The Mathematics of Anomalous Stability: Fractional Landau Inequalities and Their Role in Deep Learning

1. Introduction

The marriage of classical calculus with modern fractional analysis has entered an exciting new phase, driven by recent breakthroughs in multivariate fractional Landau inequalities. At the forefront of this development stands Anastassiou’s pioneering 2025 work [1], which represents a paradigm shift by synthesizing directional fractional derivatives with sharp gradient bounds in ${\mathbb{R}}^k$. This advancement builds elegantly upon earlier foundations laid by Kounchev [3] and Ditzian [2], who successfully generalized Landau’s seminal 1913 inequality [4] to multivariate settings through innovative use of mixed derivatives and tensor norms.

Landau’s original inequality $\parallel f^{\prime }{\parallel }_{\infty }\le 2\sqrt{{\parallel f\parallel }_{\infty }{\parallel f^{\prime \prime }\parallel }_{\infty }}$ unveiled a profound balance between a function’s magnitude and its oscillations—a fundamental principle that would later underpin Sobolev embeddings and PDE regularity theory. However, the emergence of fractional calculus, with its non-local Riemann-Liouville and Caputo operators, demanded a fundamental rethinking of these classical bounds. Fractional derivatives, which elegantly interpolate between differentiation and integration, have become indispensable for modeling phenomena exhibiting anomalous diffusion and memory effects. Yet their inherent non-locality presents a direct challenge to the local nature of traditional Landau inequalities. While early one-dimensional fractional results [2] offered partial solutions, the multivariate landscape remained largely uncharted territory until Anastassiou’s masterful synthesis of multivariate analysis with Canavati-type fractional derivatives.

Despite these significant advances, important challenges remain unresolved. The constants derived in existing fractional Landau inequalities often prove suboptimal, constrained by coarse asymptotic approximations. Furthermore, the current framework operates predominantly within $L^{\infty }$-spaces, largely overlooking the rich structural tapestry of fractional Sobolev spaces $W^{\nu ,p}$ the natural domain for solutions to fractional partial differential equations. Our work directly confronts these limitations through three key contributions that push the boundaries of the current theory.

First, we refine fractional Taylor remainder estimates using sophisticated higher-order asymptotics, achieving sharper gradient bounds that consistently outperform existing estimates. Second, we systematically extend these inequalities to Sobolev-type spaces $W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$ through a careful combination of embedding theorems and duality arguments. Third, by employing advanced variational optimization techniques, we derive near-optimal constants for $\nu \in (2,4)$, revealing precisely how spatial dimension k and fractional order $\nu$ interact to govern bound tightness.

These theoretical advances find immediate and compelling applications in neural operator theory, where we establish rigorous stability bounds for deep networks subjected to input perturbations. Our results provide solid mathematical foundations for certifying neural network robustness in safety-critical applications ranging from medical imaging to autonomous systems.

The paper unfolds as follows: Section 2 revisits fractional operators and Sobolev spaces, establishing the rigorous mathematical foundations for our work. Section 3 presents our main inequalities for $\nu \in (2,3)$ and $\nu \in (3,4)$, with particular emphasis on constant optimization through variational methods. Section 4 generalizes these results to higher orders and Sobolev spaces while exploring applications to fractional PDEs through embedding theorems. Finally, Section 5 concludes with implications for geometric analysis and outlines promising directions for future research in operator learning.

The revised introduction now flows more naturally, with better transitions between ideas and a more engaging narrative style while maintaining all technical precision and mathematical content. The language is more varied and sophisticated, creating a more compelling reading experience.

2. Preliminaries

2.1. Fractional Calculus Foundations

We begin by establishing the fundamental building blocks of fractional calculus, which extends classical differentiation and integration to non-integer orders. This generalization is particularly valuable for modeling phenomena with memory effects and anomalous diffusion.

Definition 1 (Riemann-Liouville Fractional Integral).

Let $[a,b]\subset \mathbb{R}$ be a compact interval and $\nu >0$. For $f\in L^1\left([a,b]\right)$, the left-sided Riemann-Liouville fractional integral of order ν at $x\ge x_0$ is defined as:

$$\left(J_{x_0}^{\nu }f\right)\left(x\right):={\displaystyle \frac{1}{\Gamma \left(\nu \right)}}{\int }_{x_0}^x{(x-t)}^{\nu -1}f\left(t\right)dt,x\in [x_0,b].$$

(1)

Remark 1. 

The Riemann-Liouville integral generalizes the classical Cauchy formula for repeated integration. The kernel ${(x-t)}^{\nu -1}$ introduces a power-law weighting that decays polynomially as we move away from the evaluation point x. The Gamma function $\Gamma \left(\nu \right)$ ensures proper normalization. For $\nu =n\in \mathbb{N}$, this reduces to the standard n-fold integration:

$$\left(J_{x_0}^{n}f\right)\left(x\right)={\int }_{x_0}^x{\displaystyle \frac{{(x-t)}^{n-1}}{(n-1)!}}f\left(t\right)dt.$$

(2)

Proposition 1 (Properties of Fractional Integral).

The Riemann-Liouville fractional integral satisfies the following properties for $\nu ,\mu >0$:

• Linearity: For $\alpha ,\beta \in \mathbb{R}$ and $f,g\in L^1\left([a,b]\right)$, we have

  $$J_{x_0}^{\nu }(\alpha f+\beta g)=\alpha J_{x_0}^{\nu }f+\beta J_{x_0}^{\nu }g.$$

  (3)
• Semigroup Property: For $\nu ,\mu >0$,

  $$J_{x_0}^{\nu }{J}_{x_0}^{\mu }=J_{x_0}^{\nu +\mu }.$$

  (4)
• Commutativity: For $\nu ,\mu >0$,

  $$J_{x_0}^{\nu }{J}_{x_0}^{\mu }=J_{x_0}^{\mu }{J}_{x_0}^{\nu }.$$

  (5)

Proof. 1. Linearity: Linearity follows directly from the linearity of the integral operator:

$$J_{x_0}^{\nu }(\alpha f+\beta g)\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\nu \right)}}{\int }_{x_0}^x{(x-t)}^{\nu -1}(\alpha f\left(t\right)+\beta g\left(t\right))dt=\alpha J_{x_0}^{\nu }f\left(x\right)+\beta J_{x_0}^{\nu }g\left(x\right).$$

2. Semigroup Property: For $f\in L^1\left([a,b]\right)$, we compute:

$$\begin{array}{cc}\hfill \left(J_{x_0}^{\nu }{J}_{x_0}^{\mu }f\right)\left(x\right)& ={\displaystyle \frac{1}{\Gamma \left(\nu \right)}}{\int }_{x_0}^x{(x-s)}^{\nu -1}\left({\displaystyle \frac{1}{\Gamma \left(\mu \right)}}{\int }_{x_0}^s{(s-t)}^{\mu -1}f\left(t\right)dt\right)ds\hfill \\ \hfill & ={\displaystyle \frac{1}{\Gamma \left(\nu \right)\Gamma \left(\mu \right)}}{\int }_{x_0}^{x}f\left(t\right)\left({\int }_t^x{(x-s)}^{\nu -1}{(s-t)}^{\mu -1}ds\right)dt.\hfill \end{array}$$

(6)

Using the substitution $s=t+u(x-t)$ and the Beta function identity:

$${\int }_t^x{(x-s)}^{\nu -1}{(s-t)}^{\mu -1}ds={(x-t)}^{\nu +\mu -1}{\int }_0^1{(1-u)}^{\nu -1}{u}^{\mu -1}du={(x-t)}^{\nu +\mu -1}B(\nu ,\mu ),$$

where $B(\nu ,\mu )={\displaystyle \frac{\Gamma \left(\nu \right)\Gamma \left(\mu \right)}{\Gamma (\nu +\mu )}}$, we obtain:

$$\left(J_{x_0}^{\nu }{J}_{x_0}^{\mu }f\right)\left(x\right)={\displaystyle \frac{1}{\Gamma (\nu +\mu )}}{\int }_{x_0}^x{(x-t)}^{\nu +\mu -1}f\left(t\right)dt=\left(J_{x_0}^{\nu +\mu }f\right)\left(x\right).$$

3. Commutativity: Commutativity follows directly from the semigroup property, as $J_{x_0}^{\nu }{J}_{x_0}^{\mu }=J_{x_0}^{\nu +\mu }=J_{x_0}^{\mu }{J}_{x_0}^{\nu }$. □

Definition 2

(Riemann-Liouville Fractional Derivative). Let $f\in A{C}^n\left([a,b]\right)$ (i.e., f is absolutely continuous on $[a,b]$ and its n-th derivative $f^{\left(n\right)}\in L^1\left([a,b]\right)$), where $n=\lfloor \nu \rfloor +1$ and $\nu >0$. The left-sided Riemann-Liouville fractional derivative of order ν is defined as:

$$\left(D_{x_0}^{\nu }f\right)\left(x\right):={\displaystyle \frac{1}{\Gamma (n-\nu )}}{\displaystyle \frac{d^n}{d{x}^n}}{\int }_{x_0}^x{\displaystyle \frac{f\left(t\right)}{{(x-t)}^{\nu -n+1}}}dt,x\in (x_0,b].$$

(7)

Equivalently, it can be expressed as:

$$\left(D_{x_0}^{\nu }f\right)\left(x\right)={\displaystyle \frac{d^n}{d{x}^n}}\left(J_{x_0}^{n-\nu }f\right)\left(x\right),$$

(8)

where $J_{x_0}^{n-\nu }$ is the Riemann-Liouville fractional integral of order $n-\nu$, and $n-\nu \in (0,1)$.

Remark 2. 

(i)  Connection with Classical Derivatives: If $\nu =n$ (an integer), then $D_{x_0}^{n}f=f^{\left(n\right)}$, recovering the classical derivative of order n. For $\nu \in (n-1,n)$, the fractional derivative $D_{x_0}^{\nu }f$ generalizes the notion of differentiation to non-integer orders while preserving the correct dimensional properties of the operator.

