Fractional Landau Inequalities with Mixed Sobolev Norms and Applications to Multiscale Analysis

1. Introduction

The classical Landau inequality $\parallel f^{\prime }{\parallel }_{\infty }\le 2\sqrt{{\parallel f\parallel }_{\infty }{\parallel f^{\prime \prime }\parallel }_{\infty }}$, first established by Landau in 1925 [4], represents a fundamental trade-off between function magnitude and oscillation that has profoundly influenced mathematical analysis for over a century. This elegant bound captures the intrinsic balance between a function’s size and its variation, laying the groundwork for Sobolev embeddings and regularity theory in partial differential equations.

The multivariate extension of Landau’s inequality was pioneered by Ditzian [2], who successfully generalized the classical result to multidimensional settings through innovative use of mixed derivatives and tensor norms. This work was further refined by Kounchev [3], who investigated extremizers and optimal constants for multivariate Landau-Kolmogorov inequalities, establishing deeper connections with polynomial approximation and interpolation theory.

Recent years have witnessed exciting developments in fractional calculus, with Anastassiou’s groundbreaking 2025 work [1] synthesizing directional fractional derivatives with sharp gradient bounds in ${\mathbb{R}}^k$. This represents a paradigm shift from classical to fractional Landau inequalities, extending the theory to non-local operators that model phenomena with memory effects and anomalous diffusion. However, existing fractional Landau inequalities operate primarily within isotropic function spaces, overlooking the rich multiscale structure present in many modern applications ranging from high-dimensional data analysis to physical systems with directional preferences.

This paper addresses this fundamental limitation by developing a comprehensive theory of fractional Landau inequalities with mixed norms. Our approach captures the anisotropic regularity patterns that arise naturally in deep neural networks with hierarchical representations, physical systems with directional heterogeneity, and high-dimensional datasets with varying scaling behaviors across coordinates. By introducing directional scaling parameters and mixed Sobolev norms, we obtain gradient bounds that adapt to the intrinsic geometry of the function space, providing a more nuanced understanding of regularity in complex systems.

Our work makes three principal contributions that bridge classical analysis with modern applications:

First, we introduce the mixed fractional Sobolev spaces $W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$, where $\alpha =({\alpha }_1,\dots ,{\alpha }_k)$ encodes directional scaling behavior. These spaces interpolate between classical isotropic Sobolev spaces and fully anisotropic Besov spaces, providing a flexible framework for multiscale analysis that generalizes the classical settings considered in [2,3,4].

Second, we establish sharp fractional Landau inequalities in these mixed-norm spaces, with constants that explicitly track dependence on both fractional order and directional scaling parameters. Our proofs combine innovative harmonic analysis techniques with fractional calculus, extending the approaches of [1] through the development of directional Littlewood-Paley theory and anisotropic maximal function estimates.

Third, we demonstrate applications to multiscale neural operators, proving stability bounds and approximation rates for deep networks processing data with heterogeneous regularity. Our results provide mathematical foundations for understanding why certain architectures excel at capturing multiscale features, offering new insights into the design and analysis of modern machine learning systems.

The paper is structured as follows: Section 2 introduces mixed fractional Sobolev spaces and develops their basic properties, establishing the mathematical foundation for our work. Section 3 presents our main inequalities with detailed proofs, including sharp constants and anisotropic dependence. Section 4 applies these results to neural operator theory, demonstrating stability under perturbations and optimal approximation rates. Section 5, results and Section 6, concludes with directions for future research, highlighting connections to geometric deep learning, fractional PDEs, and high-dimensional approximation theory.

2. Mixed Fractional Sobolev Spaces and Anisotropic Analysis

2.1. Anisotropic Scaling and Geometric Foundations

The study of anisotropic function spaces, differential operators, and harmonic analysis relies fundamentally on a geometric framework in which different spatial coordinates scale at different rates. Such a framework naturally emerges in diverse contexts, including kinetic equations, degenerate elliptic operators, multiscale diffusion, and the analysis of neural operators constructed with heterogeneous receptive fields. In all these settings, the underlying geometry is no longer dictated by the classical Euclidean dilation $x\mapsto \lambda x$, but rather by a more general family of dilations that respects the intrinsic anisotropies of the system.

To formalize these ideas, we introduce the anisotropic scaling structure, a parameterized family of dilations that encode how each coordinate direction is stretched under rescaling. This leads to a non-Euclidean notion of distance, a modified dimension, and a compatible notion of homogeneity. Once these quantities are established, they become the foundation for the harmonic analysis developed later in the text, including Littlewood–Paley theory, fractional operators, heat kernels, and Sobolev-type norms in the anisotropic regime.

Definition 1

(Anisotropic Scaling Structure). Let $\alpha =({\alpha }_1,\dots ,{\alpha }_k)\in {(0,\infty )}^k$ be a scaling vector defining the anisotropic geometry. The associated anisotropic dilation group ${\left\{T_{\lambda }^{\alpha }\right\}}_{\lambda >0}$ is defined by:

$$T_{\lambda }^{\alpha }f\left(x\right)=f({\lambda }^{{\alpha }_1}{x}_1,\dots ,{\lambda }^{{\alpha }_k}{x}_k),\lambda >0.$$

(1)

A function f is said to have anisotropic homogeneity δ if

$$f\left(T_{\lambda }^{\alpha }x\right)={\lambda }^{\delta }f\left(x\right),$$

(2)

in which case δ is called its anisotropic degree.

The anisotropic dimension associated with the scaling vector α is

$$d_{\alpha }=\sum _{i=1}^k{\alpha }_i^{-1},$$

which plays the role of an effective dimensional exponent in integration, Fourier analysis, and kernel estimates.

Finally, the anisotropic distance is defined by:

$${\rho }_{\alpha }\left(x\right)={\left(\sum _{i=1}^k{|x_i|}^{2/{\alpha }_i}\right)}^{1/2},$$

(3)

which is homogeneous with respect to the dilations $T_{\lambda }^{\alpha }$ in the sense that ${\rho }_{\alpha }\left(T_{\lambda }^{\alpha }x\right)=\lambda {\rho }_{\alpha }\left(x\right)$.

The objects introduced above behave in a coherent way under anisotropic rescaling, providing a robust algebraic and analytical structure. The next proposition summarizes the most fundamental properties of the dilation group, all of which will be used repeatedly in the analysis of anisotropic kernels, Sobolev norms, and semigroup characterizations.

Proposition 1

(Anisotropic Scaling Properties). The anisotropic dilation group satisfies:

• Group Structure: $T_{\lambda }^{\alpha }{T}_{\mu }^{\alpha }=T_{\lambda \mu }^{\alpha },{\left(T_{\lambda }^{\alpha }\right)}^{-1}=T_{1/\lambda }^{\alpha }.$
• Jacobian Determinant: $|det\left(D{T}_{\lambda }^{\alpha }\right)|={\lambda }^{d_{\alpha }}.$
• Scaling of Lebesgue Measure:

  $${\int }_{{\mathbb{R}}^k}f\left(T_{\lambda }^{\alpha }x\right)dx={\lambda }^{-d_{\alpha }}{\int }_{{\mathbb{R}}^k}f\left(x\right)dx.$$
• Fourier Transform Relation:

  $$\mathcal{F}[f\circ T_{\lambda }^{\alpha }]\left(\xi \right)={\lambda }^{-d_{\alpha }}\mathcal{F}\left[f\right]\left(T_{1/\lambda }^{\alpha }\xi \right).$$

Proof. 

The group structure follows immediately from composition of dilations. The Jacobian matrix of $T_{\lambda }^{\alpha }$ is diagonal with entries ${\lambda }^{{\alpha }_i}{\delta }_{ij}$, so the determinant is ${\prod }_{i=1}^k{\lambda }^{{\alpha }_i}={\lambda }^{d_{\alpha }}$. The scaling of Lebesgue measure then follows from the change-of-variables formula. For the Fourier relation, compute directly:

$$\begin{array}{cc}\hfill \mathcal{F}[f\circ T_{\lambda }^{\alpha }]\left(\xi \right)& ={\int }_{{\mathbb{R}}^k}f({\lambda }^{{\alpha }_1}{x}_1,\dots ,{\lambda }^{{\alpha }_k}{x}_k)e^{-2\pi ix·\xi }dx\hfill \\ \hfill & ={\lambda }^{-d_{\alpha }}{\int }_{{\mathbb{R}}^k}f\left(y\right)e^{-2\pi i\left(T_{1/\lambda }^{\alpha }y\right)·\xi }dy\hfill \\ \hfill & ={\lambda }^{-d_{\alpha }}\mathcal{F}\left[f\right]\left(T_{1/\lambda }^{\alpha }\xi \right),\hfill \end{array}$$

which completes the proof. □

3. Mixed Fractional Sobolev Spaces: Rigorous Construction

The purpose of this section is to establish a complete and mathematically rigorous framework for mixed fractional Sobolev spaces of anisotropic type. These spaces, denoted $W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$, are designed to encode directional fractional regularity by assigning distinct smoothness exponents along each coordinate axis. Such an anisotropic perspective naturally emerges in degenerate elliptic problems, kinetic equations, anisotropic diffusion, and multiscale models governed by heterogeneous geometric scaling.

The central idea is that fractional differentiability should be measured direction by direction, with each directional order determined by the scaling vector $\alpha =({\alpha }_1,\dots ,{\alpha }_k)$. Along the i-th axis one considers a fractional order $\nu /{\alpha }_i$, so that the Sobolev geometry remains consistent with the intrinsic anisotropy. This leads to a family of one-dimensional Gagliardo-type seminorms, each capturing the fractional variation in a single coordinate direction.

We formally construct $W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$ as the completion of $C_c^{\infty }\left({\mathbb{R}}^k\right)$ under a mixed norm combining: (i) the $L^p$-norm; and (ii) a set of directional fractional seminorms. This produces a robust function space tailored to anisotropic partial differential equations, harmonic analysis, and the spectral theory of anisotropic operators.

A fundamental analytic feature is that this mixed norm admits several equivalent formulations. Depending on the application, one may work with spatial increments, frequency localization, Bessel potential techniques, or heat semigroup estimates. The four equivalent characterizations are presented below and proven in Theorem 1. Each provides a distinct analytical viewpoint, yet all describe the same underlying function space.

We now introduce the space through directional Gagliardo seminorms, followed by the main equivalence theorem.

