Generalized Newton-Leibniz formula and the embedding of the Sobolev functions with dominating mixed smoothness into Hölder spaces

1.   Prelude

Let $W_p^1(\mathbb{R}^n), 1\leq p\leq \infty$ be a Sobolev space of weakly differentiable functions $u\in L_p(\mathbb{R}^n)$ with first order weak derivatives in $L_p(\mathbb{R}^n), i = 1, ..., n$. Originally discovered in the celebrated paper ^[1], the concept of Sobolev spaces became a trailblazing idea in many fields of mathematics. The goal of this paper is to analyze embedding of $W_p^1(\mathbb{R}^n)$ into Hölder spaces $C^{0, \mu}(\mathbb{R}^n), 0\leq \mu \leq 1$ ^[2]. A standard notation will be employed for the embedding of Banach spaces:

● $B_1 \hookrightarrow B_2$ means bounded embedding of $B_1$ into $B_2$, i.e., $B_1\subset B_2$, and

● $B_1 \Subset B_2$ denotes compact embedding of $B_1$ into $B_2$, meaning that $B_1 \hookrightarrow B_2$, and every bounded subset of $B_1$ is precompact in $B_2$.

If $n = 1$, the equivalency class of elements of $W_p^1(\mathbb{R})$ always contain an absolutely continuous element, which is Hölder continuous with exponent $1-p^{-1}$, if $p > 1$, i.e., there is a bounded embedding

The embedding (1.1) easily follows from the Newton-Leibniz formula

via the application of the Hölder inequality and compactness argument. The embedding (1.1) fails to be true if $n\geq 2$ and $p\leq n$. However, there is a bounded embedding ^[3]

Hence, stretching the integrability exponent $p$ beyond space dimension $n$ implies the Hölder continuity. In particular, elements of the Hilbert space $H^1(\mathbb{R}^n) = W_2^1(\mathbb{R}^n)$, are not continuous in general, if $n\geq 2$. The main goal of this paper is to express the continuity of elements of $W_p^1(\mathbb{R}^n)$ in terms of weak differentiability requirements.

Problem 1.1. What are the minimal weak differentiability requirements on elements of $W_p^1(\mathbb{R}^n)$ ($1\leq p \leq n$) to be continuous? In terms of weak differentiability requirements, what is the largest subspace of $W_p^1(\mathbb{R}^n)$ embedded into Hölder spaces for all $p\geq 1$?

The paper reveals that the anticipated subspace is the Sobolev-Nikol'skii space

equipped with the norm

The space $S_p^1(\mathbb{R}^n)$ is a special case of Sobolev spaces with dominating mixed smoothness. The class was introduced by Nikol'skii in ^[4,5]. There is a vast literature on the analysis of these spaces. We refer to ^[6,7,8,9] and the references therein.

The main result of this paper is twofold. First, we introduce and prove a generalization of the celebrated Newton-Leibniz formula to $n$-dimensional rectangles (or $n$-rectangles). Then by using the new formula as a tool, we present a surprisingly simple and elegant proof of the embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces. The proof resembles the proof of the embedding (1.1) in the one-dimensional case by using generalized Newton-Leibniz formula, Hölder inequality, and iterative application of the Sobolev trace embedding results. In particular, we prove that the generalized Newton-Leibniz formula is preserved in space $S_p^1(\mathbb{R}^n)$. Counterexamples support the claim that in terms of weak differentiability requirements, $S_p^1$ is the largest class of Lebesgue's integrable and weakly differentiable functions in $\mathbb{R}^n$ which preserve generalized Newton-Leibniz formula, and upgrades the pointwise regularity to Hölder continuity.

2.   Notations

● $C^0(\mathbb{R}^n)$ is a Banach space of continuous and bounded functions with the norm

● For $k\in \mathbb{N}$, $C^k(\mathbb{R}^n)$ is a Banach space of $k$ times continuously differentiable functions, with all derivatives of order $k$ bounded, and with the norm

where $D^ju$ is a tensor of rank $j$, dimension $n$, and

● $SC^1(\mathbb{R}^n)$ is a Banach space with the norm

The following standard notation will be used for Hölder spaces:

● For $0\leq \gamma \leq 1$, Hölder space $C^{0, \gamma}(\mathbb{R}^n)$ is the Banach space of elements $u\in C^0(\mathbb{R}^n)$ with finite norm

where

The space $C^{0, 0}(\mathbb{R}^n)$ is equivalent to $C^{0}(\mathbb{R}^n)$.