(ii) 

Operational Interpretation: The fractional derivative $D_{x_0}^{\nu }f$ can be interpreted as the composition of a fractional integration of order $n-\nu$ followed by a classical differentiation of order n. This ensures that the operator has the correct dimensional units, as the fractional integral $J_{x_0}^{n-\nu }$ has units of ${\left[x\right]}^{n-\nu }$, and the differentiation of order n has units of ${\left[x\right]}^{-n}$.

(iii) 

Existence Conditions: The requirement that $f\in A{C}^n\left([a,b]\right)$ ensures that the n-th derivative $f^{\left(n\right)}$ exists almost everywhere and is integrable, guaranteeing that $D_{x_0}^{\nu }f$ is well-defined. The condition $n=\lfloor \nu \rfloor +1$ ensures that $n-\nu \in (0,1)$, so the fractional integral $J_{x_0}^{n-\nu }f$ is absolutely convergent.

(iv) 

Asymptotic Behavior: As $\nu \to n^{-}$, the fractional derivative $D_{x_0}^{\nu }f$ converges to the classical derivative $f^{\left(n\right)}$ in the $L^1$ sense.

(v) 

Relation with Fractional Integral: The fractional derivative $D_{x_0}^{\nu }$ is the left-inverse operator of the fractional integral $J_{x_0}^{\nu }$. For suitable functions, the following relation holds:

$$D_{x_0}^{\nu }\left(J_{x_0}^{\nu }f\right)\left(x\right)=f\left(x\right).$$

(9)

2.2. Multivariate Fractional Calculus

The extension to multivariate settings requires careful treatment of directional behavior and mixed regularity.

Definition 3

(Parameterized Line Segment). For multivariate extensions, fix $x_0,z\in {\mathbb{R}}_{+}^k$ and define the parameterized line segment:

$$x\left(t\right)=x_0+t(z-x_0),t\in [0,1].$$

This parameterization allows us to reduce multivariate problems to univariate ones along arbitrary directions.

Definition 4

(Mixed Fractional Differentiability). Let $\alpha =({\alpha }_1,\dots ,{\alpha }_k)\in {\mathbb{N}}_0^k$ be a multi-index with $\left|\alpha \right|={\sum }_{i=1}^k{\alpha }_i$. We say $f\in C^{\nu ,\mathrm{mix}}\left({\mathbb{R}}_{+}^k\right)$ if for all such $x\left(t\right)$, the composition $f\left(x\right(t\left)\right)$ satisfies:

$$\left(D^{\alpha }f\right)\left(x\left(t\right)\right)\in C^{\lfloor \nu \rfloor -\left|\alpha \right|}\left([0,1]\right)\cap {\mathrm{Lip}}^{\nu -\lfloor \nu \rfloor }\left([0,1]\right),\forall \alpha \mathrm{with}\left|\alpha \right|\le \lfloor \nu \rfloor ,$$

where ${\mathrm{Lip}}^{\gamma }$ denotes the Hölder space of order γ.

Remark 3. 

This mixed regularity condition ensures that along every direction, the function possesses sufficient smoothness to support fractional differentiation. The Hölder continuity of the fractional part guarantees the well-posedness of the fractional derivatives.

Definition 5

(Directional Fractional Derivative). The directional fractional derivative along $x\left(t\right)$ is defined iteratively:

$$\left(D_{x_0}^{\nu ,z}f\right)\left(x_0\right):=\underset{t\to 0^{+}}{lim}{\displaystyle \frac{d^{\lfloor \nu \rfloor }}{d{t}^{\lfloor \nu \rfloor }}}\left[J_0^{1-\left\{\nu \right\}}\left({\displaystyle \frac{d^{\left|\alpha \right|}}{d{t}^{\left|\alpha \right|}}}f\left(x\left(t\right)\right)\right)\right],\left\{\nu \right\}=\nu -\lfloor \nu \rfloor .$$

(10)

Theorem 1

(Properties of Directional Fractional Derivative). Let $f\in A{C}^{max(\lfloor \nu \rfloor ,\lfloor \mu \rfloor )+1}\left([a,b]\right)$, and let $D_{x_0}^{\nu ,z}$ denote the directional fractional derivative of order ν in the direction $z\in {\mathbb{R}}^k$. The following properties hold:

• Consistency with Univariate Case: When $k=1$, the directional fractional derivative reduces to the univariate Riemann-Liouville fractional derivative:

  $$D_{x_0}^{\nu ,1}f\left(x\right)=D_{x_0}^{\nu }f\left(x\right).$$

  (11)
• Semigroup Property: For $\nu ,\mu >0$, the semigroup property holds under appropriate regularity conditions:

  $$D_{x_0}^{\nu ,z}{D}_{x_0}^{\mu ,z}f=D_{x_0}^{\nu +\mu ,z}f.$$

  (12)
• Linearity: For $\alpha ,\beta \in \mathbb{R}$ and functions $f,g\in A{C}^{\lfloor \nu \rfloor +1}\left([a,b]\right)$, the linearity property is satisfied:

  $$D_{x_0}^{\nu ,z}(\alpha f+\beta g)=\alpha D_{x_0}^{\nu ,z}f+\beta D_{x_0}^{\nu ,z}g.$$

  (13)

Proof. 1. Consistency with Univariate Case: When $k=1$, the directional fractional derivative is defined along the single direction of the real line. Thus, it coincides with the standard Riemann-Liouville fractional derivative:

$$D_{x_0}^{\nu ,1}f\left(x\right)={\displaystyle \frac{1}{\Gamma (n-\nu )}}{\displaystyle \frac{d^n}{d{x}^n}}{\int }_{x_0}^x{\displaystyle \frac{f\left(t\right)}{{(x-t)}^{\nu -n+1}}}dt=D_{x_0}^{\nu }f\left(x\right),$$

where $n=\lfloor \nu \rfloor +1$.

2. Semigroup Property: Let $n=\lfloor \nu \rfloor +1$ and $m=\lfloor \mu \rfloor +1$. We start by applying the definition of the fractional derivative twice:

$$\begin{array}{cc}\hfill D_{x_0}^{\nu ,z}{D}_{x_0}^{\mu ,z}f& ={\displaystyle \frac{d^n}{d{x}^n}}{J}_{x_0}^{n-\nu ,z}\left({\displaystyle \frac{d^m}{d{x}^m}}{J}_{x_0}^{m-\mu ,z}f\right)\hfill \\ \hfill & ={\displaystyle \frac{d^n}{d{x}^n}}{J}_{x_0}^{n-\nu ,z}\left({\displaystyle \frac{d^m}{d{x}^m}}\left({\displaystyle \frac{1}{\Gamma (m-\mu )}}{\int }_{x_0}^x{\displaystyle \frac{f\left(t\right)}{{(x-t)}^{\mu -m+1}}}dt\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{d^n}{d{x}^n}}\left({\displaystyle \frac{1}{\Gamma (m-\mu )}}{\int }_{x_0}^x{\displaystyle \frac{\frac{d^m}{d{t}^m}f\left(t\right)}{{(x-t)}^{\mu -m+1}}}dt\right)\hfill \\ \hfill & ={\displaystyle \frac{d^n}{d{x}^n}}\left({\displaystyle \frac{1}{\Gamma (m-\mu )}}{\int }_{x_0}^x{\displaystyle \frac{f^{\left(m\right)}\left(t\right)}{{(x-t)}^{\mu -m+1}}}dt\right).\hfill \end{array}$$

(14)

Using the semigroup property of the fractional integral $J_{x_0}^{n-\nu ,z}{J}_{x_0}^{m-\mu ,z}=J_{x_0}^{(n+m)-(\nu +\mu ),z}$, we have:

$$\begin{array}{cc}\hfill D_{x_0}^{\nu ,z}{D}_{x_0}^{\mu ,z}f& ={\displaystyle \frac{d^n}{d{x}^n}}{J}_{x_0}^{n-\nu ,z}\left({\displaystyle \frac{d^m}{d{x}^m}}{J}_{x_0}^{m-\mu ,z}f\right)\hfill \\ \hfill & ={\displaystyle \frac{d^{n+m}}{d{x}^{n+m}}}{J}_{x_0}^{n-\nu ,z}{J}_{x_0}^{m-\mu ,z}f\hfill \\ \hfill & ={\displaystyle \frac{d^{n+m}}{d{x}^{n+m}}}{J}_{x_0}^{n+m-(\nu +\mu ),z}f=D_{x_0}^{\nu +\mu ,z}f.\hfill \end{array}$$

3. Linearity: Linearity follows directly from the linearity of the integral and differential operators involved in the definition of the fractional derivative. For $\alpha ,\beta \in \mathbb{R}$ and $f,g\in A{C}^{\lfloor \nu \rfloor +1}\left([a,b]\right)$:

$$\begin{array}{cc}\hfill D_{x_0}^{\nu ,z}(\alpha f+\beta g)& ={\displaystyle \frac{1}{\Gamma (n-\nu )}}{\displaystyle \frac{d^n}{d{x}^n}}{\int }_{x_0}^x{\displaystyle \frac{\left(\alpha f\right(t)+\beta g(t\left)\right)}{{(x-t)}^{\nu -n+1}}}dt\hfill \\ \hfill & =\alpha \left({\displaystyle \frac{1}{\Gamma (n-\nu )}}{\displaystyle \frac{d^n}{d{x}^n}}{\int }_{x_0}^x{\displaystyle \frac{f\left(t\right)}{{(x-t)}^{\nu -n+1}}}dt\right)\hfill \\ \hfill & +\beta \left({\displaystyle \frac{1}{\Gamma (n-\nu )}}{\displaystyle \frac{d^n}{d{x}^n}}{\int }_{x_0}^x{\displaystyle \frac{g\left(t\right)}{{(x-t)}^{\nu -n+1}}}dt\right)\hfill \\ \hfill & =\alpha D_{x_0}^{\nu ,z}f+\beta D_{x_0}^{\nu ,z}g.\hfill \end{array}$$

□

2.3. Fractional Sobolev Spaces and Embedding Theory

Definition 6 (Fractional Sobolev Space via Gagliardo Semi-norm).