Definition 2

(Mixed Fractional Sobolev Space). Let $\nu >0$, $1\le p<\infty$, and $\alpha =({\alpha }_1,\dots ,{\alpha }_k)$ with ${\alpha }_i>0$. The mixed fractional Sobolev space $W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$ is the completion of $C_c^{\infty }\left({\mathbb{R}}^k\right)$ under the norm

$${\parallel f\parallel }_{W_{\alpha }^{\nu ,p}}={\parallel f\parallel }_{L^p\left({\mathbb{R}}^k\right)}+\sum _{i=1}^k{\left[f\right]}_{W_i^{\nu /{\alpha }_i,p}},$$

(4)

where the directional fractional seminorm along axis $e_i$ is given by

$${\left[f\right]}_{W_i^{\nu /{\alpha }_i,p}}={\left({\int }_{{\mathbb{R}}^k}{\int }_{\mathbb{R}}{\displaystyle \frac{|f(x+h{e}_i)-f\left(x\right){|}^p}{{|h|}^{1+p\nu /{\alpha }_i}}}dhdx\right)}^{1/p}.$$

(5)

The anisotropic geometry can also be expressed through the quasi-distance

$${\rho }_{\alpha }\left(x\right)=\sum _{i=1}^k{|x_i|}^{{\alpha }_i}.$$

(6)

This distance induces an anisotropic Gagliardo integral:

$${\mathcal{G}}_{\nu ,p}\left(f\right)={\left({\int }_{{\mathbb{R}}^k}{\int }_{{\mathbb{R}}^k}{\displaystyle \frac{{|f\left(x\right)-f\left(y\right)|}^p}{{\rho }_{\alpha }{(x-y)}^{k+p\nu }}}dydx\right)}^{1/p}.$$

(7)

The frequency-based (Littlewood–Paley) characterization uses directional dyadic projections $P_j^{\left(i\right)}$, leading to

$${∥2^{j\nu }\sum _{i=1}^k{P}_j^{\left(i\right)}f∥}_{{\ell }_j^p{L}_x^p}.$$

(8)

The potential-theoretic characterization relies on the anisotropic operator

$${\Delta }_{\alpha }=\sum _{i=1}^k{(-{\partial }_i^2)}^{1/{\alpha }_i}.$$

(9)

This leads to a Bessel-potential-type norm

$$\parallel {(I-{\Delta }_{\alpha })}^{\nu /2}{f\parallel }_{L^p}.$$

(10)

Finally, the semigroup formulation involves the anisotropic heat flow $e^{t{\Delta }_{\alpha }}$:

$${\left({\int }_0^{\infty }{t}^{-\nu /2}{\parallel f-e^{t{\Delta }_{\alpha }}f\parallel }_{L^p}^p{\displaystyle \frac{dt}{t}}\right)}^{1/p}.$$

(11)

Theorem 1

(Equivalent Characterizations). Let $\nu >0$, $1\le p<\infty$, and ${\alpha }_i>0$. For every $f\in L^p\left({\mathbb{R}}^k\right)$, the following quantities are finite and mutually equivalent:

$$\begin{array}{cc}\hfill {\parallel f\parallel }_{W_{\alpha }^{\nu ,p}}& \leftrightarrow {\parallel f\parallel }_{L^p}+{\mathcal{G}}_{\nu ,p}\left(f\right),\hfill \end{array}$$

(12)

$$\begin{array}{cc}\hfill & \leftrightarrow {\parallel f\parallel }_{L^p}+{∥2^{j\nu }\sum _{i=1}^k{P}_j^{\left(i\right)}f∥}_{{\ell }_j^p{L}_x^p},\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & \leftrightarrow \parallel {(I-{\Delta }_{\alpha })}^{\nu /2}{f\parallel }_{L^p},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \leftrightarrow {\parallel f\parallel }_{L^p}+{\left({\int }_0^{\infty }{t}^{-\nu /2}{\parallel f-e^{t{\Delta }_{\alpha }}f\parallel }_{L^p}^p{\displaystyle \frac{dt}{t}}\right)}^{1/p}.\hfill \end{array}$$

(15)

Thus, each of the formulations (4)–(11) defines the same function space $W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$.

Theorem 2

(Anisotropic Embedding Theorems). Let $1\le p<q\le \infty$, $\nu >0$, and let $d_{\alpha }$ denote the anisotropic homogeneous dimension associated with the scaling vector α. Then the following embeddings hold:

• (Anisotropic Sobolev Embedding). If

  $$\nu >d_{\alpha }\left({\displaystyle \frac{1}{p}}-{\displaystyle \frac{1}{q}}\right),$$

  (16)

  then

  $$W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)\hookrightarrow L^q\left({\mathbb{R}}^k\right).$$

  (17)

  Equivalently, $W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$ continuously embeds into $L^q$ whenever the fractional smoothness ν dominates the anisotropic integrability gap.
• (Anisotropic Hölder Embedding). If

  $$\nu >{\displaystyle \frac{d_{\alpha }}{p}},$$

  (18)

  then

  $$W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)\hookrightarrow C_{\alpha }^{0,\gamma }\left({\mathbb{R}}^k\right),\gamma =\nu -{\displaystyle \frac{d_{\alpha }}{p}},$$

  (19)

  where $C_{\alpha }^{0,\gamma }$ denotes the anisotropic Hölder space endowed with the quasi-norm determined by the homogeneous distance ${\rho }_{\alpha }$.
• (Compact Embedding on Bounded Domains). Let $\mathsf{\Omega }\subset {\mathbb{R}}^k$ be bounded with Lipschitz boundary. If $\nu p<d_{\alpha }$, let

  $$p^{*}={\displaystyle \frac{d_{\alpha }p}{d_{\alpha }-\nu p}}$$

  (20)

  be the anisotropic critical Sobolev exponent. Then the embedding

  $$W_{\alpha }^{\nu ,p}\left(\mathsf{\Omega }\right)\hookrightarrow L^q\left(\mathsf{\Omega }\right)$$

  (21)

  is compact for every $1\le q<p^{*}$. In the limiting case $\nu p=d_{\alpha }$, compactness holds for all $1\le q<\infty$.

Proof. 

We sketch the argument for each embedding, emphasizing the role of anisotropic geometry.

• (1) Sobolev Embedding. The proof relies on the anisotropic Gagliardo–Nirenberg–Sobolev inequality adapted to the homogeneous structure induced by $\alpha$. For $f\in W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$ one has

  $${\parallel f\parallel }_{L^q\left({\mathbb{R}}^k\right)}\le C\parallel {\nabla }_{\alpha }^{\nu }{f\parallel }_{L^p\left({\mathbb{R}}^k\right)}^{\theta }{\parallel f\parallel }_{L^p\left({\mathbb{R}}^k\right)}^{1-\theta },\theta ={\displaystyle \frac{d_{\alpha }(p^{-1}-q^{-1})}{\nu }},$$

  (22)

  which is valid precisely under the condition $\theta \in (0,1)$, equivalent to (16). The inequality follows from anisotropic scaling arguments and interpolation adapted to the homogeneous quasi-distance.
• (2) Hölder Embedding. Assuming $\nu >d_{\alpha }/p$, Morrey’s anisotropic inequality yields, for all $x,y\in {\mathbb{R}}^k$,

  $$|f\left(x\right)-f\left(y\right)|\le C{\rho }_{\alpha }{(x-y)}^{\nu -d_{\alpha }/p}{\parallel {\nabla }_{\alpha }^{\nu }f\parallel }_{L^p\left({\mathbb{R}}^k\right)}.$$

  (23)

  Since ${\rho }_{\alpha }$ is the homogeneous distance compatible with the anisotropic dilation group, this establishes Hölder continuity with exponent $\gamma =\nu -d_{\alpha }/p$, proving (19).
• (3) Compact Embedding. Let $\mathsf{\Omega }$ be bounded. The anisotropic Fréchet–Kolmogorov compactness theorem asserts that precompactness in $L^q\left(\mathsf{\Omega }\right)$ follows if one shows:

  (i)

  uniform $L^q$-boundedness,

  (ii)

  equicontinuity under anisotropic translations:

  $${\parallel f(·+h)-f(·)\parallel }_{L^q\left(\mathsf{\Omega }\right)}\to 0\mathrm{as}{\rho }_{\alpha }\left(h\right)\to 0,$$

  (iii)

  vanishing of the mass near the boundary, which holds since $\mathsf{\Omega }$ is bounded and Lipschitz.

Conditions (i) and (ii) follow from the same inequalities used in (1) and (2), combined with the compactness of the embedding $W_{\alpha }^{\nu ,p}\left(\mathsf{\Omega }\right)\subset L^q\left(\mathsf{\Omega }\right)$ for subcritical $q<p^{*}$, where $p^{*}$ is given by (20). Thus the embedding (21) is compact.

□

3.1. Algebraic and Interpolation Properties

Theorem 3

(Anisotropic Algebra Property). Let $1<p<\infty$ and $\nu >d_{\alpha }/p$, where $d_{\alpha }$ is the anisotropic homogeneous dimension. Then the anisotropic Sobolev space $W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$ is a Banach algebra under pointwise multiplication. In particular, there exists $C=C(\nu ,p,\alpha )>0$ such that

$${\parallel fg\parallel }_{W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)}\le {C\parallel f\parallel }_{W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)}{\parallel g\parallel }_{W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)}.$$

(24)

Moreover, for any smooth bounded nonlinearity $F\in C_b^{\infty }\left(\mathbb{R}\right)$, one has

$${\parallel F\left(f\right)\parallel }_{W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)}\le C_F\left(1+{\parallel f\parallel }_{W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)}\right).$$

(25)

Proof. 

The proof follows by adapting the classical paraproduct calculus to the anisotropic Littlewood–Paley structure induced by the scaling vector $\alpha$.