● For $k\in \mathbb{N}$, $0\leq \gamma \leq 1$, Hölder space $C^{k, \gamma}(\mathbb{R}^n)$ is a subspace of $C^k(\mathbb{R}^n)$ with finite norm

Throughout the paper we use standard notations for $L_p(Q), 1\leq p \leq \infty$ spaces; the following standard notations are used for Sobolev spaces ^[2]:

● For $k\in \mathbb{N}$, $1\leq p \leq \infty$, Sobolev space $W_p^k(\mathbb{R}^n)$ is the Banach space of measurable functions on $\mathbb{R}^n$ with finite norm

● For ${\bf s} = (s_1, ..., s_n) \in \mathbb{Z}_+^n, 1\leq p \leq \infty$, anisotropic Sobolev space $W_p^{\bf s}(\mathbb{R}^n)$ is the Banach space of measurable functions on $\mathbb{R}^n$ with finite norm

Note that the size of the vector $\bf s$ coincides with the dimension of the space. In particular, for $1\leq k\leq n$, and fixed $j\in \{1, ..., k\}$, we consider Sobolev spaces $W_p^{{\bf s}}(\mathbb{R}^k)$ of the weakly $x_j$-differentiable functions on $\mathbb{R}^k$, where ${\bf s} = (s_i)_{i = 1}^k \in \mathbb{Z}_+^k$ and $s_i = \delta_{ij}$ is a Kronecker symbol.

● For ${\bf k} = (k_1, ..., k_n)\in \mathbb{Z}_+^n, 1\leq p \leq \infty$, Sobolev space $S_p^{{\bf k}}(\mathbb{R}^n)$ with dominating mixed derivatives is a Banach space of measurable functions on $\mathbb{R}^n$ with the finite norm

If $k_1 = \cdots = k_n = k\in\mathbb{N}$, we shall write $S_p^{k}(\mathbb{R}^n) = S_p^{{\bf k}}(\mathbb{R}^n)$.

● Let $Q\subset \mathbb{R}^n$ be a bounded domain. For ${\bf k} = (k_1, ..., k_n)\in \mathbb{Z}_+^n, 1\leq p \leq \infty$, Sobolev space $S_p^{{\bf k}}(Q)$ with dominating mixed derivatives is defined as

and with

where the infimum is taken over all $g\in S_p^{\bf k}(\mathbb{R}^n)$ such that its restriction $g|_Q$ to $Q$ coincides with $f$ in the space of distributions ${\cal D}'(Q)$. If $k_1 = \cdots = k_n = k\in\mathbb{N}$, we shall write $S_p^{k}(Q) = S_p^{{\bf k}}(Q)$.

3.   Main results

3.1.   Generalized Newton-Leibniz formula

Let $x, x'\in \mathbb{R}^n$ with $x_i < x'_i, i = \overline{1, n}$ are fixed and $P$ be $n$-rectangle

with vertex $x$ (or $x'$) called a bottom (or top) corner of $P$. For any subset $\{i_1, ..., i_k\}\subset \{1, ..., n\}, k = \overline{1, n}$, let

be a $k$-rectangle with bottom corner $x$. Note that $P_{i_1\dots i_k}$ is invariant with respect to permutation of multi-index $i_1\cdots i_k$, and it coincides with $P$ if $k = n$.

The following is the generalization of the celebrated Newton-Leibniz formula:

Theorem 3.1. Any function $u\in SC^1(\mathbb{R}^n)$ satisfies the following generalized Newton-Leibniz formula:

If $n = 1$, (3.2) coincides with the Newton-Leibniz formula (1.2). Note that for $\forall k$ there are ${n}\choose {k}$ integrals in (3.2) along all $k$-rectangles $P_{i_1\dots i_k}$ with bottom corner $x$. Therefore, altogether there are

integrals in (3.2) along all sub-rectangles of $P$ with bottom corner at $x$.