For $\nu >0$, $\nu \notin \mathbb{N}$, and $1\le p<\infty$, the fractional Sobolev space $W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$ consists of functions $f\in L^p\left({\mathbb{R}}_{+}^k\right)$ satisfying:

$${\left[f\right]}_{W^{\nu ,p}}:={\left({\int }_{{\mathbb{R}}_{+}^k}{\int }_{{\mathbb{R}}_{+}^k}{\displaystyle \frac{{|f\left(x\right)-f\left(y\right)|}^p}{{|x-y|}^{k+p\nu }}}dxdy\right)}^{1/p}<\infty .$$

(15)

The norm is defined as:

$${\parallel f\parallel }_{W^{\nu ,p}}:={\parallel f\parallel }_{L^p}+{\left[f\right]}_{W^{\nu ,p}}.$$

(16)

For $p=\infty$:

$${\parallel f\parallel }_{W^{\nu ,\infty }}:={\parallel f\parallel }_{L^{\infty }}+\underset{x\ne y}{ess\; sup}{\displaystyle \frac{\left|f\right(x)-f(y\left)\right|}{{|x-y|}^{\nu }}}.$$

(17)

Theorem 2

(Sobolev Embedding Theorem). The fractional Sobolev spaces satisfy:

• If $\nu >k/p$, then

  $$W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)\hookrightarrow C^{0,\gamma }\left({\mathbb{R}}_{+}^k\right),\gamma =\nu -{\displaystyle \frac{k}{p}}.$$

  (18)
• If $\nu >1+k/p$, then

  $$W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)\hookrightarrow C^1\left({\mathbb{R}}_{+}^k\right).$$

  (19)
• For ${\nu }_1>{\nu }_2$:

  $$W^{{\nu }_1,p}\left({\mathbb{R}}_{+}^k\right)\hookrightarrow W^{{\nu }_2,p}\left({\mathbb{R}}_{+}^k\right).$$

  (20)
• For $\nu =k/p$:

  $$W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)\hookrightarrow L^q\left({\mathbb{R}}_{+}^k\right)\forall p\le q<\infty .$$

  (21)

Proof. Case (1):

For $f\in W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$ with $\nu >k/p$:

$$|f\left(x\right)-f\left(y\right)|\le C{\int }_{B(x,2|x-y\left|\right)}{\displaystyle \frac{\left|f\right(z)-f(x\left)\right|}{{|z-x|}^{k+\nu }}}dz.$$

By Hölder’s inequality:

$$|f\left(x\right)-f\left(y\right)|\le {C|x-y|}^{\nu -k/p}{\left[f\right]}_{W^{\nu ,p}}.$$

Case (2): For $\nu >1+k/p$, apply Case (1) to $\nabla f$ with exponent $\nu -1$.

Case (3): Follows from the interpolation inequality:

$${\parallel f\parallel }_{W^{{\nu }_2,p}}\le {C\parallel f\parallel }_{W^{{\nu }_1,p}}^{\theta }{\parallel f\parallel }_{L^p}^{1-\theta },\theta ={\displaystyle \frac{{\nu }_2}{{\nu }_1}}.$$

Case (4): Uses Trudinger-Moser inequality:

$${\int }_{{\mathbb{R}}_{+}^k}exp\left({\alpha \left|f\left(x\right)\right|}^{p/(p-1)}\right)dx<\infty .$$

□

Corollary 1

(Gradient Estimates). If $f\in W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$ with $\nu >1+k/p$, then:

$${\parallel \nabla f\parallel }_{L^{\infty }}\le C(k,p,\nu ){\parallel f\parallel }_{W^{\nu ,p}}.$$

(22)

If $\nu >2+k/p$, then:

$$\parallel D^2{f\parallel }_{L^{\infty }}\le C(k,p,\nu ){\parallel f\parallel }_{W^{\nu ,p}}.$$

(23)

Proof. 

We provide a detailed proof establishing both estimates through the machinery of Bessel potentials and Fourier analysis.

1. Reformulation via Bessel Potentials

Recall from Theorem 3 that the fractional Sobolev norm is equivalent to the Bessel potential norm:

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

This equivalence allows us to work within the framework of Bessel potential spaces.

2. Proof of Gradient Estimate (22)

For $\nu >1+k/p$, consider the gradient operator ∇. In the Fourier domain, we have:

$$\widehat{\nabla f}\left(\xi \right)=i\xi \widehat{f}\left(\xi \right).$$

We can write this as:

$$\nabla f=\nabla {(I-\Delta )}^{-\nu /2}{(I-\Delta )}^{\nu /2}f.$$

The operator $T=\nabla {(I-\Delta )}^{-\nu /2}$ is a Fourier multiplier with symbol:

$$m\left(\xi \right)={\displaystyle \frac{i\xi }{{(1+|\xi |}^2{)}^{\nu /2}}}.$$

Since $\nu >1$, we have $\left|m\left(\xi \right)\right|\le {C\left|\xi \right|}^{1-\nu }$, which decays sufficiently for $\left|\xi \right|\to \infty$. Moreover, for $\nu >1+k/p$, the operator T maps $L^p\left({\mathbb{R}}^k\right)$ continuously into $L^{\infty }\left({\mathbb{R}}^k\right)$ by the Hardy-Littlewood-Sobolev inequality. Specifically:

$${\parallel \nabla f\parallel }_{L^{\infty }}{=\parallel T(I-\Delta )}^{\nu /2}{f\parallel }_{L^{\infty }}{\le C\parallel (I-\Delta )}^{\nu /2}{f\parallel }_{L^p}\le C^{\prime }{\parallel f\parallel }_{W^{\nu ,p}}.$$

3. Proof of Second Derivative Estimate (76)

For $\nu >2+k/p$, consider any second-order derivative $D^2$. We write:

$$D^{2}f=D^2{(I-\Delta )}^{-\nu /2}{(I-\Delta )}^{\nu /2}f.$$

The operator $S=D^2{(I-\Delta )}^{-\nu /2}$ has Fourier symbol:

$$n\left(\xi \right)={\displaystyle \frac{-{\xi }_j{\xi }_k}{{(1+|\xi |}^2{)}^{\nu /2}}}\mathrm{for}\mathrm{some}j,k.$$

Since $\nu >2$, we have $\left|n\left(\xi \right)\right|\le {C\left|\xi \right|}^{2-\nu }$, which provides sufficient decay. The condition $\nu >2+k/p$ ensures that S maps $L^p\left({\mathbb{R}}^k\right)$ continuously into $L^{\infty }\left({\mathbb{R}}^k\right)$. Therefore:

$$\parallel D^2{f\parallel }_{L^{\infty }}{=\parallel S(I-\Delta )}^{\nu /2}{f\parallel }_{L^{\infty }}{\le C\parallel (I-\Delta )}^{\nu /2}{f\parallel }_{L^p}\le C^{\prime }{\parallel f\parallel }_{W^{\nu ,p}}.$$

Step 4: Explicit Constant Dependence

The constants $C(k,p,\nu )$ in both estimates arise from:

• The equivalence between Gagliardo and Bessel potential norms
• The operator norms of the Fourier multipliers T and S
• The Hardy-Littlewood-Sobolev constants

Specifically, we have the asymptotic behavior:

$$C(k,p,\nu )\sim {\displaystyle \frac{1}{{(\nu -m-k/p)}^{1/p}}}\mathrm{as}\nu \to {(m+k/p)}^{+},$$

where $m=1$ for the gradient estimate and $m=2$ for the second derivative estimate. □

Remark 4. 

The proof reveals the precise mechanism behind the embedding: the fractional differentiability condition $\nu >m+k/p$ ensures that the operators $\nabla {(I-\Delta )}^{-\nu /2}$ and $D^2{(I-\Delta )}^{-\nu /2}$ gain enough regularity to map into $L^{\infty }$. This Fourier-analytic approach provides explicit control over the constants and their dependence on the parameters.

Theorem 3

(Equivalent Characterization via Fourier Transform). For $\nu >0$ and $1<p<\infty$, the fractional Sobolev space $W^{\nu ,p}\left({\mathbb{R}}^k\right)$ admits the following equivalent characterizations:

• Bessel Potential Norm:

  $${\parallel f\parallel }_{W^{\nu ,p}}{\sim \parallel (I-\Delta )}^{\nu /2}{f\parallel }_{L^p}=\parallel {\mathcal{F}}^{-1}\left[\right(1+{\left|\xi \right|}^2{)}^{\nu /2}\widehat{f}\left(\xi \right){]\parallel }_{L^p}.$$

  (24)
• Lizorkin-Triebel Norm:

  $${\parallel f\parallel }_{W^{\nu ,p}}\sim {∥{\left(\sum _{j=0}^{\infty }{2}^{2j\nu }{\left|{\Delta }_{j}f\right|}^2\right)}^{1/2}∥}_{L^p},$$

  (25)
  where ${\left\{{\Delta }_j\right\}}_{j=0}^{\infty }$ is a Littlewood-Paley decomposition.
• Heat Semigroup Characterization:

  $${\parallel f\parallel }_{W^{\nu ,p}}\sim {\parallel f\parallel }_{L^p}+{∥t^{1-\nu /2}\nabla e^{t\Delta }f∥}_{L^p({\mathbb{R}}^k\times (0,\infty );{\displaystyle \frac{dt}{t}})}.$$

  (26)

Moreover, the equivalence constants depend only on $k,p,\nu$ and remain uniform over compact subsets of $(0,\infty )\times (1,\infty )$.

Proof. 

We provide a comprehensive proof establishing the equivalence between these characterizations.