• 1. Anisotropic paraproduct decomposition. Let ${\left\{{\Delta }_j^{\alpha }\right\}}_{j\in \mathbb{Z}}$ be the anisotropic dyadic blocks associated with the homogeneous dilation group. For $f,g\in W_{\alpha }^{\nu ,p}$, the Bony decomposition reads

  $$fg=\mathsf{\Pi }(f,g)+\mathsf{\Pi }(g,f)+{\mathsf{\Pi }}_0(f,g),$$

  (26)

  where the anisotropic paraproducts are

  $$\mathsf{\Pi }(f,g)=\sum _j{S}_{j-1}^{\alpha }f{\Delta }_j^{\alpha }g,{\mathsf{\Pi }}_0(f,g)=\sum _{|j-j^{\prime }|\le 1}{\Delta }_j^{\alpha }f{\Delta }_{j^{\prime }}^{\alpha }g.$$

  (27)
• 2. Estimates for the paraproducts. Using anisotropic Bernstein inequalities,

  $$\parallel {\Delta }_j^{\alpha }{f\parallel }_{L^{\infty }}\le C{2}^{j{d}_{\alpha }/p}{\parallel {\Delta }_j^{\alpha }f\parallel }_{L^p},$$

  (28)

  and boundedness of the anisotropic Hardy–Littlewood maximal operator, one obtains

  $${\parallel \mathsf{\Pi }(f,g)\parallel }_{W_{\alpha }^{\nu ,p}}\le {C\parallel f\parallel }_{L^{\infty }}{\parallel g\parallel }_{W_{\alpha }^{\nu ,p}},$$

  (29)

  and similarly for $\mathsf{\Pi }(g,f)$.

The condition $\nu >d_{\alpha }/p$ ensures, via the anisotropic Sobolev embedding, that

$$W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)\hookrightarrow L^{\infty }\left({\mathbb{R}}^k\right),$$

(30)

which is critical for the algebra property.

The resonant term ${\mathsf{\Pi }}_0(f,g)$ is controlled by the anisotropic fractional Leibniz rule:

$$\parallel {\mathsf{\Pi }}_0{(f,g)\parallel }_{W_{\alpha }^{\nu ,p}}\le {C\parallel f\parallel }_{W_{\alpha }^{\nu ,p}}{\parallel g\parallel }_{W_{\alpha }^{\nu ,p}}.$$

(31)

Combining (29), (30), and (31) in the decomposition (26) yields the algebra estimate (24).

• 3. Composition estimates. For $F\in C_b^{\infty }$, Taylor expansion and the algebra property yield

  $$F\left(f\right)=F\left(0\right)+F^{\prime }\left(0\right)f+R\left(f\right),$$

  (32)

  where the remainder satisfies

  $${\parallel R\left(f\right)\parallel }_{W_{\alpha }^{\nu ,p}}\le C_F{\parallel f\parallel }_{W_{\alpha }^{\nu ,p}}^2.$$

  (33)

Applying (24) and grouping the terms gives the desired bound (25).

□

Theorem 4

(Anisotropic Interpolation). Let $0<\theta <1$, $1\le p_0,p_1<\infty$, and ${\nu }_0,{\nu }_1>0$. Then the real interpolation space between anisotropic Sobolev spaces satisfies

$${[W_{\alpha }^{{\nu }_0,p_0}\left({\mathbb{R}}^k\right),W_{\alpha }^{{\nu }_1,p_1}\left({\mathbb{R}}^k\right)]}_{\theta }=W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right),$$

(34)

where

$$\nu =(1-\theta ){\nu }_0+\theta {\nu }_1,{\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\theta }{p_0}}+{\displaystyle \frac{\theta }{p_1}}.$$

The norms on both sides are equivalent, with constants depending only on $({\nu }_0,{\nu }_1,p_0,p_1,\theta ,\alpha )$.

Proof. 

The proof proceeds through the anisotropic Littlewood–Paley decomposition and the structural equivalence

$$W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)\equiv B_{p,2;\alpha }^{\nu }\left({\mathbb{R}}^k\right),$$

(35)

where $B_{p,2;\alpha }^{\nu }$ denotes the anisotropic Besov space defined by

$${\parallel f\parallel }_{B_{p,2;\alpha }^{\nu }}\sim {\left(\sum _{j=0}^{\infty }{2}^{2\nu j}{\parallel {\Delta }_j^{\alpha }f\parallel }_{L^p}^2\right)}^{1/2}.$$

1. Interpolation of Besov scales. For anisotropic Besov spaces defined via the scaling matrix $A_{\alpha }=\mathrm{diag}(2^{1/{\alpha }_1},\dots ,2^{1/{\alpha }_k}),$ the classical real-interpolation identity holds:

$${[B_{p_0,2;\alpha }^{{\nu }_0},B_{p_1,2;\alpha }^{{\nu }_1}]}_{\theta }=B_{p,2;\alpha }^{\nu },$$

(36)

where $\nu$ and p satisfy the affine relations above. This follows from the reiteration theorems for the K-functional and the quasi-homogeneous nature of the anisotropic dyadic decomposition, which preserves interpolation structure.

2. Transfer to Sobolev spaces. Using the characterization (35), we obtain

$${[W_{\alpha }^{{\nu }_0,p_0},W_{\alpha }^{{\nu }_1,p_1}]}_{\theta }={[B_{p_0,2;\alpha }^{{\nu }_0},B_{p_1,2;\alpha }^{{\nu }_1}]}_{\theta }.$$

Applying the Besov interpolation identity (61) yields

$${[B_{p_0,2;\alpha }^{{\nu }_0},B_{p_1,2;\alpha }^{{\nu }_1}]}_{\theta }=B_{p,2;\alpha }^{\nu }.$$

Finally, using again (35) we conclude

$$B_{p,2;\alpha }^{\nu }=W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right),$$

establishing (34). □

This enhanced mathematical framework provides a rigorous foundation for the analysis of functions with anisotropic regularity, establishing the fundamental properties of mixed fractional Sobolev spaces and their various characterizations.

3.2. Mixed Fractional Derivatives and Their Analysis

3.2.1. Rigorous Foundations of Directional Fractional Calculus

Definition 3

(Anisotropic Riemann-Liouville Fractional Derivative). For $\nu >0$, $n=\lfloor \nu \rfloor +1$, and direction $e_i$, the anisotropic Riemann-Liouville fractional derivative of order ν with scaling ${\alpha }_i$ is defined as:

$$D_i^{\nu ,{\alpha }_i}f\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-\nu /{\alpha }_i)}}{\displaystyle \frac{d^n}{d{x}_i^n}}{\int }_{-\infty }^{x_i}{\displaystyle \frac{f(x_1,\dots ,t,\dots ,x_k)}{{(x_i-t)}^{\nu /{\alpha }_i-n+1}}}dt.$$

(37)

Equivalently, using fractional integral notation:

$$D_i^{\nu ,{\alpha }_i}f\left(x\right)={\displaystyle \frac{d^n}{d{x}_i^n}}{J}_i^{n-\nu /{\alpha }_i}f\left(x\right),$$

(38)

where $J_i^{\mu }$ is the directional fractional integral:

$$J_i^{\mu }f\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\mu \right)}}{\int }_{-\infty }^{x_i}{\displaystyle \frac{f(x_1,\dots ,t,\dots ,x_k)}{{(x_i-t)}^{1-\mu }}}dt.$$

(39)

Definition 4

(Anisotropic Caputo Fractional Derivative). For $\nu >0$, $n=\lfloor \nu \rfloor +1$, the anisotropic Caputo fractional derivative is defined as:

$${}^{C}D_i^{\nu ,{\alpha }_i}f\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-\nu /{\alpha }_i)}}{\int }_{-\infty }^{x_i}{\displaystyle \frac{f^{\left(n\right)}(x_1,\dots ,t,\dots ,x_k)}{{(x_i-t)}^{\nu /{\alpha }_i-n+1}}}dt,$$

(40)

where $f^{\left(n\right)}$ denotes the n-th classical derivative in the $x_i$ direction.

Theorem 5

(Equivalence and Properties). The anisotropic fractional derivatives satisfy the following properties:

• Consistency with Classical Derivatives : For $\nu =m\in \mathbb{N}$,

  $$D_i^{m,{\alpha }_i}f{=}^C{D}_i^{m,{\alpha }_i}f={\displaystyle \frac{{\partial }^{m}f}{\partial x_i^m}}.$$

  (41)
• Relation between Riemann-Liouville and Caputo :

  $${}^{C}D_i^{\nu ,{\alpha }_i}f\left(x\right)=D_i^{\nu ,{\alpha }_i}f\left(x\right)-\sum _{k=0}^{n-1}{\displaystyle \frac{f^{\left(k\right)}(-\infty )}{\mathsf{\Gamma }(k-\nu /{\alpha }_i+1)}}{x}_i^{k-\nu /{\alpha }_i}.$$

  (42)
• Fourier Transform Characterization :

  $$\mathcal{F}\left[D_i^{\nu ,{\alpha }_i}f\right]\left(\xi \right)={\left(i{\xi }_i\right)}^{\nu /{\alpha }_i}\widehat{f}\left(\xi \right)+(\mathit{boundary}\mathit{terms}),$$

  (43)

  where the complex power is defined via the principal branch.
• Semigroup Property : For $\nu ,\mu >0$ with $\nu +\mu <n$,

  $$D_i^{\nu ,{\alpha }_i}{D}_i^{\mu ,{\alpha }_i}f=D_i^{\nu +\mu ,{\alpha }_i}f,$$

  (44)

  provided f has sufficient decay at infinity.

Proof. 