Having generalized the Newton-Leibniz formula in the class of smooth functions, we can now formulate the major problem of classical analysis generated by the Newton-Leibniz formula:

Problem 3.1. What is the largest class of Lebesgue integrable and weakly differentiable functions in $\mathbb{R}^n$ which preserve generalized Newton-Leibniz formula?

In Theorem 3.2 we prove that the formula (3.2) remains valid in spaces $S_p^1(\mathbb{R}^n), 1\leq p \leq \infty$, where the right-hand side is understood as trace integrals of respective weak derivatives. Counterexamples constructed in Section 3.3 support the claim that in terms of weak differentiability requirements $S_p^1(\mathbb{R}^n)$ is the largest class of Lebesgue's integrable and weakly differentiable functions in $\mathbb{R}^n$ which preserve generalized Newton-Leibniz formula. The formula (3.2) is a key to prove the Hölder continuity of elements of $S_p^1(\mathbb{R}^n)$.

Remark 3.1. Some variant of the formula (3.2) is proved in ^[10] in the class of tensor product space, i.e., linear cover of the space of separable (or product form) functions equipped with special norm consisting of some algebraic combination of one-dimensional $W^{1, p}$ norms selected in a way to guarantee the $L^p$-boundedness of mixed derivatives of product functions.

3.2.   Embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces

Theorem 3.2. The following bounded embedding holds

The equivalency class of every element of $S_p^1(\mathbb{R}^n)$ possesses a representative in $C^{0, 1-\frac{1}{p}}(\mathbb{R}^n)$, which satisfies the generalized Newton-Leibniz formula (3.2), where $P\subset \mathbb{R}^n$ is an $n$-rectangle with bottom and top corner at $x$ and $x'$ respectively. In particular, $\forall k = 1, ..., n-1$ and $1\leq i_1 < \cdots < i_k\leq n$

in the sense of traces.

Corollary 3.1. For $k\in \mathbb{N}$ the following bounded embedding holds

The following sharp embedding result holds for the anisotropic Sobolev spaces with dominating mixed smoothness:

Corollary 3.2. Let ${\bf k} = (k_1, ..., k_n)\in \mathbb{N}^n, 1\leq p \leq \infty$, and $u\in S_p^{{\bf k}}(\mathbb{R}^n)$. Then $\forall m = 1, ..., n$ and $\forall 1\leq i_1 < i_2 < \cdots < i_m\leq n$

Corollary 3.3. Let $Q\subset \mathbb{R}^n$ be a bounded domain. For $k\in \mathbb{N}$ the following bounded and compact embeddings hold

3.3.   Counterexamples

The goal of this section is to provide counterexamples to support the claim that $S_p^1(\mathbb{R}^n)$ is a subspace of $W_p^1(\mathbb{R}^n), 1\leq p \leq n$ with minimal increase of the weak differentiability requirements in order that every equivalency class has an element which is

● Hölder continuous;

● satisfy generalized Newton-Leibniz formula (3.2);

● satisfy the trace regularity (3.4);

Example 3.1. Consider a function

where $B(0, 1)\subset \mathbb{R}^n$ is a unit ball with center $0$. It is discontinuous at $0$. Direct calculation demonstrates that for arbitrary $k\in\{1, ..., n-1\}$ we have

for all $1\leq i_1\leq \cdots \leq i_k\leq n$. However, we have

Since for all $k\in\{1, ..., n-1\}$ all $k$-th order weak derivatives are in $L_1(B(0, 1))$, by using standard extension theorem ^[2] function $u$ can be extended to $\mathbb{R}^n$ by possessing the regularity

for all $1\leq i_1\leq \cdots \leq i_k\leq n$. From (3.11) it follows that the $n$-th order mixed derivative of the extended function is not integrable, i.e.,

It can also be verified that if $P$ is an $n$-rectangle with bottom corner at $0$, then $\forall k = 1, ..., n-1$ and $1\leq i_1 < \cdots < i_k\leq n$, we have

Clearly, Newton-Leibniz formula (3.2) is not satisfied at the origin. Hence, the extended function presents the desired counterexample when the $n$-th order mixed derivative is removed from the definition of the space $S_p^1(\mathbb{R}^n)$.