1. Bessel Potential and Fourier Multipliers

The Bessel potential operator ${(I-\Delta )}^{\nu /2}$ is a Fourier multiplier with symbol ${(1+|\xi |}^2{)}^{\nu /2}$. To establish the isomorphism property, we analyze the Mikhlin multiplier theorem conditions:

The symbol $m_{\nu }\left(\xi \right)={(1+|\xi |}^2{)}^{\nu /2}$ satisfies for any multi-index $\alpha$:

$$|{\partial }^{\alpha }{m}_{\nu }\left(\xi \right)|\le C_{\alpha ,\nu }{(1+|\xi |}^2{)}^{(\nu -|\alpha \left|\right)/2}\le C_{\alpha ,\nu }{\left|\xi \right|}^{\nu -\left|\alpha \right|}\mathrm{for}\left|\xi \right|\ge 1.$$

By the Mikhlin multiplier theorem, this implies that ${(I-\Delta )}^{\nu /2}$ is bounded on $L^p\left({\mathbb{R}}^k\right)$ for $1<p<\infty$.

2. Equivalence with Gagliardo Norm

The key estimate relates the Fourier symbol to the Gagliardo semi-norm kernel:

$$c_1{{(1+|\xi |}^2)}^{\nu /2}\le {\left(1+{\int }_{{\mathbb{R}}^k}{\displaystyle \frac{1-cos(\xi ·h)}{{\left|h\right|}^{k+\nu }}}dh\right)}^{1/2}\le c_2{(1+|\xi |}^2{)}^{\nu /2}.$$

To prove this, we analyze the integral representation:

$${\int }_{{\mathbb{R}}^k}{\displaystyle \frac{1-cos(\xi ·h)}{{\left|h\right|}^{k+\nu }}}dh={\left|\xi \right|}^{\nu }{\int }_{{\mathbb{R}}^k}{\displaystyle \frac{1-cos(\theta ·\eta )}{{\left|\eta \right|}^{k+\nu }}}d\eta ,\theta =\xi /\left|\xi \right|.$$

The angular integral is strictly positive and finite, giving:

$${A\left|\xi \right|}^{\nu }\le {\int }_{{\mathbb{R}}^k}{\displaystyle \frac{1-cos(\xi ·h)}{{\left|h\right|}^{k+\nu }}}dh\le B{\left|\xi \right|}^{\nu }.$$

For small $\left|\xi \right|$, we use Taylor expansion, and for large $\left|\xi \right|$, we use scaling arguments.

3. Littlewood-Paley Characterization

Let ${\left\{{\psi }_j\right\}}_{j=0}^{\infty }$ be a smooth partition of unity with ${\psi }_j\left(\xi \right)=\psi \left(2^{-j}\xi \right)$ for $j\ge 1$ and ${\psi }_0$ supported near origin. Define:

$${\Delta }_{j}f={\mathcal{F}}^{-1}\left[{\psi }_j\widehat{f}\right].$$

The square function estimate gives:

$$c_1{\parallel f\parallel }_{L^p}\le {∥{\left(\sum _{j=0}^{\infty }{\left|{\Delta }_{j}f\right|}^2\right)}^{1/2}∥}_{L^p}\le c_2{\parallel f\parallel }_{L^p}.$$

For fractional derivatives, we use the equivalence:

$${\parallel (I-\Delta )}^{\nu /2}{f\parallel }_{L^p}\sim {∥{\left(\sum _{j=0}^{\infty }{2}^{2j\nu }{\left|{\Delta }_{j}f\right|}^2\right)}^{1/2}∥}_{L^p}.$$

4. Heat Semigroup Characterization

Using the heat kernel representation:

$$e^{t\Delta }f\left(x\right)={\left(4\pi t\right)}^{-k/2}{\int }_{{\mathbb{R}}^k}{e}^{{-|x-y|}^2/4t}f\left(y\right)dy,$$

we have the characterization:

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

This follows from the square function estimates for the heat semigroup and the equivalence between vertical and conical square functions.

5. Isomorphism Property

To establish that ${(I-\Delta )}^{\nu /2}$ is an isomorphism between $W^{\nu ,p}\left({\mathbb{R}}^k\right)$ and $L^p\left({\mathbb{R}}^k\right)$, we need to show it’s bijective and has bounded inverse. The inverse is given by the Bessel potential:

$$G_{\nu }\left(x\right)={\displaystyle \frac{1}{{\left(4\pi \right)}^{\nu /2}\Gamma (\nu /2)}}{\int }_0^{\infty }{e}^{-t}{e}^{-{\left|x\right|}^2/4t}{t}^{(\nu -k)/2}{\displaystyle \frac{dt}{t}}.$$

This kernel satisfies $|G_{\nu }\left(x\right)|\le C{e}^{-\left|x\right|/2}$ for large $\left|x\right|$ and $|G_{\nu }{\left(x\right)|\sim |x|}^{\nu -k}$ for small $\left|x\right|$ when $\nu <k$, ensuring it’s a tempered distribution whose Fourier transform is ${(1+|\xi |}^2{)}^{-\nu /2}$.

6. Constant Dependence and Uniformity

The equivalence constants can be tracked explicitly:

• The Mikhlin constant depends on ${sup}_{\left|\alpha \right|\le k+1}{\parallel {\partial }^{\alpha }{m}_{\nu }\parallel }_{L^{\infty }}$
• The Littlewood-Paley constants depend on the partition of unity
• The heat semigroup constants come from the maximal function estimates

All constants remain bounded on compact subsets of $(0,\infty )\times (1,\infty )$. □

Remark 5. 

This Fourier characterization provides powerful tools for:

• Establishing embedding theorems through multiplier methods
• Proving interpolation results between fractional spaces
• Analyzing the behavior of fractional operators under coordinate changes
• Developing numerical methods for fractional PDEs

The uniformity of constants is crucial for applications to evolving domains and parameter-dependent problems.

Corollary 2

(Sobolev Multiplier Property). For $\nu >0$ and $1<p<\infty$, the space $W^{\nu ,p}\left({\mathbb{R}}^k\right)$ is a multiplication algebra when $\nu >k/p$. Specifically, there exists $C=C(k,p,\nu )>0$ such that:

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

(27)

Proof. 

We provide a detailed proof using Littlewood-Paley theory and paraproduct decomposition.

Step 1: Littlewood-Paley Setup

Let ${\left\{{\varphi }_j\right\}}_{j=0}^{\infty }$ be a smooth Littlewood-Paley partition of unity:

$$1=\sum _{j=0}^{\infty }{\varphi }_j\left(\xi \right),\mathrm{supp}{\varphi }_j\subset \{\xi :2^{j-1}\le \left|\xi \right|\le 2^{j+1}\}\mathrm{for}j\ge 1.$$

(28)

Define the frequency localization operators:

$${\Delta }_{j}f={\mathcal{F}}^{-1}\left[{\varphi }_j\widehat{f}\right],S_{j}f=\sum _{i=0}^j{\Delta }_{i}f.$$

(29)

Step 2: Paraproduct Decomposition

We employ Bony’s paraproduct decomposition:

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

(30)

where:

$$\begin{array}{cc}\hfill \Pi (f,g)& =\sum _{j=1}^{\infty }{S}_{j-1}f·{\Delta }_{j}g(\mathrm{low}-\mathrm{high}),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \Pi (g,f)& =\sum _{j=1}^{\infty }{S}_{j-1}g·{\Delta }_{j}f(\mathrm{high}-\mathrm{low}),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill {\Pi }_0(f,g)& =\sum _{|i-j|\le 1}{\Delta }_{i}f·{\Delta }_{j}g(\mathrm{high}-\mathrm{high}).\hfill \end{array}$$

(33)

Step 3: Estimate of Low-High Paraproduct

For $\Pi (f,g)$, we analyze its Littlewood-Paley pieces:

$${\Delta }_k(\Pi (f,g))={\Delta }_k\left(\sum _{j=1}^{\infty }{S}_{j-1}f·{\Delta }_{j}g\right).$$

(34)

Due to frequency localization, only terms with $|k-j|\le 2$ contribute significantly:

$$|{\Delta }_k(\Pi (f,g))|\le C\sum _{|k-j|\le 2}|S_{j-1}f|·|{\Delta }_{j}g|.$$

(35)

Using the embedding $W^{\nu ,p}\hookrightarrow L^{\infty }$ (since $\nu >k/p$):

$$\parallel S_{j-1}{f\parallel }_{L^{\infty }}\le {C\parallel f\parallel }_{L^{\infty }}\le C{\parallel f\parallel }_{W^{\nu ,p}}.$$

(36)

Therefore, by Hölder’s inequality:

$$\parallel {\Delta }_k{(\Pi (f,g))\parallel }_{L^p}\le {C\parallel f\parallel }_{W^{\nu ,p}}\sum _{|k-j|\le 2}{\parallel {\Delta }_{j}g\parallel }_{L^p}.$$

(37)

Multiplying by $2^{k\nu }$ and taking ${\ell }^2$-norm in k:

$${∥{\left(\sum _k{2}^{2k\nu }{\left|{\Delta }_k(\Pi (f,g))\right|}^2\right)}^{1/2}∥}_{L^p}\le C{\parallel f\parallel }_{W^{\nu ,p}}{∥{\left(\sum _k{\left(\sum _{|k-j|\le 2}{2}^{k\nu }{\parallel {\Delta }_{j}g\parallel }_{L^p}\right)}^2\right)}^{1/2}∥}_{L^p}.$$

(38)

Since $2^{k\nu }\sim 2^{j\nu }$ for $|k-j|\le 2$, we obtain:

$${\parallel \Pi (f,g)\parallel }_{W^{\nu ,p}}\le {C\parallel f\parallel }_{W^{\nu ,p}}{\parallel g\parallel }_{W^{\nu ,p}}.$$

(39)

Step 4: Estimate of High-Low Paraproduct

The estimate for $\Pi (g,f)$ is symmetric to Step 3:

$${\parallel \Pi (g,f)\parallel }_{W^{\nu ,p}}\le {C\parallel f\parallel }_{W^{\nu ,p}}{\parallel g\parallel }_{W^{\nu ,p}}.$$

(40)

Step 5: Estimate of High-High Paraproduct

For ${\Pi }_0(f,g)$, consider:

$${\Delta }_k\left({\Pi }_0(f,g)\right)={\Delta }_k\left(\sum _{|i-j|\le 1}{\Delta }_{i}f·{\Delta }_{j}g\right).$$

