let $\omega\in\Lambda^{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$ with $A\in\{B, F\}$. Under the conditions of Proposition 3, we can represent

with coefficients

in the interpretation of (2.34). We are interested in a similar decomposition of $W^\alpha_t\omega$ in terms of molecules.

Let

based on (2.29), where $\psi_F, \; \psi_M\in C^u(\mathbb{R})$ are the Daubechies wavelets as in (2.25) and (2.26). According to the case $\alpha = 1$, cf. [25, Subsection 2.4.2], the functions $b^j_{G, m}(x, t)$ are called $\alpha$-caloric wavelets. As already mentioned, we show that after a slight modification, they are molecules in the sense of Definition 5 for appropriately chosen parameters $N, \, K, \, L$. We put

Proposition 5. Let $\alpha > 0, \ d\geq0$ and $\big(b^j_{G, m}\big)_d$ have the meaning of (3.18). Let $u\in \mathbb {N}$ such that

Then, there exists $L > n$ such that $\big(b^j_{G, m}\big)_d$ are $(u, 1, L)$-molecules according to Definition 5 for some $c > 0$ and any fixed $t$ with $2^jt^{1/2\alpha}\geq1$.

Proof. $\underline {{\rm{Step\;1:}}}$ We prove the vanishing moment conditions for $b^j_{G, m}(x, t)_d$ first. Let $\beta$ be a lattice point on $\mathbb {N}_0^n$ such that $|\beta| < u$. We have

In (3.20), we used the fact that because of $G\in G^*$, there exists $l\in\{1, \cdots, n\}$ such that $G_l = M$. Hence, using the moment condition (2.26) and $0\leq\beta_l < u$, at least one of the factors is equal to zero, which can be seen by elementary calculations.

$\underline {{\rm{Step\;2:}}}$ Now, we prove that there exists an $L > 0$ satisfying (2.42) such that

Due to (3.17) we may assume $m = 0$. Let $|\zeta| = 0$ and consider

$j\in \mathbb {N}_0$, $G\in G^*$ and $2^jt^{1/2\alpha}\geq1$. We rewrite

where $G^\alpha$ is defined as in (3.3). Apparently, it holds

We expand $G^\alpha$ in a Taylor polynomial about the origin with a remainder term of order $u$ (which is possible according to Lemma 2) and substitute it into (3.22). Because of the moment conditions, terms of order less than $u$ vanish such that we have the estimate

where we used the boundedness of the derivatives of $G^\alpha$ in (3.23). The integrand in (3.23) is zero outside a ball of radius $c\, 2^{-j}t^{-1/2\alpha}$ centered at the origin. Hence, we obtain

for all $x\in \mathbb{R}^{n}$. On the other hand, it follows from (3.22) and Lemma 1 that

Since $2^jt^{1/2\alpha}$ and $|y|\leq c2^{-j}t^{-1/2\alpha}$, we have $1+|y|\leq c$.

Thus,

Let $0 < \varepsilon < 1$. Combining (3.24) and (3.26), we have

Since $2^{j}t^{1/2\alpha}\geq1$, it holds

Hence,

Replacing $t^{1/2\alpha}x$ by $x$ in (3.28) yields the estimate

We define $g(\varepsilon): = (1-\varepsilon)(u+n)-2\alpha\varepsilon = u+n-\varepsilon(u+n+2\alpha)$ with $0 < \varepsilon < 1$. Obviously, the graph of $g$ is a strictly decreasing straight line. Because of $u > 0$, it holds that

Hence,

According to (3.19), we have

Hence, there exists a uniquely determined $\varepsilon\in(0, 1)$, more precisely,

such that $g(\varepsilon) = d$. For this choice of $\varepsilon$, we put $L: = \big(n+2\alpha\big)\varepsilon > n.$ Inserting this in (3.29) leads finally to the estimate

$\underline {{\rm{Step\;3:}}}$ Let now $1\leq |\zeta|\leq u$. Recall that

The derivatives $D^{\zeta}_{x}$ can be shifted to $\Psi^j_{G, 0}$ and we get

Note that $\left(D^{\zeta}\prod_{r = 1}^{n}\psi_{G_r}\right)(2^jy_r)$ fulfills the vanishing moment condition for $|\zeta|\leq u$, which can be seen by iterative integration by parts. Thus, we obtain, using the same arguments as in the case $\zeta = 0$,

To be in correlation with the Definition 5, the choice of $N = 1$ leads directly to $N\leq u.$ The condition on $L$ as discussed in Step 2 remains unchanged. We conclude from (3.32) and (3.33) that $b^j_{G, m}(\cdot; t)_d$ are $(u, N, L)$-molecules according to Definition 5. □

Remark 7. Note that throughout the previous proof, we assumed $m = 0$. This is due to the fact that the case $m\neq0$ can be transformed to $m = 0$ by applying a change of variable.