We provide detailed proofs for key properties:

(1) Consistency: For $\nu =m\in \mathbb{N}$, $n=m+1$, and:

$$D_i^{m,{\alpha }_i}f\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(1\right)}}{\displaystyle \frac{d^{m+1}}{d{x}_i^{m+1}}}{\int }_{-\infty }^{x_i}{(x_i-t)}^{0}f(x_1,\dots ,t,\dots ,x_k)dt={\displaystyle \frac{{\partial }^{m}f}{\partial x_i^m}}.$$

(2) Fourier Transform: Using the Fourier transform of the fractional integral:

$$\mathcal{F}\left[J_i^{\mu }f\right]\left(\xi \right)={(-i{\xi }_i)}^{-\mu }\widehat{f}\left(\xi \right),$$

we obtain for the derivative:

$$\mathcal{F}\left[D_i^{\nu ,{\alpha }_i}f\right]\left(\xi \right)={(-i{\xi }_i)}^n{(-i{\xi }_i)}^{-(n-\nu /{\alpha }_i)}\widehat{f}\left(\xi \right)={\left(i{\xi }_i\right)}^{\nu /{\alpha }_i}\widehat{f}\left(\xi \right).$$

(3) Semigroup Property: Let $n_{\nu }=\lfloor \nu \rfloor +1$, $n_{\mu }=\lfloor \mu \rfloor +1$. Then:

$$\begin{array}{cc}\hfill D_i^{\nu ,{\alpha }_i}{D}_i^{\mu ,{\alpha }_i}f& ={\displaystyle \frac{d^{n_{\nu }}}{d{x}_i^{n_{\nu }}}}{J}_i^{n_{\nu }-\nu /{\alpha }_i}\left({\displaystyle \frac{d^{n_{\mu }}}{d{x}_i^{n_{\mu }}}}{J}_i^{n_{\mu }-\mu /{\alpha }_i}f\right)\hfill \\ \hfill & ={\displaystyle \frac{d^{n_{\nu }+n_{\mu }}}{d{x}_i^{n_{\nu }+n_{\mu }}}}{J}_i^{n_{\nu }-\nu /{\alpha }_i}{J}_i^{n_{\mu }-\mu /{\alpha }_i}f\hfill \\ \hfill & ={\displaystyle \frac{d^{n_{\nu +\mu }}}{d{x}_i^{n_{\nu +\mu }}}}{J}_i^{n_{\nu +\mu }-(\nu +\mu )/{\alpha }_i}f=D_i^{\nu +\mu ,{\alpha }_i}f.\hfill \end{array}$$

□

3.2.2. Mixed Fractional Derivatives and Commutation Properties

Definition 5

(Mixed Fractional Derivative). For multi-index $\beta =({\beta }_1,\dots ,{\beta }_k)\in {(0,\infty )}^k$ and scaling vector α, the mixed fractional derivative is defined as:

$$D_{\alpha }^{\beta }f=D_1^{{\beta }_1,{\alpha }_1}{D}_2^{{\beta }_2,{\alpha }_2}\dots D_k^{{\beta }_k,{\alpha }_k}f.$$

(45)

The mixed fractional Sobolev norm is given by:

$${\parallel f\parallel }_{W_{\alpha }^{\beta ,p}}={\parallel f\parallel }_{L^p}+\sum _{i=1}^k{\parallel D_i^{{\beta }_i,{\alpha }_i}f\parallel }_{L^p}.$$

(46)

Theorem 6

(Generalized Commutation Relations). Let $\beta ,\gamma \in {(0,\infty )}^k$ be fractional multi-indices and let

$$D_i^{{\beta }_i,{\alpha }_i}\mathit{denote}\mathit{the}\mathit{anisotropic}\mathit{Riemann}–\mathit{Liouville}\mathit{fractional}\mathit{derivative}\mathit{in}\mathit{direction}{e}_i.$$

Assume f is sufficiently regular so that all expressions below are well-defined. Then:

• Diagonal Commutation. For derivatives acting along the same coordinate direction $e_i$, one has

  $$D_i^{{\beta }_i,{\alpha }_i}{D}_i^{{\gamma }_i,{\alpha }_i}=D_i^{{\gamma }_i,{\alpha }_i}{D}_i^{{\beta }_i,{\alpha }_i}=D_i^{{\beta }_i+{\gamma }_i,{\alpha }_i},$$

  (47)

  whenever ${\beta }_i+{\gamma }_i\notin \mathbb{N}$ so that no cancellation with integer-order derivatives occurs.
• Off-Diagonal Commutation. If $i\ne j$, then the anisotropic fractional derivatives commute:

  $$D_i^{{\beta }_i,{\alpha }_i}{D}_j^{{\gamma }_j,{\alpha }_j}f=D_j^{{\gamma }_j,{\alpha }_j}{D}_i^{{\beta }_i,{\alpha }_i}f,$$

  (48)

  for all $f\in W_{\alpha }^{\beta +\gamma ,p}\left({\mathbb{R}}^k\right)$, since the operators act on distinct variables and have independent integral kernels.
• Anisotropic Scaling Law. Let $T_{\lambda }^{\alpha }f\left(x\right):=f({\lambda }^{1/{\alpha }_1}{x}_1,\dots ,{\lambda }^{1/{\alpha }_k}{x}_k)$ denote the anisotropic scaling operator. Then

  $$D_{\alpha }^{\beta }\left(T_{\lambda }^{\alpha }f\right)={\lambda }^{{\sum }_{i=1}^k{\beta }_i/{\alpha }_i}{T}_{\lambda }^{\alpha }\left(D_{\alpha }^{\beta }f\right).$$

  (49)
• Fractional Leibniz Rule. For ${\beta }_i\in (0,1)$ and sufficiently smooth $f,g$, the one-direction fractional derivative satisfies

  $$D_i^{{\beta }_i,{\alpha }_i}\left(fg\right)=\sum _{k=0}^{\infty }\left({\displaystyle \genfrac{}{}{0pt}{}{{\beta }_i/{\alpha }_i}{k}}\right)\left(D_i^{{\beta }_i-k,{\alpha }_i}f\right)\left({\partial }_i^{k}g\right),$$

  (50)

  where the generalized binomial coefficient is defined via Gamma functions:

  $$\left({\displaystyle \genfrac{}{}{0pt}{}{{\beta }_i/{\alpha }_i}{k}}\right)={\displaystyle \frac{\mathsf{\Gamma }({\beta }_i/{\alpha }_i+1)}{\mathsf{\Gamma }(k+1)\mathsf{\Gamma }({\beta }_i/{\alpha }_i-k+1)}}.$$

  (51)

Proof. 

(1) Diagonal commutation. For Riemann–Liouville derivatives, the representation

$$D_i^{{\beta }_i,{\alpha }_i}={\displaystyle \frac{d^{n_i}}{d{x}_i^{n_i}}}{J}_i^{n_i-{\beta }_i/{\alpha }_i},n_i=\lceil {\beta }_i/{\alpha }_i\rceil ,$$

(52)

together with the semigroup property of fractional integrals

$$J_i^a{J}_i^b=J_i^{a+b},$$

(53)

implies the commutation identity (47) under the stated non-integer condition.

(2) Off-diagonal commutation. Since $D_i^{{\beta }_i,{\alpha }_i}$ and $D_j^{{\gamma }_j,{\alpha }_j}$ operate on different coordinates, their kernels factorize:

$$D_i^{{\beta }_i,{\alpha }_i}{D}_j^{{\gamma }_j,{\alpha }_j}f\left(x\right)={\displaystyle \frac{{\partial }^{n_i}}{\partial x_i^{n_i}}}{J}_i^{n_i-{\beta }_i/{\alpha }_i}{\displaystyle \frac{{\partial }^{n_j}}{\partial x_j^{n_j}}}{J}_j^{n_j-{\gamma }_j/{\alpha }_j}f\left(x\right),$$

(54)

and all operators commute pairwise, giving (48).

(3) Scaling law. For each coordinate direction,

$$D_i^{{\beta }_i,{\alpha }_i}\left(T_{\lambda }^{\alpha }f\right)\left(x\right)={\lambda }^{{\beta }_i/{\alpha }_i}{T}_{\lambda }^{\alpha }\left(D_i^{{\beta }_i,{\alpha }_i}f\right)\left(x\right),$$

(55)

which follows from a direct change of variables in the fractional integral. Multiplying over $i=1,\dots ,k$ yields (49).

(4) Fractional Leibniz rule. For ${\beta }_i\in (0,1)$, one has the integral representation

$$D_i^{{\beta }_i,{\alpha }_i}\left(fg\right)\left(x\right)={\displaystyle \frac{1}{\mathsf{\Gamma }(1-{\beta }_i/{\alpha }_i)}}{\displaystyle \frac{d}{d{x}_i}}{\int }_{-\infty }^{x_i}{\displaystyle \frac{f\left(t\right)g\left(t\right)}{{(x_i-t)}^{{\beta }_i/{\alpha }_i}}}dt.$$

(56)

Expanding $g\left(t\right)$ in its Taylor series at $x_i$ and applying the generalized binomial theorem to the kernel yields the series expansion (50). The coefficients match (51). □

Theorem 7

(Anisotropic Interpolation). Let $0<\theta <1$, $1\le p_0,p_1<\infty$, and ${\nu }_0,{\nu }_1>0$. Then the real interpolation of mixed anisotropic fractional Sobolev spaces satisfies

$${[W_{\alpha }^{{\nu }_0,p_0}\left({\mathbb{R}}^k\right),W_{\alpha }^{{\nu }_1,p_1}\left({\mathbb{R}}^k\right)]}_{\theta }=W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right),$$

(57)

where the interpolated smoothness and integrability indices are given by

$$\nu =(1-\theta ){\nu }_0+\theta {\nu }_1,$$

(58)

and

$${\displaystyle \frac{1}{p}}={\displaystyle \frac{1-\theta }{p_0}}+{\displaystyle \frac{\theta }{p_1}}.$$

(59)

Proof. 

The proof relies on the identification of the mixed anisotropic Sobolev spaces with their anisotropic Besov counterparts:

$$W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)\equiv B_{p,p,\alpha }^{\nu }\left({\mathbb{R}}^k\right),$$

(60)

where $B_{p,p,\alpha }^{\nu }$ denotes the anisotropic Besov space associated with the scaling vector $\alpha$.

Using this equivalence, the result follows from the real interpolation theory of anisotropic Besov spaces. In particular, the anisotropic Littlewood–Paley decomposition ${\left\{{\Delta }_j^{\alpha }\right\}}_{j\in \mathbb{Z}}$ satisfies the standard interpolation identity

$${[B_{p_0,p_0,\alpha }^{{\nu }_0},B_{p_1,p_1,\alpha }^{{\nu }_1}]}_{\theta }=B_{p,p,\alpha }^{\nu },$$

(61)

with parameters $(\nu ,p)$ as in (58)–(59).

Since the Littlewood–Paley pieces scale according to the anisotropy dictated by $\alpha$, the proof reduces to standard K-functional estimates, adapted to the directional scaling. Combining (60) and (61) yields the desired identity (57). □

Theorem 8

(Compactness in Mixed Fractional Spaces). Let $\mathsf{\Omega }\subset {\mathbb{R}}^k$ be a bounded domain with Lipschitz boundary. For $1<p<\infty$ and β with ${min}_i{\beta }_i>0$, the embedding:

$$W_{\alpha }^{\beta ,p}\left(\mathsf{\Omega }\right)\hookrightarrow L^p\left(\mathsf{\Omega }\right)$$

(62)

is compact. Moreover, if ${\nu }_{\mathrm{eff}}>d_{\alpha }/p$, then the embedding into $C\left(\overline{\mathsf{\Omega }}\right)$ is also compact.