Example 3.2. Consider a function

It is discontinuous at $0$. Along the each hyperplane $\{x_k = 0\}$ it is zero, but for $\forall C > 0$

Direct calculation demonstrates that for arbitrary $k\in\{1, ..., n-1\}$ we have

for all $1\leq i_1\leq \cdots \leq i_k\leq n$. However, we have

Since for all $k\in\{1, ..., n-1\}$ all $k$-th order weak derivatives are in $L_1(B(0, 1))$, by using standard extension theorem ^[2] function $u$ can be extended to $\mathbb{R}^n$ by possessing the regularity

for all $1\leq i_1\leq \cdots \leq i_k\leq n$. From (3.18) it follows that the $n$-th order mixed derivative of the extended function is not integrable, i.e.

It can also be verified that if $P$ is an $n$-rectangle with bottom corner at $0$, then $\forall k = 1, ..., n-1$ and $1\leq i_1 < \cdots < i_k\leq n$, we have

In particular, this implies that the Newton-Leibniz formula is satisfied in any $k$-rectangle $P_{i_1\dots i_k}$ with $k\leq n-1$ if we assign $u(0) = 0$. However, Newton-Leibniz formula is not satisfied in an $n$-rectangle $P$ due to the fact that

Example 3.3. Consider a function

with $k\in\{1, ..., n-1\}$. It is discontinuous along all the coordinates axis $x_1, ..., x_k$. For arbitrary $s\in\{1, ..., k\}$ the weak derivatives

do not exist. All the other mixed derivatives which involve differentiation with respect to any other variables $x_l$ with $l\in\{k+1, ..., n\}$ exists and equal to zero. Clearly, the Newton-Leibniz formula is not satisfied in any $n$-rectangle whose interior intersects any of the coordinate axis $x_1, ..., x_k$.

4.   Proof of main results

Proof of Theorem 3.1. Assuming that $u\in SC^1(\mathbb{R}^n)$, we prove (3.2) by induction in terms of the space dimension $n$. If $n = 1$, it coincides with the Newton-Leibniz formula. Assume that (3.2) is true, and demonstrate that it is true if $n$ is replaced with $n+1$. Let $x, x'\in \mathbb{R}^{n+1}$ with $x_i < x'_i, i = \overline{1, n+1}$, are fixed. We have

where $\tilde{x} = (x_1, ..., x_n)$. Applying (3.2) to the first term and the Newton-Leibniz formula to the second term in (4.1), we derive

Applying the Newton-Leibniz formula to all but the last integrand, we have

which imply that

where we use the same notation for the $(n+1)$-rectangle $P$, as well as its corresponding sub-rectangles in $\mathbb{R}^{n+1}$. Indeed, divide all $2^{n+1}-1$ sub-rectangles of $P$ with the bottom corner at $x$ into two groups depending on whether or not the edge $p_{n+1}$ joining vertices $x$ and $(\tilde{x}, x'_{n+1})$ is contained in it. The first two terms on the right-hand side of (4.3) consist of all $2^n$ terms of (4.4) with integrals along sub-rectangles containing the edge $p_{n+1}$, and the last term on the right-hand side of (4.3) is identical with the remaining $2^n-1$ integrals in (4.4) along sub-rectangles which do not contain the edge $p_{n+1}$. This completes the proof by induction. □

Proof of Theorem 3.2. First, we prove the Theorem assuming that $1\leq p < \infty$. The proof will be pursued in four steps.