3.3.   Main result

In this section, we prove the estimate (1.1) which is the key estimate to derive existence and uniqueness results for several nonlinear heat and Navier-Stokes equations.

Theorem 1. Let $\ 1\leq p < \infty, 1\leq q\leq\infty, s\in \mathbb{R}, \ \alpha > 0, \ d\geq0$ and $-n < \rho < 0$. Then there exists a constant $C > 0$ such that

for all $0 < t\leq 1$ and for all $\omega\in \rho\mathit{{-}}A^{s}_{p, q}(\mathbb{R}^{n}).$

Proof. $\underline {{\rm{Step\;1:}}}$ We assume first $s > \frac{|\rho|}{p}$. Let $\omega\in \Lambda^{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$. Then by Proposition 3 we have the wavelet representation

with $(\lambda^{j, G}_m)\in \Lambda^{\rho}a^{s}_{p, q}$, where we choose sufficiently smooth wavelets $\Psi^j_{G, m}$. More precisely, we assume for $u\in \mathbb {N}$, according to (2.25) and (3.19), respectively, that

Let $k\in \mathbb {N}$. We split (3.35) as follows:

Applying the Gauss-Weierstrass semigroup on the previous representation yields

We consider the second summand on the righthand side of (3.37) and assume $2^{-2\alpha k} < t\leq 2^{-2\alpha(k-1)}$. Since $j > k$, it follows $2^jt^{1/2\alpha}\geq 2^{j-k}\geq 1.$ Applying Proposition 5, we want to derive a molecular representation of $W^\alpha_t\omega_k$. We have

where $\mu^{j, G}_m = c^{-1}2^{-jd}t^{-d/2\alpha}\lambda^{j, G}_m$ and $b^j_{G, m}(\cdot, t)_d$ has the meaning of (3.18). We set

and $\mu^j_m = 0$, if $j = 0, \dots, k$. For $j > k$, the choice of $\{\mu^j_m\}_{j, m}\in \Lambda^{\rho}\overline{a}^{s+d}_{p, q}$ depends on $\overline{a} = \overline{b}$ or $\overline{a} = \overline{f}$ (according to Definition 6). It follows from Proposition 5 that $\{b_{j, m}\}_{j, m}$ with $j\in \mathbb {N}\ \text{and}\ m\in \mathbb{Z}^n$ are $(u, 1, L)$-molecules in the sense of Definition 5, where $L > n$ has the meaning as in Step 2, Proposition 5.

In order to show that $W^\alpha_t\omega_k\in \Lambda^{\rho}A^{s+d}_{p, q}(\mathbb{R}^{n})$ and that

we use Proposition 4 with $N = 1, K = u$, $L$ as above and $s+d$ in place of $s$. It remains to be shown the estimate

where $\mu^* = \{\mu^{j}_m:\ j\in \mathbb {N}_0, \ m\in \mathbb{Z}^{n}\}$ and

If $a = b$, we have

Thus,

If a = f, we have

Thus,

Estimates (3.42) and (3.43) imply (3.40) and thus, (3.39).

Now, we consider the first term on the righthand side of (3.37), i.e., $j\leq k$ and assume $A = B$. Then,

Since $j\leq k$, we have $2^{jd}\leq 2^{kd}$. This leads to

Reasoning in the same way but now with the $F$-spaces and the $\Lambda^{\rho}f^{s}_{p, q}$ sequences, we also obtain

Hence, it follows from (3.44) and (3.45) that

We assumed at the very beginning that $s > \frac{|\rho|}{p}$. Hence, $s+d > \frac{|\rho|}{p}$ and

is well-defined.