Proof. 

The proof uses the anisotropic Fréchet-Kolmogorov theorem. We need to verify:

• Uniform Boundedness: ${sup}_{f\in \mathcal{F}}{\parallel f\parallel }_{W_{\alpha }^{\beta ,p}}<\infty$.
• Equicontinuity: For every $ϵ>0$, there exists $\delta >0$ such that:

  $${\parallel f(·+h)-f\parallel }_{L^p}<ϵ\mathrm{for}\mathrm{all}f\in \mathcal{F},\parallel h\parallel <\delta .$$
• Uniform Decay: ${lim}_{R\to \infty }{sup}_{f\in \mathcal{F}}{\parallel f\parallel }_{L^p({\mathbb{R}}^k\setminus B_R)}=0$.

The fractional differentiability ensures equicontinuity via the estimate:

$${\parallel f(·+h)-f\parallel }_{L^p}\le C\sum _{i=1}^k|h_i{|}^{{\beta }_i/{\alpha }_i}{\parallel D_i^{{\beta }_i,{\alpha }_i}f\parallel }_{L^p}.$$

The compactness follows from the Arzelà-Ascoli theorem in the continuous case. □

3.2.3. Applications to Partial Differential Equations

Theorem 9

(Well-posedness for Mixed Fractional Elliptic Equations). Let $\beta =({\beta }_1,\dots ,{\beta }_k)$ with ${\beta }_i>0$, and let

$$D_i^{2{\beta }_i,{\alpha }_i}={(-{\partial }_i^2)}^{{\beta }_i/{\alpha }_i}$$

denote the anisotropic fractional derivative associated with the scaling exponent ${\alpha }_i$. Consider the mixed fractional elliptic problem

$$\sum _{i=1}^k{(-1)}^{\lfloor {\beta }_i\rfloor +1}{D}_i^{2{\beta }_i,{\alpha }_i}u\left(x\right)+V\left(x\right)u\left(x\right)=f\left(x\right)\mathrm{in}{\mathbb{R}}^k,$$

(63)

where (i) $V\in L^{\infty }\left({\mathbb{R}}^k\right)$, (ii) $V\left(x\right)\ge V_0>0$ a.e., and (iii) $f\in L^2\left({\mathbb{R}}^k\right)$.

Define the multi-order anisotropic Sobolev index

$${\parallel \xi \parallel }_{\alpha ,\beta }^2=\sum _{i=1}^k{|{\xi }_i|}^{2{\beta }_i/{\alpha }_i}.$$

(64)

Then the operator

$${\mathcal{L}}_{\alpha }^{\beta }u=\sum _{i=1}^k{D}_i^{2{\beta }_i,{\alpha }_i}u+Vu$$

(65)

defines a bounded, coercive bilinear form on $W_{\alpha }^{\beta ,2}\left({\mathbb{R}}^k\right)$, and the PDE (63) admits a unique weak solution

$$u\in W_{\alpha }^{\beta ,2}\left({\mathbb{R}}^k\right),$$

which satisfies the energy estimate

$${\parallel u\parallel }_{W_{\alpha }^{\beta ,2}\left({\mathbb{R}}^k\right)}\le C(\beta ,\alpha ,V_0){\parallel f\parallel }_{L^2\left({\mathbb{R}}^k\right)}.$$

(66)

Proof. 

1. Bilinear form and weak formulation. Define the bilinear form associated with ${\mathcal{L}}_{\alpha }^{\beta }$:

$$B[u,v]=\sum _{i=1}^k{\int }_{{\mathbb{R}}^k}{D}_i^{{\beta }_i,{\alpha }_i}u\left(x\right)D_i^{{\beta }_i,{\alpha }_i}v\left(x\right)dx+{\int }_{{\mathbb{R}}^k}V\left(x\right)u\left(x\right)v\left(x\right)dx.$$

(67)

The weak solution is defined by

$$B[u,v]={\int }_{{\mathbb{R}}^k}f\left(x\right)v\left(x\right)dx,v\in W_{\alpha }^{\beta ,2}.$$

(68)

2. Coercivity. Using the Fourier transform representation

$$\widehat{D_i^{{\beta }_i,{\alpha }_i}u}\left(\xi \right)={|{\xi }_i|}^{{\beta }_i/{\alpha }_i}\widehat{u}\left(\xi \right),$$

we obtain

$$\sum _{i=1}^k{\int }_{{\mathbb{R}}^k}|D_i^{{\beta }_i,{\alpha }_i}{u|}^{2}dx={\int }_{{\mathbb{R}}^k}{\parallel \xi \parallel }_{\alpha ,\beta }^2{|\widehat{u}\left(\xi \right)|}^{2}d\xi .$$

(69)

Thus,

$$B[u,u]\ge {\int }_{{\mathbb{R}}^k}{\parallel \xi \parallel }_{\alpha ,\beta }^2|\widehat{u}{\left(\xi \right)|}^{2}d\xi +V_0{\parallel u\parallel }_{L^2}^2\ge c{\parallel u\parallel }_{W_{\alpha }^{\beta ,2}}^2,$$

(70)

for some $c=c(\beta ,\alpha ,V_0)>0$. Hence B is coercive.

3. Boundedness. Since $V\in L^{\infty }$, one has

$$|B[u,v]|\le {C\parallel u\parallel }_{W_{\alpha }^{\beta ,2}}{\parallel v\parallel }_{W_{\alpha }^{\beta ,2}},$$

(71)

by Cauchy–Schwarz and the definition of the anisotropic Sobolev norm.

4. Existence and uniqueness via Lax–Milgram. The form B is continuous and coercive on $W_{\alpha }^{\beta ,2}$, and the linear functional

$$\ell \left(v\right)={\int }_{{\mathbb{R}}^k}fvdx$$

is bounded on $L^2\subset W_{\alpha }^{\beta ,2}$. Therefore, the Lax–Milgram theorem yields a unique $u\in W_{\alpha }^{\beta ,2}$ solving (68). Coercivity immediately implies the energy estimate (66). □

This provides a rigorous functional-analytic framework for mixed anisotropic fractional elliptic equations, ensuring well-posedness under natural structural assumptions and making explicit the role of anisotropic scaling in the construction of the associated Sobolev spaces.

4. Main Results: Mixed Fractional Landau Inequalities

4.1. Sharp Inequalities for Mixed Norms

Theorem 10

(Mixed Fractional Landau Inequality). Let $f\in W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$ with $\nu \in (1,2)$, $1<p<\infty$, and scaling vector α. Then for any $i=1,\dots ,k$:

$$\parallel D_i^1{f\parallel }_{L^{\infty }}\le C(\nu ,p,\alpha ){\parallel f\parallel }_{L^{\infty }}^{1-1/\nu }{\left(\sum _{j=1}^k{\parallel D_j^{\nu ,{\alpha }_j}f\parallel }_{L^p}^{{\alpha }_j/\nu }\right)}^{1/\nu },$$

(72)

where the constant $C(\nu ,p,\alpha )$ satisfies:

$$C(\nu ,p,\alpha )\le {\displaystyle \frac{2^{1-1/\nu }}{\mathsf{\Gamma }{(2-\nu )}^{1/\nu }}}{\left({\displaystyle \frac{p^{\prime }}{p^{\prime }\nu -1}}\right)}^{1/p^{\prime }}{\left(\sum _{j=1}^k{\alpha }_j^{-1}\right)}^{1/\nu }\kappa (p,\alpha ),$$

(73)

with $1/p+1/p^{\prime }=1$ and $\kappa (p,\alpha )$ the optimal constant from the anisotropic Hardy-Littlewood inequality.

Proof. 

We provide a comprehensive proof with detailed estimates.

1. Anisotropic Fractional Taylor Expansion with Remainder Estimate

Fix direction $e_i$ and consider the anisotropic fractional Taylor expansion:

$$f(x+h{e}_i)=f\left(x\right)+h{D}_i^{1}f\left(x\right)+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\nu \right)}}{\int }_0^h{(h-t)}^{\nu -1}{D}_i^{\nu ,{\alpha }_i}f(x+t{e}_i)dt+R_{\nu }(x,h),$$

(74)

where the remainder satisfies the sharp estimate:

$$|R_{\nu }(x,h)|\le {\displaystyle \frac{h^{\nu }}{\mathsf{\Gamma }(\nu +1)}}\underset{0\le t\le h}{sup}|D_i^{\nu ,{\alpha }_i}f(x+t{e}_i)|.$$

(75)

2. Pointwise Estimate via Maximal Functions

Rearranging and applying the triangle inequality:

$$\begin{array}{cc}\hfill |h{D}_i^{1}f\left(x\right)|& \le |f(x+h{e}_i)-f\left(x\right)|+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\nu \right)}}{\int }_0^h{(h-t)}^{\nu -1}|D_i^{\nu ,{\alpha }_i}f(x+t{e}_i)|dt+|R_{\nu }(x,h)|\hfill \\ \hfill & \le {2\parallel f\parallel }_{L^{\infty }}+{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\nu \right)}}{\int }_0^h{(h-t)}^{\nu -1}|D_i^{\nu ,{\alpha }_i}f(x+t{e}_i)|dt.\hfill \end{array}$$

(76)

Define the anisotropic fractional maximal function:

$$M_i^{\nu ,{\alpha }_i}f\left(x\right)=\underset{h>0}{sup}{\displaystyle \frac{1}{h^{\nu /{\alpha }_i}}}{\int }_0^h|D_i^{\nu ,{\alpha }_i}f(x+t{e}_i)|dt.$$

(77)

Then equation (76) implies:

$$|D_i^{1}f\left(x\right)|\le {\displaystyle \frac{{2\parallel f\parallel }_{L^{\infty }}}{h}}+{\displaystyle \frac{h^{\nu -1}}{\mathsf{\Gamma }\left(\nu \right)}}{M}_i^{\nu ,{\alpha }_i}f\left(x\right).$$

(78)

3. Optimal Scaling and Interpolation

Optimize over $h>0$ by considering the function:

$$\varphi \left(h\right)={\displaystyle \frac{{2\parallel f\parallel }_{L^{\infty }}}{h}}+{\displaystyle \frac{h^{\nu -1}}{\mathsf{\Gamma }\left(\nu \right)}}{M}_i^{\nu ,{\alpha }_i}f\left(x\right).$$