Step 1. Prove that for $u \in S_p^1(\mathbb{R}^n)$, each of the $2^n-1$ integrals on the right-hand side of (3.2) is finite, and in particular, (3.4) is satisfied. Existence of the integral with $k = n$ on the right hand side of (3.2) follows from the definition of $S_p^1(\mathbb{R}^n)$ and Hölder inequality. We prove the existence of the remaining $2^n-2$ trace integrals in (3.2) by mathematical induction and Sobolev trace embedding result. First, we demonstrate that the claim is true if $k = n-1$. Then we show that the claim is true for any $k < n-1$, provided it is true for $k+1$. Indeed, if $k = n-1$, for each of the $n$ integrals we select a unique integer $j$ satisfying

and define a multi-index ${\bf s} = (s_1, ..., s_n) \in \mathbb{Z}_+^n$, where $s_i = \delta_{ij}$ is a Kronecker symbol. We have

Note that $(n-1)$-rectangle $P_{i_1 \cdots i_{n-1}}$ is a boundary of $P$ on the hyperplane $\eta_j = x_j$. Existence of the trace

is a consequence of the Sobolev trace embedding result:

For completeness, we present a proof of (4.8). Consider a function

which satisfy

Assuming that $u\in SC^1(P)$, we have

If $p > 1$, by using Young's inequality and (4.10), from (4.11) it follows

where $C = \max(p-1+|x_j'-x_j|^{-1}; 1)$. If $p = 1$, (4.12) follows directly from (4.10) and (4.11). In general, we can approximate $u\in S_p^1(\mathbb{R}^n)$ with the sequence $u^\epsilon = u*\phi^\epsilon \in C^\infty_{loc}(\mathbb{R}^n)$, where $\phi^\epsilon$ is a standard rescaled mollifier, and derive (4.12) for $u^\epsilon$. Since $u^\epsilon$ converges to $u$ in the norm given on the right-hand side of (4.12), it is so in the norm of the left-hand side as well, and passing to the limit as $\epsilon \to 0$, (4.12), (4.8) and (4.7) follow. Hence, $n$ relations of (3.4) with $k = n-1$ are established. Next, we prove that the claim is true for $k$ if it is so for $k+1$. For any of the ${{n}\choose {k}}$ integrals in (3.2) along the $k$-dimensional prism $P_{i_1 \cdots i_k}$ we select any integer $j$ satisfying (4.5), and define a multiindex ${\bf s} = (s_1, ..., s_{k+1}) \in \mathbb{Z}_+^{k+1}$, where $s_i = \delta_{ij}$ is a Kronecker symbol. Noting that $P_{i_1 \cdots i_{k+1}}$ is invariant with respect to permutations of the multi-index $i_1 \cdots i_{k+1}$, and due to the induction assumption we have

$k$-rectangle $P_{i_1 \cdots i_{k}}$ is a boundary of $(k+1)$-rectangle $P_{i_1 \cdots i_{k} j}$ on the hyperplane $x_j = const$. Sobolev trace embedding result implies:

The proof of (4.14) is identical to the proof of (4.8). Hence, (3.4) is proved for all $k$-dimensional integrals.

Step 2. In this step we prove that any $u\in S_p^1(\mathbb{R}^n)\cap SC^1_{loc}(\mathbb{R}^n), p > 1$ satisfies the estimate

for all $x, x'\in\mathbb{R}^n$. Similarly, any $u\in S_1^1(\mathbb{R}^n)\cap SC^1_{loc}(\mathbb{R}^n)$ satisfy the estimate

for all $x, x'\in\mathbb{R}^n$. Note that the estimate (4.16) is a formal limit of the estimate (4.15) as $p\to 1$.

To prove (4.15) (or (4.16)) without loss of generality we can assume that $x_i < x'_i, i = \overline{1, n}$. Indeed, if $x_i\neq x'_i, i = \overline{1, n}$, then we can transform the space via finitely many translations

and note that the space $S_p^1(\mathbb{R}^n)$ is invariant under this transformation. Then we can apply (4.15) (or (4.16)) to the $\epsilon$-mollification of the transformed function $\tilde{u}(\tilde{x}) = u(\tilde{x})$, and passing to limit as $\epsilon \to 0$ deduce (4.15) (or (4.16)) for $\tilde{u}$. Applying inverse transformation of (4.17) implies (4.15) (or (4.16)) for $u$. If, on the other side $x_i = x'_i$ for some $i$, we can replace $x'_i$ with $x'_i+\delta$, prove (4.15) (or (4.16)) and pass to limit as $\delta\to 0$.