Applying Minkowski's inequality and the translation invariance of the spaces, we can estimate

where the constant is independent of $t$. Together with (3.44) and (3.45), we achieve

Since $2^{-k2\alpha} < t\leq 2^{-2\alpha(k-1)}$, we have the equivalence $t^{-d/2\alpha}\sim 2^{kd}.$ Together with the wavelet characterization of $\omega$, this leads to the estimate

where $0 < t\leq1$ and $s > \frac{|\rho|}{p}, \ d\geq0$.

$\underline {{\rm{Step\;2:}}}$ Let $s\leq\frac{|\rho|}{p}$.

We choose $\sigma > 0$ such that $s+\sigma > \frac{|\rho|}{p}$. Let $\omega\in \Lambda^{\rho}A^{s}_{p, q}(\mathbb{R}^{n})$. Recall from part (ⅱ) of Definition 7 that

where $I_{\sigma}\omega\in\Lambda^{\rho}A^{s+\sigma}_{p, q}.$ Let $d\geq0$.

Then,

Finally, the corresponding result for the spaces $\Lambda_\rho A^s_{p, q}(\mathbb{R}^{n})$ follows by real interpolation according to Remark 3, part (ⅵ). This completes the proof of (3.34). □

3.4.   Final remarks

Let us give a short outlook in view of possible applications. As mentioned in the introduction, estimates of type (1.1) play a significant role in the analysis of (fractional) evolution equations e.g., heat, Navier-Stokes, quasi-geostrophic, Keller-Segel or Burger's equations. We refer to the approach developed and elaborated in the monographs ^[25,26,27] which is related to the classical Gauss-Weierstrass semigroup (i.e., $\alpha = 1$ in (1.1)). Refined mathematical models, for example, in physics and chemotaxis, suggest and require us to replace the Laplacian by the fractional Laplacian $(-\Delta)^\alpha$ in related (nonlinear) evolution equations. As far as the study of corresponding Cauchy problems is concerned, let us mention, for example, the papers ^{[5,7,15,16,17,31]}. For example, let us consider the Cauchy problem

where $0 < T \leq \infty$, $2\leq n \in \mathbb {N}$, $\alpha > 0$ and

stands for the nonlinear term. It serves as a scalar model case for fractional Navier-Stokes equations. For further types of nonlinear terms, we refer to the abovementioned papers. The standard approach to prove the existence and uniqueness of mild solutions is to consider the related fixed point problem $T_{u_0}u = u$, where the operator $T_{u_0}$ is defined as

in appropriate function spaces. We are interested in vector-valued weighted Lebesgue spaces

as solution spaces. Here, $1\leq v < \infty$ (usual modification if $v = \infty$), $b\in \mathbb{R}$, $0 < T\leq\infty$, and $X$ is an appropriately chosen Banach space according to given initial data. For initial data belonging to Besov or Triebel-Lizorkin spaces $A^{s_0}_{p, q}(\mathbb{R}^{n})$, this has been investigated in the hyper-dissipative case $\alpha\in \mathbb {N}$ in ^[1,2,3] as well as in ^[4] in the case of fractional $\alpha$, where $X = A^s_{p, q}(\mathbb{R}^{n})$. Here, parameters $s, \; b,$ and $v$ depend on $\alpha, \; p, \; s_0$ and the dimension $n$.

4.   Conclusions

The smoothing property (1.1) paves the way to deal with Cauchy problems of the above type for initial data belonging to Morrey smoothness spaces $\rho\text{-}A^{s_0}_{p, q}(\mathbb{R}^{n})$. As far as the case $\alpha = 1$ is concerned, partial results can be found in [26, Chapters 4 and 5] for hybrid spaces $\Lambda^\rho A^{s_0}_{p, q}(\mathbb{R}^{n})$ (see also Remark 3, part (ⅳ)). We intend to consider the general case of fractional $\alpha$ and certain classes of nonlinear terms $f$ in (3.48) in forthcoming papers.

5.   Author contributions

All authors contributed equally to this work. All authors have read and approved the final version of the manuscript for publication.

Acknowledgments

We are grateful to the referees for careful reading and for their useful comments and suggestions, which improved the quality of the paper.

Conflict of interest

All authors declare no conflicts of interest in this paper.