(79)

The critical point satisfies:

$${\varphi }^{\prime }\left(h^{*}\right)=-{\displaystyle \frac{{2\parallel f\parallel }_{L^{\infty }}}{{\left(h^{*}\right)}^2}}+{\displaystyle \frac{(\nu -1){\left(h^{*}\right)}^{\nu -2}}{\mathsf{\Gamma }\left(\nu \right)}}{M}_i^{\nu ,{\alpha }_i}f\left(x\right)=0,$$

(80)

yielding:

$$h^{*}={\left({\displaystyle \frac{{2\mathsf{\Gamma }\left(\nu \right)\parallel f\parallel }_{L^{\infty }}}{(\nu -1)M_i^{\nu ,{\alpha }_i}f\left(x\right)}}\right)}^{1/\nu }.$$

(81)

Substituting back gives the sharp pointwise bound:

$$|D_i^{1}f\left(x\right)|\le {\displaystyle \frac{2^{1-1/\nu }\nu }{{(\nu -1)}^{1-1/\nu }\mathsf{\Gamma }{\left(\nu \right)}^{1/\nu }}}{\parallel f\parallel }_{L^{\infty }}^{1-1/\nu }{\left(M_i^{\nu ,{\alpha }_i}f\left(x\right)\right)}^{1/\nu }.$$

(82)

4. Anisotropic Maximal Function Estimates

We now bound the maximal function in mixed norms. Consider the anisotropic Hardy-Littlewood maximal operator:

$$M_{\alpha }g\left(x\right)=\underset{r>0}{sup}{\displaystyle \frac{1}{|B_{\alpha }(x,r)|}}{\int }_{B_{\alpha }(x,r)}|g\left(y\right)|dy,$$

(83)

where $B_{\alpha }(x,r)=\{y\in {\mathbb{R}}^k:{\rho }_{\alpha }(x-y)<r\}$ is the anisotropic ball.

By the anisotropic maximal theorem:

$$\parallel M_{\alpha }{g\parallel }_{L^p}\le C(p,\alpha ){\parallel g\parallel }_{L^p},1<p\le \infty .$$

(84)

For the directional maximal function, we have the embedding:

$$M_i^{\nu ,{\alpha }_i}f\left(x\right)\le C\sum _{j=1}^k(M_{\alpha }|D_j^{\nu ,{\alpha }_j}f{{|}^{{\alpha }_j/\nu }\left(x\right))}^{\nu /{\alpha }_j}.$$

(85)

Taking $L^p$ norms and applying Hölder’s inequality:

$$\begin{array}{cc}\hfill \parallel M_i^{\nu ,{\alpha }_i}{f\parallel }_{L^p}& \le C\sum _{j=1}^k\parallel (M_{\alpha }|D_j^{\nu ,{\alpha }_j}{f|}^{{\alpha }_j/\nu }{{)}^{\nu /{\alpha }_j}\parallel }_{L^p}\hfill \\ \hfill & \le C\sum _{j=1}^k\parallel M_{\alpha }|D_j^{\nu ,{\alpha }_j}f{{|}^{{\alpha }_j/\nu }\parallel }_{L^{p{\alpha }_j/\nu }}^{\nu /{\alpha }_j}\hfill \\ \hfill & \le C\sum _{j=1}^k\parallel |D_j^{\nu ,{\alpha }_j}f{{|}^{{\alpha }_j/\nu }\parallel }_{L^{p{\alpha }_j/\nu }}^{\nu /{\alpha }_j}\hfill \\ \hfill & =C\sum _{j=1}^k{\parallel D_j^{\nu ,{\alpha }_j}f\parallel }_{L^p}^{{\alpha }_j/\nu }.\hfill \end{array}$$

(86)

5. Synthesis and Constant Optimization

Combining (82) and (86), we obtain:

$$\parallel D_i^1{f\parallel }_{L^{\infty }}\le C_{\nu }{\parallel f\parallel }_{L^{\infty }}^{1-1/\nu }{\left(\sum _{j=1}^k{\parallel D_j^{\nu ,{\alpha }_j}f\parallel }_{L^p}^{{\alpha }_j/\nu }\right)}^{1/\nu },$$

(87)

where:

$$C_{\nu }={\displaystyle \frac{2^{1-1/\nu }\nu }{{(\nu -1)}^{1-1/\nu }\mathsf{\Gamma }{\left(\nu \right)}^{1/\nu }}}·C(p,\alpha ).$$

(88)

The explicit dependence on $\alpha$ comes from the anisotropic maximal constant $C(p,\alpha )$, which scales as ${\left({\sum }_{j=1}^k{\alpha }_j^{-1}\right)}^{1/\nu }$ by the geometry of anisotropic balls. □

Theorem 11

(Anisotropic Hardy–Littlewood Maximal Inequality). Let $M_{\alpha }$ denote the anisotropic Hardy–Littlewood maximal operator associated with the anisotropic balls

$$B_{\alpha }(x,r):=\left\{y\in {\mathbb{R}}^k:|y_i-x_i|<r^{1/{\alpha }_i}\mathrm{for}\mathrm{all}i=1,\dots ,k\right\}.$$

(89)

Define

$$M_{\alpha }f\left(x\right):=\underset{r>0}{sup}{\displaystyle \frac{1}{|B_{\alpha }(x,r)|}}{\int }_{B_{\alpha }(x,r)}|f\left(y\right)|dy.$$

(90)

Then for every $1<p\le \infty$, the operator $M_{\alpha }$ is bounded on $L^p\left({\mathbb{R}}^k\right)$:

$$\parallel M_{\alpha }{f\parallel }_{L^p}\le C(p,\alpha ){\parallel f\parallel }_{L^p},$$

(91)

where the constant satisfies the sharp structural estimate

$$C(p,\alpha )\le {\displaystyle \frac{p}{p-1}}{\left(\prod _{i=1}^k{\alpha }_i^{-1}\right)}^{1/p}{\omega }_{d_{\alpha }},$$

(92)

and ${\omega }_{d_{\alpha }}$ denotes the measure of the unit anisotropic ball $B_{\alpha }(0,1)$, whose volume scales as

$$|B_{\alpha }(0,r)|={\omega }_{d_{\alpha }}{r}^{d_{\alpha }},d_{\alpha }:=\sum _{i=1}^k{\displaystyle \frac{1}{{\alpha }_i}}.$$

(93)

Proof. 

The proof follows the classical Hardy–Littlewood argument, adapted to the anisotropic quasi-metric induced by the $\alpha$-scaling.

1. Anisotropic Vitali Covering. Given any family of anisotropic balls $\left\{B_{\alpha }(x,r_x)\right\}$, one constructs a disjoint subfamily $\left\{B_{\alpha }(x_j,r_{x_j})\right\}$ such that the original family is covered by finitely overlapped dilates $\left\{B_{\alpha }(x_j,3r_{x_j})\right\}$. The overlap number is bounded by a constant depending only on $({\alpha }_1,\dots ,{\alpha }_k)$ and the homogeneous dimension $d_{\alpha }$ defined in (93).

2. Weak-$(1,1)$ Estimate. Let $\lambda >0$ and define the level set

$$E_{\lambda }:=\{x\in {\mathbb{R}}^k:M_{\alpha }f\left(x\right)>\lambda \}.$$

(94)

By the covering lemma, there exist disjoint balls $B_{\alpha }(x_j,r_j)$ such that

$$E_{\lambda }\subset \bigcup _j{B}_{\alpha }(x_j,3r_j).$$

For each j,

$$\lambda <{\displaystyle \frac{1}{|B_{\alpha }(x_j,r_j)|}}{\int }_{B_{\alpha }(x_j,r_j)}|f\left(y\right)|dy,$$

(95)

so summing over the disjoint subcollection yields

$$|E_{\lambda }|\le {\displaystyle \frac{C\left(\alpha \right)}{\lambda }}{\parallel f\parallel }_{L^1},$$

(96)

establishing the anisotropic weak-$(1,1)$ bound.

3. Strong $L^p$ boundedness. The weak-$(1,1)$ estimate (96) and the trivial $L^{\infty }$ bound

$$\parallel M_{\alpha }{f\parallel }_{L^{\infty }}\le {\parallel f\parallel }_{L^{\infty }}$$

(97)

allow the application of the Marcinkiewicz interpolation theorem. This gives the $L^p$ estimate (91) with the constant in (92), whose structure reflects the anisotropic volume scaling (93). □

Corollary 1

(Higher-Order Mixed Landau Inequalities). Let $\nu \in (m,m+1)$ with $m\in \mathbb{N}$, and suppose $f\in W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$ for some $1<p<\infty$. Then each coordinate derivative of integer order m satisfies the anisotropic Landau-type bound

$$\parallel D_i^m{f\parallel }_{L^{\infty }}\le C(m,\nu ,p,\alpha ){\parallel f\parallel }_{L^{\infty }}^{1-m/\nu }{\left(\sum _{j=1}^k{\parallel D_j^{\nu ,{\alpha }_j}f\parallel }_{L^p}^{{\alpha }_j/\nu }\right)}^{m/\nu }.$$

(98)

Proof. 

Since $\nu >m$, the anisotropic Sobolev embedding implies

$$W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)\hookrightarrow W_{\alpha }^{m,\infty }\left({\mathbb{R}}^k\right).$$

(99)

Applying Theorem 11 to each intermediate fractional derivative of order $1,2,\dots ,m$ and interpolating between the extremal norms yields the mixed-power inequality (98). The exponents ${\alpha }_j/\nu$ arise from the anisotropic directional scaling inherent to the definition of $W_{\alpha }^{\nu ,p}$. □

4.2. Applications to Multiscale Neural Operators

Theorem 12

(Stability of Multiscale Neural Operators). Let ${\mathcal{N}}_{\theta }:{\mathbb{R}}^k\to \mathbb{R}$ be a neural operator with L layers, where each layer l has the form:

$$x_{l+1}={\sigma }_l(W_l{x}_l+b_l),l=0,\dots ,L-1,$$

(100)

with anisotropic weight constraints $\parallel W_l^i{\parallel }_{op}\le {\lambda }_i^{1/{\alpha }_i}$ for each directional component. Assume the activation functions satisfy ${\sigma }_l\in C_b^{1,1}\left(\mathbb{R}\right)$ with $\parallel {\sigma }_l^{\prime }{\parallel }_{L^{\infty }}\le 1$ and $\parallel {\sigma }_l^{\prime \prime }{\parallel }_{L^{\infty }}\le K$. Then for input perturbations $\delta x$ with ${\parallel \delta x\parallel }_{\infty }\le ϵ$:

$$\parallel {\mathcal{N}}_{\theta }(x+\delta x)-{\mathcal{N}}_{\theta }{\left(x\right)\parallel }_{L^{\infty }}\le CLϵ\left(\prod _{l=1}^L\underset{i}{max}{\lambda }_i^{1/{\alpha }_i}\right){\parallel {\mathcal{N}}_{\theta }\parallel }_{W_{\alpha }^{1,\infty }},$$

(101)

where the constant C depends on K and the scaling vector α.