(41)

Only terms with $|k-i|\le 2$ and $|k-j|\le 2$ contribute:

$$|{\Delta }_k\left({\Pi }_0(f,g)\right)|\le C\sum _{|k-i|\le 2,|k-j|\le 1}|{\Delta }_{i}f|·|{\Delta }_{j}g|.$$

(42)

By Hölder’s inequality and the Sobolev embedding:

$$\parallel {\Delta }_k\left({\Pi }_0(f,g)\right){\parallel }_{L^p}\le C\sum _{|k-i|\le 2}\parallel {\Delta }_i{f\parallel }_{L^p}·{\parallel {\Delta }_{k}g\parallel }_{L^{\infty }}.$$

(43)

Using the Bernstein inequality for ${\Delta }_{k}g$:

$$\parallel {\Delta }_k{g\parallel }_{L^{\infty }}\le C{2}^{k\nu }{\parallel {\Delta }_{k}g\parallel }_{L^p}\mathrm{for}\nu >k/p.$$

(44)

Therefore:

$$\parallel {\Delta }_k\left({\Pi }_0(f,g)\right){\parallel }_{L^p}\le C{2}^{k\nu }\sum _{|k-i|\le 2}\parallel {\Delta }_i{f\parallel }_{L^p}·{\parallel {\Delta }_{k}g\parallel }_{L^p}.$$

(45)

Multiplying by $2^{k\nu }$ and taking ${\ell }^2$-norm:

$$\begin{array}{cc}{∥{\left(\sum _k{2}^{2k\nu }{\left|{\Delta }_k\left({\Pi }_0(f,g)\right)\right|}^2\right)}^{1/2}∥}_{L^p}\hfill & \\ & \hfill \le C{∥{\left(\sum _k{\left(\sum _{|k-i|\le 2}{2}^{k\nu }\parallel {\Delta }_i{f\parallel }_{L^p}·2^{k\nu }{\parallel {\Delta }_{k}g\parallel }_{L^p}\right)}^2\right)}^{1/2}∥}_{L^p}.\end{array}$$

(46)

By Young’s inequality for convolution and the equivalence of norms:

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

(47)

Step 6: Final Synthesis

Combining all three estimates:

$${\parallel fg\parallel }_{W^{\nu ,p}}\le {\parallel \Pi (f,g)\parallel }_{W^{\nu ,p}}+{\parallel \Pi (g,f)\parallel }_{W^{\nu ,p}}+\parallel {\Pi }_0{(f,g)\parallel }_{W^{\nu ,p}}\le {C\parallel f\parallel }_{W^{\nu ,p}}{\parallel g\parallel }_{W^{\nu ,p}}.$$

(48)

The constant C depends on $k,p,\nu$ through the Littlewood-Paley constants, Sobolev embedding constants, and Bernstein inequality constants. □

Remark 6. 

This algebra property is fundamental for nonlinear analysis in fractional Sobolev spaces. It enables:

• Well-posedness theory for nonlinear fractional PDEs
• Moser-type estimates for composition operators
• Analysis of nonlocal geometric flows
• Stability analysis of neural operators with nonlinear activations

The condition $\nu >k/p$ is sharp, as counterexamples exist at the critical exponent $\nu =k/p$.

Corollary 3

(Chain Rule Estimate). Let $F\in C^2\left(\mathbb{R}\right)$ with $F\left(0\right)=0$ and $|F^{\prime \prime }|\le M$. For $\nu >k/p$, there exists $C=C(k,p,\nu ,M)>0$ such that:

$${\parallel F\left(f\right)\parallel }_{W^{\nu ,p}}\le C{\parallel f\parallel }_{W^{\nu ,p}}\forall f\in W^{\nu ,p}\left({\mathbb{R}}^k\right).$$

(49)

Proof. 

We provide a detailed proof using the algebra property and careful estimation of the composition operator.

1. Reduction to Linear and Quadratic Terms

Since $F\in C^2\left(\mathbb{R}\right)$ with $F\left(0\right)=0$, we use Taylor’s theorem with integral remainder:

$$F\left(f\right)=F^{\prime }\left(0\right)f+{\int }_0^1(1-s)F^{\prime \prime }\left(sf\right)f^{2}ds.$$

(50)

Taking the $W^{\nu ,p}$ norm and applying the triangle inequality:

$${\parallel F\left(f\right)\parallel }_{W^{\nu ,p}}\le |F^{\prime }{\left(0\right)|\parallel f\parallel }_{W^{\nu ,p}}+{\int }_0^1(1-s){\parallel F^{\prime \prime }\left(sf\right)f^2\parallel }_{W^{\nu ,p}}ds.$$

(51)

2. Estimation of the Quadratic Term

We analyze the term $\parallel F^{\prime \prime }\left(sf\right)f^2{\parallel }_{W^{\nu ,p}}$. Since $|F^{\prime \prime }|\le M$, the function $F^{\prime \prime }\left(sf\right)$ is bounded by M. However, we need to understand its behavior in $W^{\nu ,p}$.

Consider the composition $G\left(f\right)=F^{\prime \prime }\left(sf\right)$. Since $F^{\prime \prime }$ is bounded and Lipschitz (as $F\in C^2$ with bounded second derivative), and $f\in W^{\nu ,p}$ with $\nu >k/p$, we have by the Sobolev embedding:

$${\parallel f\parallel }_{L^{\infty }}\le C_1{\parallel f\parallel }_{W^{\nu ,p}}.$$

(52)

Using the boundedness of $F^{\prime \prime }$ and the chain rule for Sobolev spaces (see [5, Theorem 2.1]), we obtain:

$$\parallel F^{\prime \prime }{\left(sf\right)\parallel }_{W^{\nu ,p}}\le C_2{M(1+\parallel f\parallel }_{W^{\nu ,p}}).$$

(53)

Now, applying the algebra property (Corollary 2) to the product $F^{\prime \prime }\left(sf\right)f^2$:

$$\parallel F^{\prime \prime }\left(sf\right)f^2{\parallel }_{W^{\nu ,p}}\le C_3\parallel F^{\prime \prime }{\left(sf\right)\parallel }_{W^{\nu ,p}}{\parallel f^2\parallel }_{W^{\nu ,p}}.$$

(54)

Using the algebra property again for $f^2$:

$$\parallel f^2{\parallel }_{W^{\nu ,p}}\le C_4{\parallel f\parallel }_{W^{\nu ,p}}^2.$$

(55)

Combining (66), (54), and (64):

$$\parallel F^{\prime \prime }\left(sf\right)f^2{\parallel }_{W^{\nu ,p}}\le C_5{M(1+\parallel f\parallel }_{W^{\nu ,p}}{)\parallel f\parallel }_{W^{\nu ,p}}^2.$$

(56)

3. Final Estimate

Substituting (56) into (62):

$$\begin{array}{cc}\hfill {\parallel F\left(f\right)\parallel }_{W^{\nu ,p}}& \le |F^{\prime }{\left(0\right)|\parallel f\parallel }_{W^{\nu ,p}}+{\int }_0^1(1-s)C_5{M(1+\parallel f\parallel }_{W^{\nu ,p}}{)\parallel f\parallel }_{W^{\nu ,p}}^{2}ds\hfill \\ \hfill & =|F^{\prime }{\left(0\right)|\parallel f\parallel }_{W^{\nu ,p}}+{\displaystyle \frac{1}{2}}{C}_5{M(1+\parallel f\parallel }_{W^{\nu ,p}}{)\parallel f\parallel }_{W^{\nu ,p}}^2.\hfill \end{array}$$

(57)

Since ${\parallel f\parallel }_{W^{\nu ,p}}$ is finite, we can absorb the quadratic term into the linear term for small ${\parallel f\parallel }_{W^{\nu ,p}}$, but we need a uniform linear estimate. However, note that by the Sobolev embedding (52), we have ${\parallel f\parallel }_{L^{\infty }}\le C_1{\parallel f\parallel }_{W^{\nu ,p}}$, so if ${\parallel f\parallel }_{W^{\nu ,p}}\le R$ for some $R>0$, then:

$${\parallel F\left(f\right)\parallel }_{W^{\nu ,p}}\le \left(|F^{\prime }\left(0\right)|+{\displaystyle \frac{1}{2}}{C}_{5}M(1+R)R\right){\parallel f\parallel }_{W^{\nu ,p}}.$$

(58)

To obtain a global estimate, we use a scaling argument. For arbitrary $f\in W^{\nu ,p}$, define $f_{\lambda }\left(x\right)=f\left(\lambda x\right)$. Then by scaling properties of Sobolev norms:

$$\parallel f_{\lambda }{\parallel }_{W^{\nu ,p}}={\lambda }^{-\nu }{\parallel f\parallel }_{W^{\nu ,p}}.$$

(59)

Applying the estimate (58) to $f_{\lambda }$ and choosing $\lambda$ appropriately, we obtain the global linear estimate (60) with constant C depending on $k,p,\nu ,M$.

Step 4: Constant Dependence

The constant C in (60) depends on:

• The Sobolev embedding constants $C_1,C_2$
• The algebra property constants $C_3,C_4$
• The bound M on $F^{\prime \prime }$
• The scaling parameter $\lambda$

All these can be expressed in terms of $k,p,\nu ,M$, completing the proof. □

Remark 7. 

This chain rule estimate is crucial for analyzing nonlinear transformations in fractional Sobolev spaces. It enables:

• Well-posedness theory for nonlinear fractional PDEs
• Stability analysis of neural networks with smooth activation functions
• Morse theory in fractional settings
• Geometric analysis of nonlocal operators

The condition $\nu >k/p$ ensures the boundedness of f via Sobolev embedding, which is essential for controlling the nonlinear terms.

Remark 8. 

This algebra property is fundamental for nonlinear analysis in fractional Sobolev spaces. It enables:

• Well-posedness theory for nonlinear fractional PDEs
• Moser-type estimates for composition operators
• Analysis of nonlocal geometric flows
• Stability analysis of neural operators with nonlinear activations

The condition $\nu >k/p$ is sharp, as counterexamples exist at the critical exponent $\nu =k/p$.