The proof of (4.15) and (4.16) under the assumption that $x_i < x'_i, i = \overline{1, n}$ is based on the generalized Newton-Leibniz formula (3.2). The following is the proof of the estimate (4.15). Let $P$ be a $n$-rectangle (3.1), and

By using Hölder inequality the integral on the right-hand side of (3.2) with $k = n$ is estimated as follows

where $|P|$ denotes volume of the $n$-rectangle $P$. For $k = 1, ..., n-1$, estimation of any of the $k$-dimensional integrals on the right-hand side of (3.2) will be pursued in $n-k$ steps. Consider typical $k$-dimensional integral in (3.2) along the $k$-rectangle $P_{i_1 \cdots i_{k}}$. The idea is based on successive application of the trace embedding result (4.14) $n-k$ times. First, we select any integer $j$ from (4.5) and assign it to the multi-index component $i_{k+1}$. By using Hölder inequality we have

Consider a function

which satisfy

We have

By using Young's inequality and (4.21), from (4.22) it follows

From (4.19), (4.23) it follows that

where $P^1_{i_1 \cdots i_k} = P_{i_1 \cdots i_k}\times (x_{i_{k+1}}, x_{i_{k+1}}'+1)$ is a $(k+1)$-rectangle. This completes one out of $n-k$ steps for the estimation of the $k$-dimensional integral in (3.2) along the $k$-rectangle $P_{i_1 \cdots i_{k}}$. In the next step, we select any integer $j$ from (4.5) with $k$ replaced with $k+1$ and assign it to multi-index component $i_{k+2}$. Then for each of the $k+1$-dimensional integrals on the right-hand side of (4.24) we derive the estimation similar to (4.23), where $P_{i_1 \cdots i_k}$ is replaced with $(k+1)$-rectangle $P^1_{i_1 \cdots i_k}$, and integration interval $(x_{i_{k+1}}, x_{i_{k+1}}'+1)$ is replaced accordingly with $(x_{i_{k+2}}, x_{i_{k+2}}'+1)$. Application of these estimations to the right-hand side of (4.24) would complete the second out of $n-k$ steps. By repeating the procedure after $m = 1, ..., n-k$ steps we derive the following estimate:

where

be a $(k+m)$-rectangle. Let us prove the estimation (4.25) by induction. If $m = 1$, the estimation (4.25) coincides with (4.24). Prove that (4.25) is true for $m+1$ if it is so for any $m < n-k$. Each of the $k+m$-dimensional integrals on the right-hand side of (4.25) satisfy the following estimate

Using (4.26), we have

Since

from (4.27) it follows

which completes the proof of (4.25) by mathematical induction. By choosing $m = n-k$ in (4.25), we derive an upper bound of the right-hand side by replacing the integration domain with $\mathbb{R}^n$:

Note that the estimation (4.29) holds for k = n as well in view of (4.18). By using (4.29) from the generalized Newton-Leibniz formula (3.2) it follows the estimate

which proves the desired estimate (4.15). The proof of the estimate (4.16) is almost identical to the proof of (4.15).

Step 3. In this step we prove

● the uniform $C^{0, 1-\frac{1}{p}}(\mathbb{R}^n)$-estimate for any $u\in S_p^1(\mathbb{R}^n)\cap SC^1_{loc}(\mathbb{R}^n), 1 < p < \infty$;

● the uniform $C^{0}(\mathbb{R}^n)$-estimate for any $u\in S_1^1(\mathbb{R}^n)\cap SC^1_{loc}(\mathbb{R}^n)$.