Proof. 

We provide a detailed layer-wise analysis with careful tracking of constants.

1. Single Layer Sensitivity Analysis

Consider a single layer $y=\sigma (Wx+b)$. For a perturbation $\delta x$, we have:

$$\begin{array}{cc}\hfill {\parallel y(x+\delta x)-y\left(x\right)\parallel }_{L^{\infty }}& \le \parallel {\sigma }^{\prime }{\parallel }_{L^{\infty }}{\parallel W\parallel }_{op}{\parallel \delta x\parallel }_{L^{\infty }}\hfill \\ \hfill & \le \underset{i}{max}{\lambda }_i^{1/{\alpha }_i}ϵ.\hfill \end{array}$$

(102)

For the derivative bound, using the chain rule and the activation function regularity:

$$\begin{array}{cc}\hfill \parallel D_i^1{y\parallel }_{L^{\infty }}& \le \parallel {\sigma }^{\prime }{\parallel }_{L^{\infty }}\parallel W^i{\parallel }_{op}\parallel D_i^1{x\parallel }_{L^{\infty }}+\parallel {\sigma }^{\prime \prime }{\parallel }_{L^{\infty }}\parallel W^i{\parallel }_{op}^2{\parallel x\parallel }_{L^{\infty }}\hfill \\ \hfill & \le {\lambda }_i^{1/{\alpha }_i}\parallel D_i^1{x\parallel }_{L^{\infty }}+K{\lambda }_i^{2/{\alpha }_i}{\parallel x\parallel }_{L^{\infty }}.\hfill \end{array}$$

(103)

2. Multi-layer Composition and Anisotropic Chain Rule

For the composition of L layers, we use the anisotropic Faa di Bruno formula. Let $F={\mathcal{N}}_{\theta }={\sigma }_L\circ W_L\circ \dots \circ {\sigma }_1\circ W_1$. Then:

$$D_i^{1}F\left(x\right)=\sum _{j_1,\dots ,j_L=1}^k\left(\prod _{l=1}^L{\displaystyle \frac{\partial {\sigma }_l}{\partial z_l}}{W}_l^{i_l{j}_{l-1}}\right)D_{j_L}^{1}x,$$

(104)

where we use the convention $j_0=i$.

Taking $L^{\infty }$ norms and using the weight constraints:

$$\begin{array}{cc}\hfill \parallel D_i^1{F\parallel }_{L^{\infty }}& \le \left(\prod _{l=1}^L{\parallel {\sigma }_l^{\prime }\parallel }_{L^{\infty }}\right)\left(\prod _{l=1}^L\underset{i_l}{max}{\parallel W_l^{i_l}\parallel }_{op}\right)\sum _{j_L=1}^k{\parallel D_{j_L}^{1}x\parallel }_{L^{\infty }}\hfill \\ \hfill & \le \left(\prod _{l=1}^L\underset{i}{max}{\lambda }_i^{1/{\alpha }_i}\right){\parallel x\parallel }_{W_{\alpha }^{1,\infty }}.\hfill \end{array}$$

(105)

3. Stability under Input Perturbations

Using the mean value theorem and the derivative bound:

$$\begin{array}{cc}\hfill |F(x+\delta x)-F(x)|& \le \underset{0\le t\le 1}{sup}|DF(x+t\delta x)\delta x|\hfill \\ \hfill & \le \sum _{i=1}^k\parallel D_i^1{F\parallel }_{L^{\infty }}|\delta x_i|\hfill \\ \hfill & \le ϵ\sum _{i=1}^k{\parallel D_i^{1}F\parallel }_{L^{\infty }}\hfill \\ \hfill & \le kϵ\left(\prod _{l=1}^L\underset{i}{max}{\lambda }_i^{1/{\alpha }_i}\right){\parallel F\parallel }_{W_{\alpha }^{1,\infty }}.\hfill \end{array}$$

(106)

4. Sobolev Norm Control via Mixed Landau Inequality

Applying Theorem 10 to the neural operator:

$${\parallel F\parallel }_{W_{\alpha }^{1,\infty }}\le C(\nu ,p,\alpha ){\parallel F\parallel }_{L^{\infty }}^{1-1/\nu }{\left(\sum _{j=1}^k{\parallel D_j^{\nu ,{\alpha }_j}F\parallel }_{L^p}^{{\alpha }_j/\nu }\right)}^{1/\nu }.$$

(107)

The fractional derivatives of F can be bounded using the anisotropic chain rule and the regularity of activations, completing the proof. □

Theorem 13

(Approximation Theory for Mixed Fractional Spaces). Let $f\in W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$ with $\nu >0$, $1<p<\infty$, and scaling vector $\alpha =({\alpha }_1,\dots ,{\alpha }_k)\in {(0,\infty )}^k$. Let $\mathcal{N}$ be a neural operator with L layers and N trainable parameters per layer. Then there exists a choice of parameters θ such that:

$$\parallel f-{\mathcal{N}}_{\theta }{\parallel }_{L^{\infty }\left({\mathbb{R}}^k\right)}\le C(\nu ,p,\alpha )L{N}^{-\nu /d_{\alpha }}{\parallel f\parallel }_{W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)},$$

(108)

where the anisotropic dimension is defined by:

$$d_{\alpha }=\sum _{i=1}^k{\alpha }_i^{-1},$$

(109)

and the constant $C(\nu ,p,\alpha )$ satisfies:

$$C(\nu ,p,\alpha )\le K(p,\alpha ){\left({\displaystyle \frac{\mathsf{\Gamma }(\nu +1)}{\nu }}\right)}^{1/\nu }{\left(\sum _{i=1}^k{\alpha }_i^{-1}\right)}^{1/2},$$

(110)

with $K(p,\alpha )$ depending only on p and the scaling ratios ${max}_i{\alpha }_i/{min}_i{\alpha }_i$.

Proof. 

We provide a detailed proof with quantitative estimates.

1. Anisotropic Partition of Unity and Localization

Construct an anisotropic partition of unity adapted to the scaling $\alpha$. For $\delta >0$, define the anisotropic cubes:

$$Q_{\delta ,m}=\prod _{i=1}^k[{\delta }^{{\alpha }_i}{m}_i,{\delta }^{{\alpha }_i}(m_i+1)],m\in {\mathbb{Z}}^k.$$

(111)

Let ${\left\{{\varphi }_{\delta ,m}\right\}}_{m\in {\mathbb{Z}}^k}$ be a smooth partition of unity subordinate to this covering, satisfying:

$$\begin{array}{cc}\hfill & \sum _{m\in {\mathbb{Z}}^k}{\varphi }_{\delta ,m}\left(x\right)=1\mathrm{for}\mathrm{all}x\in {\mathbb{R}}^k,\hfill \end{array}$$

(112)

$$\begin{array}{cc}\hfill & \parallel {\varphi }_{\delta ,m}{\parallel }_{L^{\infty }}\le 1,{\parallel D_i^1{\varphi }_{\delta ,m}\parallel }_{L^{\infty }}\le C{\delta }^{-{\alpha }_i},\hfill \end{array}$$

(113)

$$\begin{array}{cc}\hfill & \mathrm{supp}{\varphi }_{\delta ,m}\subset {\tilde{Q}}_{\delta ,m},|{\tilde{Q}}_{\delta ,m}|\le C{\delta }^{d_{\alpha }}.\hfill \end{array}$$

(114)

2. Local Mixed Fractional Taylor Approximation

For each $m\in {\mathbb{Z}}^k$, let $x_m$ be the center of $Q_{\delta ,m}$. Consider the mixed fractional Taylor polynomial:

$$P_{\delta ,m}\left(x\right)=\sum _{{|\beta |}_{\alpha }\le \nu }{\displaystyle \frac{D_{\alpha }^{\beta }f\left(x_m\right)}{{\prod }_{i=1}^k\mathsf{\Gamma }({\beta }_i/{\alpha }_i+1)}}\prod _{i=1}^k{(x_i-x_{m,i})}^{{\beta }_i/{\alpha }_i},$$

(115)

where ${|\beta |}_{\alpha }={\sum }_{i=1}^k{\beta }_i/{\alpha }_i$ is the anisotropic degree.

Using the mixed fractional Landau inequality (Theorem 10), we obtain the local approximation error:

$$\parallel f-P_{\delta ,m}{\parallel }_{L^{\infty }\left(Q_{\delta ,m}\right)}\le C_1(\nu ,\alpha ){\delta }^{\nu }\sum _{i=1}^k{\parallel D_i^{\nu ,{\alpha }_i}f\parallel }_{L^p\left(Q_{\delta ,m}\right)}^{{\alpha }_i/\nu }.$$

(116)

3. Neural Network Representation and Parameter Counting

Each local polynomial $P_{\delta ,m}$ can be approximated by a neural network with ReLU^k activation. By standard approximation theory [5], there exists a neural network ${\mathcal{N}}_{\theta ,m}$ with:

$$\parallel P_{\delta ,m}-{\mathcal{N}}_{\theta ,m}{\parallel }_{L^{\infty }\left(Q_{\delta ,m}\right)}\le ϵ,$$

(117)

using at most $N_m=O\left({ϵ}^{-d_{\alpha }/\nu }\right)$ parameters per layer.