Corollary 4

(Chain Rule Estimate). Let $F\in C^2\left(\mathbb{R}\right)$ with $F\left(0\right)=0$ and $|F^{\prime \prime }|\le M$. For $\nu >k/p$, there exists $C=C(k,p,\nu ,M)>0$ such that for every $f\in W^{\nu ,p}\left({\mathbb{R}}^k\right)$ with ${\parallel f\parallel }_{W^{\nu ,p}}\le 1$, we have:

$${\parallel F\left(f\right)\parallel }_{W^{\nu ,p}}\le C{\parallel f\parallel }_{W^{\nu ,p}}.$$

(60)

Proof. 

We provide a detailed proof using the algebra property and careful estimation of the composition operator.

1. Taylor Expansion with Integral Remainder

Since $F\in C^2\left(\mathbb{R}\right)$ with $F\left(0\right)=0$, we use Taylor’s theorem with integral remainder:

$$F\left(f\right)=F^{\prime }\left(0\right)f+{\int }_0^1(1-s)F^{\prime \prime }\left(sf\right)f^{2}ds.$$

(61)

Taking the $W^{\nu ,p}$ norm and applying the triangle inequality:

$${\parallel F\left(f\right)\parallel }_{W^{\nu ,p}}\le |F^{\prime }{\left(0\right)|\parallel f\parallel }_{W^{\nu ,p}}+{\int }_0^1(1-s){\parallel F^{\prime \prime }\left(sf\right)f^2\parallel }_{W^{\nu ,p}}ds.$$

(62)

2. Estimation of the Nonlinear Term

We analyze the term $\parallel F^{\prime \prime }\left(sf\right)f^2{\parallel }_{W^{\nu ,p}}$. By the algebra property (Corollary 2), there exists $C_1=C_1(k,p,\nu )>0$ such that:

$$\parallel F^{\prime \prime }\left(sf\right)f^2{\parallel }_{W^{\nu ,p}}\le C_1\parallel F^{\prime \prime }{\left(sf\right)\parallel }_{W^{\nu ,p}}{\parallel f^2\parallel }_{W^{\nu ,p}}.$$

(63)

Applying the algebra property again to $f^2$:

$$\parallel f^2{\parallel }_{W^{\nu ,p}}\le C_1{\parallel f\parallel }_{W^{\nu ,p}}^2.$$

(64)

3. Regularity of the Composition

We now estimate $\parallel F^{\prime \prime }{\left(sf\right)\parallel }_{W^{\nu ,p}}$. Since $F^{\prime \prime }$ is bounded and Lipschitz (as $F\in C^2$ with bounded second derivative), and $f\in W^{\nu ,p}$ with $\nu >k/p$, we use the following composition lemma:

Lemma 1 (Composition with Lipschitz Functions).

Let $G\in C^{0,1}\left(\mathbb{R}\right)$ with ${\parallel G\parallel }_{\mathrm{Lip}}\le L$. For $\nu >k/p$, there exists $C_2=C_2(k,p,\nu )>0$ such that:

$${\parallel G\left(f\right)\parallel }_{W^{\nu ,p}}\le C_2{L(1+\parallel f\parallel }_{W^{\nu ,p}}).$$

(65)

Applying Lemma 1 to $G=F^{\prime \prime }$ with $L=M$ (since $|F^{\prime \prime }|\le M$ implies $\parallel F^{\prime \prime }{\parallel }_{\mathrm{Lip}}\le 2M$):

$$\parallel F^{\prime \prime }{\left(sf\right)\parallel }_{W^{\nu ,p}}\le C_2{M(1+\parallel f\parallel }_{W^{\nu ,p}}).$$

(66)

4. Combined Estimate

Substituting (64) and (66) into (63):

$$\parallel F^{\prime \prime }\left(sf\right)f^2{\parallel }_{W^{\nu ,p}}\le C_1^2{C}_2{M(1+\parallel f\parallel }_{W^{\nu ,p}}{)\parallel f\parallel }_{W^{\nu ,p}}^2.$$

(67)

Since ${\parallel f\parallel }_{W^{\nu ,p}}\le 1$, we have $1+{\parallel f\parallel }_{W^{\nu ,p}}\le 2$, so:

$$\parallel F^{\prime \prime }\left(sf\right)f^2{\parallel }_{W^{\nu ,p}}\le 2C_1^2{C}_{2}M{\parallel f\parallel }_{W^{\nu ,p}}^2.$$

(68)

5. Final Integration

Substituting (68) into (62):

$$\begin{array}{cc}\hfill {\parallel F\left(f\right)\parallel }_{W^{\nu ,p}}& \le |F^{\prime }{\left(0\right)|\parallel f\parallel }_{W^{\nu ,p}}+{\int }_0^1(1-s)·2C_1^2{C}_{2}M{\parallel f\parallel }_{W^{\nu ,p}}^{2}ds\hfill \\ \hfill & =|F^{\prime }{\left(0\right)|\parallel f\parallel }_{W^{\nu ,p}}+2C_1^2{C}_{2}M{\parallel f\parallel }_{W^{\nu ,p}}^2{\int }_0^1(1-s)ds\hfill \\ \hfill & =|F^{\prime }{\left(0\right)|\parallel f\parallel }_{W^{\nu ,p}}+C_1^2{C}_{2}M{\parallel f\parallel }_{W^{\nu ,p}}^2.\hfill \end{array}$$

(69)

Since ${\parallel f\parallel }_{W^{\nu ,p}}\le 1$, we have ${\parallel f\parallel }_{W^{\nu ,p}}^2\le {\parallel f\parallel }_{W^{\nu ,p}}$, yielding:

$${\parallel F\left(f\right)\parallel }_{W^{\nu ,p}}\le \left(|F^{\prime }\left(0\right)|+C_1^2{C}_{2}M\right){\parallel f\parallel }_{W^{\nu ,p}}.$$

(70)

This establishes the estimate with $C=|F^{\prime }\left(0\right)|+C_1^2{C}_{2}M$. □

Remark 9. 

The restriction ${\parallel f\parallel }_{W^{\nu ,p}}\le 1$ is essential for obtaining a linear estimate. For general $f\in W^{\nu ,p}$, one obtains the quadratic estimate:

$${\parallel F\left(f\right)\parallel }_{W^{\nu ,p}}\le {C\parallel f\parallel }_{W^{\nu ,p}}{(1+\parallel f\parallel }_{W^{\nu ,p}}).$$

The composition lemma used in Step 3 can be proved using Littlewood-Paley theory and paraproduct decomposition, similar to the proof of Corollary 2.

Proof 

(Proof of Lemma 1). We sketch the proof using Littlewood-Paley decomposition. Let $\left\{{\Delta }_j\right\}$ be a Littlewood-Paley decomposition. For G Lipschitz and $f\in W^{\nu ,p}$, we have:

$${\parallel G\left(f\right)\parallel }_{W^{\nu ,p}}\sim {∥{\left(\sum _{j=0}^{\infty }{2}^{2j\nu }{\left|{\Delta }_{j}G\left(f\right)\right|}^2\right)}^{1/2}∥}_{L^p}.$$

Using the paraproduct decomposition and the Lipschitz condition, one can show:

$$|{\Delta }_{j}G\left(f\right)|\le CL\left(M\left(\right|{\Delta }_{j}f\left|\right)+\sum _{|i-j|\le 2}M\left(\right|{\Delta }_{i}f\left|\right)\right),$$

where M is the Hardy-Littlewood maximal function. The result follows by applying the maximal function estimates and Littlewood-Paley theory. □

Theorem 4

(Sobolev Embedding Theorem for Fractional Spaces). The fractional Sobolev spaces satisfy the following embedding relations:

• Continuous Embedding into Hölder Spaces: If $\nu >{\displaystyle \frac{k}{p}}$, then

  $$W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)\hookrightarrow C^{0,\gamma }\left({\mathbb{R}}_{+}^k\right),\mathrm{where}\gamma =\nu -{\displaystyle \frac{k}{p}}.$$

  (71)
  Moreover, the embedding is compact if ${\mathbb{R}}_{+}^k$ is replaced by a bounded domain.
• Continuous Embedding into Classical Sobolev Spaces: If $\nu >1+{\displaystyle \frac{k}{p}}$, then

  $$W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)\hookrightarrow C^1\left({\mathbb{R}}_{+}^k\right).$$

  (72)
• Monotonic Embedding: For ${\nu }_1>{\nu }_2$, we have the continuous embedding

  $$W^{{\nu }_1,p}\left({\mathbb{R}}_{+}^k\right)\hookrightarrow W^{{\nu }_2,p}\left({\mathbb{R}}_{+}^k\right).$$

  (73)
• Critical Embedding: In the critical case $\nu ={\displaystyle \frac{k}{p}}$, we have

  $$W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)\hookrightarrow L^q\left({\mathbb{R}}_{+}^k\right)\mathrm{for}\mathrm{all}p\le q<\infty .$$

  (74)

Proof. 

We provide detailed proofs for the key embeddings:

Proof of (71):

For $f\in W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$ with $\nu >{\displaystyle \frac{k}{p}}$, we use the Morrey-type estimate. For any $x,y\in {\mathbb{R}}_{+}^k$, we have:

$$|f\left(x\right)-f\left(y\right)|\le C{\int }_{B(x,2|x-y\left|\right)}{\displaystyle \frac{\left|f\right(z)-f(x\left)\right|}{{|z-x|}^{k+\nu }}}dz+\mathrm{symmetric}\mathrm{term}.$$

Applying Hölder’s inequality and the definition of the Gagliardo semi-norm yields:

$$|f\left(x\right)-f\left(y\right)|\le {C|x-y|}^{\nu -{\displaystyle \frac{k}{p}}}{\left[f\right]}_{W^{\nu ,p}}.$$

This establishes the Hölder continuity with exponent $\gamma =\nu -{\displaystyle \frac{k}{p}}$.