Assume $p > 1$ and fix $x, x'\in \mathbb{R}^n$ such that $|x-x'|\leq 1$. From (4.15) it follows that

Hence, we have

Now fix $x\in\mathbb{R}^n$. By using (4.32) and Hölder inequality we deduce

where $\Gamma_n$ is a volume of the unit ball in $\mathbb{R}^n$. Hence, we have

From (4.34) it follows that

From (4.32) and (4.35) we deduce the following Hölder seminorm estimate for $u$:

Finally, (4.34), (4.36) imply the following Hölder norm estimate for $u\in S_p^1(\mathbb{R}^n)\cap SC^1_{loc}(\mathbb{R}^n), 1 < p < \infty$:

If $p = 1$ from the estimate (4.16) with the similar argument as in (4.33) we derive the following $C^{0}(\mathbb{R}^n)$-estimate for any $u\in S_1^1(\mathbb{R}^n)\cap SC^1_{loc}(\mathbb{R}^n)$:

Step 4. We complete the proof of the embedding (3.3) by using estimates (4.37), (4.38) and smooth approximation of elements of $S_p^1(\mathbb{R}^n)$. Given $u\in S_p^1(\mathbb{R}^n), 1\leq p < \infty$, we select a sequence $v_m\in C_0^\infty(\mathbb{R}^n)$ such that

For example, the sequence $v_m$ can be given explicitly as in ^[11] (Lemma 23):

where $u^{\frac{1}{m}} = u*\phi^{\frac{1}{m}} \in C^\infty_{loc}(\mathbb{R}^n)\cap S_p^1(\mathbb{R}^n)$ is the $\frac{1}{m}$-mollification of $u$, $\phi^{\frac{1}{m}}$ is a standard rescaled mollifier, $\eta \in C_0^\infty(\mathbb{R}^n)$ be a compactly supported function which equals 1 near the origin. If $p > 1$, then by applying the estimate (4.37) to $v_m$, we have

Equivalently, we have

for all $m, l\geq 1$, whence there exists a function $u_*\in C^{0, 1-\frac{1}{p}}(\mathbb{R}^n)$ such that

From (4.39) it follows that $u_* = u$, a.e. on $\mathbb{R}^n$, so that $u_*$ is in the equivalency class of $u$. Passing to limit as $m\to \infty$, from (4.40) it also follows that

which proves the bounded embedding (3.3). Step 1 of the proof implies that the traces of $u_*$ satisfy (3.4), and each of them is an $L_p$-limit of the corresponding sequence of traces of $v_m$. Therefore, writing (3.2) for $v_m$, and passing to limit as $m\to\infty$, it follows that $u_*$ satisfies the generalized Newton-Leibniz formula (3.2).

Proof in the case $p = 1$ is identical by using an estimate (4.38). This completes the proof of the theorem in the case $1\leq p < \infty$.

Assume that $p = \infty$. In this case, the embedding (3.3) is not new, and it is contained in the well-known fact that ^[2]

Since $S_\infty^1(\mathbb{R}^n)$ is a subspace of $W_\infty^1(\mathbb{R}^n)$, its elements are bounded and Lipschitz continuous functions and the embedding (3.3) holds. The assertion that $u$ satisfies (3.2) follows from the proof given for the case $p < \infty$. It only remains to show that (3.4) holds with $p = \infty$. Note that from the given proof it follows that (3.4) holds for any $p < \infty$. In particular for the smoothing sequence $u^\epsilon = u*\phi^\epsilon \in S_\infty^1(\mathbb{R}^n)\cap C_{loc}^\infty(\mathbb{R}^n)$ all the traces indicated on the left hand side of (3.4) are uniformly bounded in $L_\infty(P_{i_1\cdots i_k})$, and converge to corresponding traces of $u$ in $L_p(P_{i_1\cdots i_k})$ with any $1 < p < \infty$. Such limits are also limits in the sense of distributions. Since $L_\infty(P_{i_1\cdots i_k})$ is a dual space of $L_1(P_{i_1\cdots i_k})$, distributional limit of the sequence bonded in $L_\infty(P_{i_1\cdots i_k})$ remains in $L_\infty(P_{i_1\cdots i_k})$. Therefore, (3.4) holds with $p = \infty$. Theorem is proved. □

Corollaries 3.1, 3.2 and 3.3 are the direct consequence of the Theorem 3.2 due to the fact that if $u\in S_p^k(\mathbb{R}^n)$, then all the weak partial derivatives of order $k-1$ are elements of $S_p^1(\mathbb{R}^n)$, and if $u\in S_p^{{\bf k}}(\mathbb{R}^n)$ the indicated partial derivative on the left hand side of (3.6) is an element of $S_p^1(\mathbb{R}^n)$. The bounded embedding (3.7) is a direct consequence of (3.5) and the definition of the space $S_p^1(Q)$. The compact embedding (3.8) follows from (3.7) and Arzela-Ascoli's theorem.