The global approximation is constructed as:

$${\mathcal{N}}_{\theta }\left(x\right)=\sum _{m\in {\mathbb{Z}}^k}{\varphi }_{\delta ,m}\left(x\right){\mathcal{N}}_{\theta ,m}\left(x\right).$$

(118)

4. Error Analysis and Parameter Optimization

The total approximation error decomposes as:

$$\begin{array}{cc}\hfill \parallel f-{\mathcal{N}}_{\theta }{\parallel }_{L^{\infty }}& \le \underset{x\in {\mathbb{R}}^k}{sup}\sum _m{\varphi }_{\delta ,m}\left(x\right)|f\left(x\right)-P_{\delta ,m}\left(x\right)|\hfill \\ \hfill & +\underset{x\in {\mathbb{R}}^k}{sup}\sum _m{\varphi }_{\delta ,m}\left(x\right)|P_{\delta ,m}\left(x\right)-{\mathcal{N}}_{\theta ,m}\left(x\right)|\hfill \\ \hfill & \le C_1{\delta }^{\nu }{\parallel f\parallel }_{W_{\alpha }^{\nu ,p}}+ϵ.\hfill \end{array}$$

(119)

The number of active cubes at any point is bounded by the overlap constant $K\left(\alpha \right)$. The total number of parameters satisfies:

$$N_{\mathrm{total}}\le C_2\left(\alpha \right){\delta }^{-d_{\alpha }}{N}_m\le C_3\left(\alpha \right){\delta }^{-d_{\alpha }}{ϵ}^{-d_{\alpha }/\nu }.$$

(120)

Optimizing by choosing $\delta \sim {ϵ}^{1/\nu }$ and $ϵ\sim N^{-\nu /d_{\alpha }}$ yields:

$$\parallel f-{\mathcal{N}}_{\theta }{\parallel }_{L^{\infty }}\le C(\nu ,p,\alpha )N^{-\nu /d_{\alpha }}{\parallel f\parallel }_{W_{\alpha }^{\nu ,p}}.$$

(121)

5. Layer Count and Architecture

The construction requires $L=O(log(1/ϵ\left)\right)$ layers to implement the partition of unity and local approximations. Since $ϵ\sim N^{-\nu /d_{\alpha }}$, we have:

$$L\le C_{4}logN.$$

(122)

The logarithmic factor is absorbed into the constant for the final bound (108). □

Theorem 14

(Sharpness of Approximation Rate). The approximation rate $N^{-\nu /d_{\alpha }}$ in Theorem 13 is sharp. Specifically, there exists a family of functions $\left\{f_{ϵ}\right\}\subset W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$ such that for any neural operator ${\mathcal{N}}_{\theta }$ with N parameters per layer:

$$\parallel f_{ϵ}-{\mathcal{N}}_{\theta }{\parallel }_{L^{\infty }}\ge c(\nu ,p,\alpha )N^{-\nu /d_{\alpha }}{\parallel f_{ϵ}\parallel }_{W_{\alpha }^{\nu ,p}},$$

(123)

where $c(\nu ,p,\alpha )>0$ depends only on $\nu ,p,$ and α.

Proof. 

The proof uses information-theoretic arguments adapted to the anisotropic setting. Consider the metric entropy of the unit ball in $W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$. By the anisotropic extension of Kolmogorov’s entropy theorem:

$${log}_{2}N(ϵ,B_1\left(W_{\alpha }^{\nu ,p}\right),L^{\infty })\asymp {ϵ}^{-d_{\alpha }/\nu },$$

(124)

where $N(ϵ,B_1,L^{\infty })$ is the covering number.

On the other hand, the class of neural operators with N parameters has VC-dimension (or similar complexity measure) bounded by $O\left(N^2\right)$. Therefore:

$${log}_{2}N(ϵ,{\mathcal{N}}_N,L^{\infty })\le C{N}^{2}log(1/ϵ),$$

(125)

where ${\mathcal{N}}_N$ denotes neural operators with N parameters.

Comparing (124) and (125) yields the sharpness of the rate $N^{-\nu /d_{\alpha }}$. □

Remark 1.

The mixed fractional Landau inequalities provide a refined analytic framework with several important implications:

• Dimensional Adaptation: The approximation rate $N^{-\nu /d_{\alpha }}$ in (108) adapts to the intrinsic anisotropic dimension $d_{\alpha }$ rather than the ambient dimension k. This is particularly beneficial when ${\alpha }_i\gg 1$ for some directions, effectively reducing the curse of dimensionality.
• Regularity-Specific Rates: Different choices of ν and α in (108) yield approximation rates tailored to the function’s specific mixed regularity pattern, enabling optimal architecture design for given problem classes.
• Multiscale Applications: The explicit dependence on scaling parameters in (110) informs the design of neural operators for multiscale problems, where different coordinates may exhibit vastly different scaling behaviors.
• Sharpness Guarantee: Theorem 14 establishes that the rate in (108) cannot be improved in general, providing fundamental limits for neural operator approximation in mixed fractional spaces.

This theoretical framework bridges harmonic analysis, approximation theory, and deep learning, offering principled guidance for neural architecture design in high-dimensional and anisotropic settings.

Corollary 2

(Isotropic Special Case). When $\alpha =(1,\dots ,1)$, we recover the classical isotropic approximation rate:

$$\parallel f-{\mathcal{N}}_{\theta }{\parallel }_{L^{\infty }}\le C(\nu ,p,k)L{N}^{-\nu /k}{\parallel f\parallel }_{W^{\nu ,p}},$$

(126)

which matches known optimal rates for Sobolev space approximation [6].

Proof. 

Substitute ${\alpha }_i=1$ for all i into Theorem 13, noting that $d_{\alpha }=k$ in this case. □

5. Results

5.1. Theoretical Framework and Main Inequalities

This work establishes a comprehensive mathematical theory of mixed fractional Landau inequalities with several fundamental advances:

Theorem 15

(Mixed Fractional Landau Inequalities). For functions in mixed fractional Sobolev spaces $W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$, the following sharp inequalities hold:

• First-Order Gradient Bound: For $\nu \in (1,2)$ and any direction $e_i$,

  $$\parallel D_i^1{f\parallel }_{L^{\infty }}\le C(\nu ,p,\alpha ){\parallel f\parallel }_{L^{\infty }}^{1-1/\nu }{\left(\sum _{j=1}^k{\parallel D_j^{\nu ,{\alpha }_j}f\parallel }_{L^p}^{{\alpha }_j/\nu }\right)}^{1/\nu },$$

  (127)

  with explicit constant dependence on the scaling vector α.
• Higher-Order Generalization: For $\nu \in (m,m+1)$ with $m\in \mathbb{N}$,

  $$\parallel D_i^m{f\parallel }_{L^{\infty }}\le C(m,\nu ,p,\alpha ){\parallel f\parallel }_{L^{\infty }}^{1-m/\nu }{\left(\sum _{j=1}^k{\parallel D_j^{\nu ,{\alpha }_j}f\parallel }_{L^p}^{{\alpha }_j/\nu }\right)}^{m/\nu }.$$

  (128)
• Anisotropic Maximal Function Control: The proofs rely on the anisotropic Hardy-Littlewood maximal inequality establishing $L^p$-boundedness with explicit dependence on α.

5.2. Applications to Neural Operator Theory

The theoretical framework yields powerful applications to deep learning:

Theorem 16

(Neural Operator Stability). Multiscale neural operators with anisotropic weight constraints exhibit controlled sensitivity:

$$\parallel {\mathcal{N}}_{\theta }(x+\delta x)-{\mathcal{N}}_{\theta }{\left(x\right)\parallel }_{L^{\infty }}\le CLϵ\left(\prod _{l=1}^L\underset{i}{max}{\lambda }_i^{1/{\alpha }_i}\right){\parallel {\mathcal{N}}_{\theta }\parallel }_{W_{\alpha }^{1,\infty }}.$$

(129)

Theorem 17

(Approximation Theory). Neural operators achieve optimal approximation rates for mixed fractional spaces:

$$\parallel f-{\mathcal{N}}_{\theta }{\parallel }_{L^{\infty }}\le C(\nu ,p,\alpha )L{N}^{-\nu /d_{\alpha }}{\parallel f\parallel }_{W_{\alpha }^{\nu ,p}},$$

(130)

where $d_{\alpha }={\sum }_{i=1}^k{\alpha }_i^{-1}$ is the anisotropic dimension.

Theorem 18

(Sharpness of Rates). The approximation rate $N^{-\nu /d_{\alpha }}$ is optimal and cannot be improved in general, as established through metric entropy arguments.

5.3. Computational Advantages

The mixed fractional framework provides significant computational benefits:

Corollary 3

(Dimensional Adaptation). When ${\alpha }_i\gg 1$ for some directions, the effective dimension $d_{\alpha }$ becomes substantially smaller than the ambient dimension k, leading to:

• Accelerated convergence rates compared to isotropic theory
• Principled guidance for neural architecture design
• Natural regularization schemes respecting problem geometry

6. Conclusions

This work establishes a comprehensive mathematical framework for fractional Landau inequalities with mixed Sobolev norms, making fundamental contributions to the analysis of functions with anisotropic regularity.

6.1. Principal Contributions

• Novel Function Spaces: Introduction of mixed fractional Sobolev spaces $W_{\alpha }^{\nu ,p}\left({\mathbb{R}}^k\right)$ providing a flexible framework for multiscale analysis.
• Sharp Inequalities: Establishment of mixed fractional Landau inequalities with explicit dependence on directional scaling parameters.
• Harmonic Analysis Foundations: Development of directional Littlewood-Paley theory and anisotropic maximal function estimates.
• Neural Operator Applications: Rigorous stability bounds and approximation rates for deep learning architectures.

6.2. Theoretical Significance

The mixed fractional approach addresses fundamental limitations of existing theory:

• Captures directional scaling behavior essential for high-dimensional problems
• Provides dimensional adaptation through anisotropic dimension $d_{\alpha }$
• Unifies fractional calculus, harmonic analysis, and deep learning theory
• Offers refined understanding of the curse of dimensionality

6.3. Future Research Directions

This work opens several promising avenues:

• Extension to anisotropic Besov and Triebel-Lizorkin spaces
• Applications to nonlinear PDEs with anisotropic diffusion
• Connections to geometric deep learning and graph neural networks
• Numerical implementation of mixed norm regularization
• Stochastic extensions for uncertainty quantification
• Further development of neural operator theory

The mixed fractional Landau inequalities provide a powerful framework bridging classical analysis and modern computational practice, offering both deep theoretical insights and practical guidance for high-dimensional and multiscale problems.