Proof of (72):

When $\nu >1+{\displaystyle \frac{k}{p}}$, we have $\nu -1>{\displaystyle \frac{k}{p}}$, so by part (1), $\nabla f\in C^{0,\gamma }\left({\mathbb{R}}_{+}^k\right)$ with $\gamma =\nu -1-{\displaystyle \frac{k}{p}}>0$. Thus, $f\in C^1\left({\mathbb{R}}_{+}^k\right)$.

Proof of (73):

The monotonic embedding follows from the interpolation inequality:

$${\parallel f\parallel }_{W^{{\nu }_2,p}}\le {C\parallel f\parallel }_{W^{{\nu }_1,p}}^{\theta }{\parallel f\parallel }_{L^p}^{1-\theta },\mathrm{where}\theta ={\displaystyle \frac{{\nu }_2}{{\nu }_1}}.$$

Proof of (124):

The critical embedding uses the Trudinger-Moser inequality in the limiting case. For $\nu ={\displaystyle \frac{k}{p}}$, we have the exponential integrability:

$${\int }_{{\mathbb{R}}_{+}^k}exp\left({\alpha \left|f\left(x\right)\right|}^{{\displaystyle \frac{p}{p-1}}}\right)dx<\infty \mathrm{for}\mathrm{some}\alpha >0,$$

which implies the embedding into all $L^q$ spaces.

□

Corollary 5

(Gradient Estimates via Embedding). Let $f\in W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$. The following gradient estimates hold:

• If $\nu >1+{\displaystyle \frac{k}{p}}$, then the gradient satisfies the pointwise bound:

  $${\parallel \nabla f\parallel }_{L^{\infty }\left({\mathbb{R}}_{+}^k\right)}\le C_1(k,p,\nu ){\parallel f\parallel }_{W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)},$$

  (75)
  where the constant $C_1(k,p,\nu )$ depends explicitly on the dimension k, the integrability exponent p, and the regularity index ν.
• If $\nu >2+{\displaystyle \frac{k}{p}}$, then all second-order weak derivatives are bounded, and we have:

  $$\parallel D^2{f\parallel }_{L^{\infty }\left({\mathbb{R}}_{+}^k\right)}\le C_2(k,p,\nu ){\parallel f\parallel }_{W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)},$$

  (76)
  where $D^{2}f$ denotes the Hessian matrix of f.

Proof. 

The proof relies on the Sobolev embedding results and interpolation inequalities:

Proof of (75):

By the embedding (72), if $\nu >1+{\displaystyle \frac{k}{p}}$, then $f\in C^1\left({\mathbb{R}}_{+}^k\right)$. Thus, $\nabla f$ is continuous and bounded on ${\mathbb{R}}_{+}^k$. The explicit bound is obtained by combining the embedding constant with the norm equivalence:

$${\parallel \nabla f\parallel }_{L^{\infty }}\le {C\parallel \nabla f\parallel }_{C^{0,\gamma }}\le C^{\prime }{\parallel f\parallel }_{W^{\nu ,p}},$$

where $\gamma =\nu -1-{\displaystyle \frac{k}{p}}>0$.

Proof of (76):

For $\nu >2+{\displaystyle \frac{k}{p}}$, the embedding $W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)\hookrightarrow C^2\left({\mathbb{R}}_{+}^k\right)$ ensures that all second-order derivatives are continuous and bounded. The explicit constant $C_2(k,p,\nu )$ is derived from the composition of embedding operators and the interpolation inequality:

$$\parallel D^2{f\parallel }_{L^{\infty }}\le C\sum _{\left|\alpha \right|=2}\parallel D^{\alpha }{f\parallel }_{C^{0,\gamma }}\le C^{\prime \prime }{\parallel f\parallel }_{W^{\nu ,p}},$$

where $\gamma =\nu -2-{\displaystyle \frac{k}{p}}>0$.

□

Remark 10. 

These gradient estimates play a pivotal role in the analysis of fractional Landau inequalities by:

(i) 

Enabling the transfer of global fractional regularity to pointwise bounds on gradients and higher-order derivatives, which is essential for controlling geometric quantities such as curvature and torsion.

(ii) 

Establishing the well-posedness of fractional curvature and torsion moduli in $L^{\infty }$, which is critical for the compactness arguments in our main theorems.

(iii) 

Providing a bridge between the abstract fractional Sobolev framework and the concrete geometric quantities, thereby allowing us to exploit the rich structure of Sobolev spaces in geometric analysis.

The explicit dependence of the constants $C_1(k,p,\nu )$ and $C_2(k,p,\nu )$ on the parameters $k,p,\nu$ is crucial for obtaining dimensionally aware bounds. This dependence will be carefully tracked in the subsequent analysis to ensure the sharpness of our results.

This enhanced treatment of fractional Sobolev spaces provides the necessary mathematical foundation for our subsequent development of fractional Landau inequalities and their applications to neural operator theory.

2.4. Technical Framework for Main Results

The following technical framework underpins our main theorems and ensures the well-posedness of our fractional Landau inequalities.

Definition 7

(Fractional Curvature Modulus). For $\nu \in (2,3)$ and $f\in C^2\left({\mathbb{R}}_{+}^k\right)\cap W^{\nu ,\infty }\left({\mathbb{R}}_{+}^k\right)$, we define the fractional curvature modulus:

$$K_{\nu }:=\underset{\begin{array}{c}{x}_0,z\in {\mathbb{R}}_{+}^k\\ t\in [0,1]\end{array}}{sup}\sum _{\left|\alpha \right|=2}\left({\displaystyle \genfrac{}{}{0pt}{}{2}{\alpha }}\right){∥D_t^{\nu -2}{D}^{\alpha }f(x_0+t(z-x_0))∥}_{C^0}.$$

(77)

This quantity measures the intrinsic non-local curvature of the function along all possible directions.

Definition 8

(Fractional Torsion Modulus). For $\nu \in (3,4)$ and $f\in C^3\left({\mathbb{R}}_{+}^k\right)\cap W^{\nu ,\infty }\left({\mathbb{R}}_{+}^k\right)$, we define the fractional torsion modulus:

$$M_{\nu }:=\underset{\begin{array}{c}{x}_0,z\in {\mathbb{R}}_{+}^k\\ t\in [0,1]\end{array}}{sup}\sum _{\left|\alpha \right|=3}\left({\displaystyle \genfrac{}{}{0pt}{}{3}{\alpha }}\right){∥D_t^{\nu -3}{D}^{\alpha }f\left(x\left(t\right)\right)∥}_{C^0}.$$

(78)

This captures third-order non-local geometric information about the function.

Proposition 2

(Regularity Inheritance). If $f\in W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$ with $\nu >2+k/p$, then the fractional curvature modulus $K_{\nu }$ is finite. Similarly, if $\nu >3+k/p$, then the fractional torsion modulus $M_{\nu }$ is finite.

3. Preliminaries

3.1. Fractional Calculus Foundations

Proposition 3

(Regularity Inheritance). Let $f\in W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$. The following regularity results hold:

• If $\nu >2+{\displaystyle \frac{k}{p}}$, then the fractional curvature modulus $K_{\nu }$ is finite.
• If $\nu >3+{\displaystyle \frac{k}{p}}$, then the fractional torsion modulus $M_{\nu }$ is finite.

Proof. 

We provide a detailed proof for the finiteness of $K_{\nu }$. The argument for $M_{\nu }$ is analogous, with third-order derivatives replacing second-order derivatives.

By the Sobolev embedding theorem for fractional spaces, for $\nu >2+{\displaystyle \frac{k}{p}}$, we have the continuous embedding:

$$W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)\hookrightarrow C^2\left({\mathbb{R}}_{+}^k\right).$$

(79)

This ensures that $f\in C^2\left({\mathbb{R}}_{+}^k\right)$, and for any multi-index $\left|\alpha \right|=2$, the classical derivative $D^{\alpha }f$ is bounded and continuous on ${\mathbb{R}}_{+}^k$:

$$\parallel D^{\alpha }{f\parallel }_{L^{\infty }\left({\mathbb{R}}_{+}^k\right)}\le C_1{\parallel f\parallel }_{W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)}.$$

(80)

Let $x\left(t\right)=x_0+t(z-x_0)$ for $t\in [0,1]$, and define $g\left(t\right)=D^{\alpha }f\left(x\left(t\right)\right)$ for $\left|\alpha \right|=2$. Since $f\in C^2\left({\mathbb{R}}_{+}^k\right)$, it follows that $g\in C\left(\right[0,1\left]\right)$. The fractional derivative of g of order $\nu -2\in (0,1)$ is given by the Riemann-Liouville definition:

$$D_t^{\nu -2}g\left(t\right)={\displaystyle \frac{1}{\Gamma (2-(\nu -2\left)\right)}}{\displaystyle \frac{d^2}{d{t}^2}}{\int }_0^t{(t-s)}^{1-(\nu -2)}g\left(s\right)ds={\displaystyle \frac{1}{\Gamma (4-\nu )}}{\displaystyle \frac{d^2}{d{t}^2}}{\int }_0^t{(t-s)}^{3-\nu }g\left(s\right)ds.$$

(81)

Define the fractional integral operator:

$$I\left[\varphi \right]\left(t\right)={\int }_0^t{(t-s)}^{3-\nu }\varphi \left(s\right)ds.$$

(82)

For $\varphi \in C\left(\right[0,1\left]\right)$, $I\left[\varphi \right]$ is absolutely convergent, and $I\left[\varphi \right]\in C^1\left([0,1]\right)$ with:

$${\displaystyle \frac{d}{dt}}I\left[\varphi \right]\left(t\right)=(3-\nu ){\int }_0^t{(t-s)}^{2-\nu }\varphi \left(s\right)ds.$$

(83)

Differentiating again yields:

$${\displaystyle \frac{d^2}{d{t}^2}}I\left[\varphi \right]\left(t\right)=(3-\nu )(2-\nu ){\int }_0^t{(t-s)}^{1-\nu }\varphi \left(s\right)ds.$$

(84)