The following remarks are added following on the recommendation of the reviewer:

Remark 4.1. Reviewer of the paper writes: The techniques used to prove the Theorem 3.2 are nice alternative to Fourier-based methods. The results of Theorem 3.2 and Corollary 3.1 are new if $p = 1$. If $1 < p < \infty$, the results can be derived from the known embedding results of Besov and Triebel-Lizorkin spaces as follows: The theorem on page 104 in ^[7] tells us that $S_p^mW(\mathbb{R}^n) = S^m_{p, 2}F(\mathbb{R}^n)$ if $m\in \mathbb{N}$ and $1 < p < \infty$. By Proposition 1.15 in ^[9] we have $S^m_{p, 2}(\mathbb{R}^n)\hookrightarrow S^m_{p, \max{(p, 2)}}B(\mathbb{R}^n) \hookrightarrow S^m_{p, \infty}B(\mathbb{R}^n)$. Next, by the theorem on p.131 in ^[7] we have $S^m_{p, \infty}B(\mathbb{R}^n)\hookrightarrow S^{m-1/p}_{\infty, \infty}B(\mathbb{R}^n)$. The latter space $S^{m-1/p}_{\infty, \infty}B(\mathbb{R}^n)$ is a dominating mixed smoothness type Hölder-Zygmund space and sometimes denoted by $S^{m-1, 1-1/p}\mathcal{C}(\mathbb{R}^n)$. Either way, relying on Theorem 3.1 from ^[12] we get the embedding $S^{m-1, 1-1/p}\mathcal{C}(\mathbb{R}^n)\hookrightarrow B_{\infty, \infty}^{m-1/p}(\mathbb{R}^n) = \mathcal{C}^{m-1, 1-1/p}(\mathbb{R}^n)$, where the last equality is a classical result (see e.g., ^[13]). In summary, we have the embedding $S^m_pW(\mathbb{R}^n)\hookrightarrow \mathcal{C}^{m-1, 1-1/p}(\mathbb{R}^n)$. Regarding the assertion on traces, the cases with $p = 1, \infty$ are novel in the literature. But the cases $1 < p < \infty$ are contained in Lemma 4.1 of ^[14].

Remark 4.2. The task of embedding large subspaces of the Sobolev spaces $W_p^m$ into Lebesgue spaces $L_q$ is touched on e.g., in ^[15] or ^[6].

5.   Conclusions

The concept of Sobolev spaces became a trailblazing idea in many fields of mathematics. The goal of this paper is to gain insight into the embedding of the Sobolev spaces into Hölder spaces-a very powerful concept that reveals the connection between weak differentiability and integrability (or weak regularity) of the function with its pointwise regularity. It is well-known that the embedding of the Sobolev space of weakly differentiable functions into Hölder spaces holds if the integrability exponent is higher than the space dimension. Otherwise speaking, one can trade one degree of weak regularity with an integrability exponent higher than the space dimension to upgrade the pointwise regularity to Hölder continuity. In this paper, the embedding of the Sobolev functions into the Hölder spaces is expressed in terms of the minimal weak differentiability requirement independent of the integrability exponent. Precisely, the question asked is what is the minimal weak regularity degree of Sobolev functions which upgrades the pointwise regularity to Hölder continuity independent of the integrability exponent. The paper presents proof of the embedding of the Sobolev spaces with dominating mixed smoothness into Hölder spaces. The new method of proof is based on the generalization of the Newton-Leibniz formula to the $n$-dimensional rectangle and inductive application of the Sobolev trace embedding results. Counterexamples demonstrate that in terms of minimal weak regularity degree the Sobolev spaces with dominating mixed smoothness present the largest class of weakly differentiable functions which preserve generalized Newton-Leibniz formula, and upgrades the pointwise regularity to Hölder continuity.

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