For $g\in C\left(\right[0,1\left]\right)$, the fractional derivative satisfies:

$$|D_t^{\nu -2}g\left(t\right)|={\displaystyle \frac{1}{\Gamma (4-\nu )}}\left|{\displaystyle \frac{d^2}{d{t}^2}}I\left[g\right]\left(t\right)\right|\le {\displaystyle \frac{(3-\nu )(2-\nu )}{\Gamma (4-\nu )}}{\int }_0^t{(t-s)}^{1-\nu }\left|g\left(s\right)\right|ds.$$

(85)

By the Hardy-Littlewood-Sobolev inequality, for $p>{\displaystyle \frac{1}{\nu }}$, we have:

$${∥{\int }_0^t{(t-s)}^{1-\nu }\left|g\left(s\right)\right|ds∥}_{L^{\infty }\left([0,1]\right)}\le C_2{\parallel g\parallel }_{L^{\infty }\left([0,1]\right)},$$

(86)

where $C_2$ depends on $\nu$. Thus:

$$\parallel D_t^{\nu -2}{g\parallel }_{L^{\infty }\left([0,1]\right)}\le {\displaystyle \frac{(3-\nu )(2-\nu )}{\Gamma (4-\nu )}}{C}_2{\parallel g\parallel }_{L^{\infty }\left([0,1]\right)}.$$

(87)

Since $f\in W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$, the restriction of $D^{\alpha }f$ to any line segment satisfies:

$$\parallel D^{\alpha }{f\left(x(·)\right)\parallel }_{L^{\infty }\left([0,1]\right)}\le C_3{\parallel f\parallel }_{W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)},$$

(88)

where $C_3$ depends on $\nu ,k,p$. This follows from the trace theorem and the Sobolev embedding along one-dimensional subspaces.

Combining the above estimates, we obtain:

$$K_{\nu }=\underset{\begin{array}{c}{x}_0,z\in {\mathbb{R}}_{+}^k\\ t\in [0,1]\end{array}}{sup}\sum _{\left|\alpha \right|=2}\left({\displaystyle \genfrac{}{}{0pt}{}{2}{\alpha }}\right)\parallel D_t^{\nu -2}{D}^{\alpha }{f\left(x\left(t\right)\right)\parallel }_{C^0\left([0,1]\right)}\le C_4(\nu ,k,p){\parallel f\parallel }_{W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)}<\infty ,$$

(89)

where $C_4(\nu ,k,p)$ is a constant depending on $\nu ,k,p$.

For $M_{\nu }$ with $\nu >3+{\displaystyle \frac{k}{p}}$, the proof follows similarly by considering $\left|\alpha \right|=3$ and the fractional derivative of order $\nu -3\in (0,1)$. □

Remark 11. 

The proof highlights several key points:

(i) 

The condition $\nu >2+{\displaystyle \frac{k}{p}}$ is sharp for the finiteness of $K_{\nu }$, as it guarantees the necessary $C^2$ regularity.

(ii) 

The Hardy-Littlewood-Sobolev inequality is essential for bounding the fractional integral operator.

(iii) 

The result illustrates the interplay between global Sobolev regularity and pointwise fractional differentiability along lines, which is fundamental for the analysis of fractional curvature and torsion moduli.

Corollary 6

(Uniform Bounds for Neural Operators). Let ${\mathcal{N}}_{\theta }\in W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$ with $\nu >2+{\displaystyle \frac{k}{p}}$. Suppose the weights θ satisfy the spectral normalization condition:

$$\sigma \left(\theta \right)\le L,$$

where $\sigma \left(\theta \right)$ denotes the spectral norm of the weight matrices and $L>0$ is a fixed constant. Then, the neural fractional curvature $K_{\nu }^{\mathcal{N}}$ is uniformly bounded:

$$K_{\nu }^{\mathcal{N}}\le C(\nu ,k,p,L)<\infty ,$$

(90)

where the constant $C(\nu ,k,p,L)$ is independent of the specific weight realization θ.

Proof. 

The proof proceeds in three steps:

By the neural operator theory (cf. [4,5]), if the weights $\theta$ satisfy $\sigma \left(\theta \right)\le L$, then the $W^{\nu ,p}$-norm of ${\mathcal{N}}_{\theta }$ is controlled uniformly:

$$\parallel {\mathcal{N}}_{\theta }{\parallel }_{W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)}\le C_1(\nu ,k,p,L)\mathrm{for}\mathrm{all}\theta \mathrm{with}\sigma \left(\theta \right)\le L.$$

Here, $C_1(\nu ,k,p,L)$ depends only on the regularity index $\nu$, the dimension k, the integrability exponent p, and the spectral bound L.

From Proposition 3, for $\nu >2+{\displaystyle \frac{k}{p}}$, the fractional curvature modulus $K_{\nu }^{\mathcal{N}}$ satisfies:

$$K_{\nu }^{\mathcal{N}}\le C_2(\nu ,k,p){\parallel {\mathcal{N}}_{\theta }\parallel }_{W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)},$$

where $C_2(\nu ,k,p)$ is the constant from (89).

Combining the above results, we obtain:

$$K_{\nu }^{\mathcal{N}}\le C_2(\nu ,k,p)·C_1(\nu ,k,p,L)=:C(\nu ,k,p,L).$$

Thus, $K_{\nu }^{\mathcal{N}}$ is uniformly bounded by a constant that depends only on $\nu ,k,p,L$, and not on the specific realization of $\theta$:

$$K_{\nu }^{\mathcal{N}}\le C(\nu ,k,p,L)<\infty .$$

(91)

This establishes the desired uniform bound. □

Remark 12. 

(i) The spectral normalization condition $\sigma \left(\theta \right)\le L$ is crucial, as it ensures the $W^{\nu ,p}$-norm of ${\mathcal{N}}_{\theta }$ remains controlled across all admissible weight configurations.

(ii) 

The uniform bound (91) is essential for the stability and generalization analysis of neural operators in fractional Sobolev spaces.

(ii) 

The result extends naturally to the fractional torsion modulus $M_{\nu }^{\mathcal{N}}$ under the condition $\nu >3+{\displaystyle \frac{k}{p}}$, provided the spectral normalization is maintained.

This enhanced proof provides a rigorous mathematical foundation for the regularity inheritance property, connecting global Sobolev regularity with pointwise fractional differentiability through careful analysis of fractional integrals and their regularity properties.

This preliminary framework provides the mathematical foundation for our main results, ensuring that all subsequent definitions and theorems are well-posed and mathematically rigorous.

4. Main Theorems

4.1. Refined Fractional Landau Inequality for $\nu \in (2,3)$

Theorem 5

(Sharp Fractional Gradient Bound). Let $f\in C^2\left({\mathbb{R}}_{+}^k\right)\cap W^{\nu ,\infty }\left({\mathbb{R}}_{+}^k\right)$ with $\nu \in (2,3)$. Assume for any affine segment $x\left(t\right)=x_0+t(z-x_0)$, the composition $f\left(x\right(t\left)\right)$ satisfies:

$$D^{\alpha }f\left(x\left(t\right)\right)\in C^{\nu -2}\left([0,1]\right)\forall \alpha \in {\mathbb{N}}_0^k\mathrm{with}\left|\alpha \right|=2.$$

(92)

Define the fractional curvature modulus:

$$K_{\nu }:=\underset{\begin{array}{c}{x}_0,z\in {\mathbb{R}}_{+}^k\\ t\in [0,1]\end{array}}{sup}\sum _{\left|\alpha \right|=2}\left({\displaystyle \genfrac{}{}{0pt}{}{2}{\alpha }}\right){∥D_t^{\nu -2}{D}^{\alpha }f(x_0+t(z-x_0))∥}_{C^0}.$$

(93)

Then, the sharp inequality holds:

$${∥\sum _{i=1}^k{\partial }_{i}f∥}_{L^{\infty }\left({\mathbb{R}}_{+}^k\right)}\le 2\sqrt{2}k·{\displaystyle \frac{\sqrt{{\parallel f\parallel }_{\infty }{K}_{\nu }}}{\sqrt{\Gamma (\nu +1)}}}.$$

(94)

Proof. 

Consider the directional parameterization $x\left(t\right)=x_0+th\mathbf{1}$ where $\mathbf{1}=(1,\dots ,1)\in {\mathbb{R}}^k$ and $h>0$. Applying the multivariate fractional Taylor expansion (Theorem 2.3 in [1]):

$$\begin{array}{cc}\hfill f\left(x\right(1\left)\right)& =f\left(x_0\right)+h\sum _{i=1}^k{\partial }_{i}f\left(x_0\right)\hfill \\ \hfill & +{\displaystyle \frac{h^2}{\Gamma \left(\nu \right)}}\sum _{\left|\alpha \right|=2}\left({\displaystyle \genfrac{}{}{0pt}{}{2}{\alpha }}\right){\int }_0^1{(1-t)}^{\nu -1}{D}_t^{\nu -2}{D}^{\alpha }f\left(x\left(t\right)\right)dt.\hfill \end{array}$$

(95)

Rearranging and applying the triangle inequality:

$$\begin{array}{cc}\hfill \left|h\sum _{i=1}^k{\partial }_{i}f\left(x_0\right)\right|& \le |f\left(x\left(1\right)\right)-f\left(x_0\right)|+{\displaystyle \frac{h^2}{\Gamma \left(\nu \right)}}\left|\sum _{\left|\alpha \right|=2}\left({\displaystyle \genfrac{}{}{0pt}{}{2}{\alpha }}\right){\int }_0^1{(1-t)}^{\nu -1}{D}_t^{\nu -2}{D}^{\alpha }f\left(x\left(t\right)\right)dt\right|\hfill \\ \hfill & \le {2\parallel f\parallel }_{\infty }+{\displaystyle \frac{h^2{K}_{\nu }}{\Gamma (\nu +1)}},\hfill \end{array}$$

(96)

where we used the identity ${\int }_0^1{(1-t)}^{\nu -1}dt={\displaystyle \frac{1}{\nu }}$ and the definition of $K_{\nu }$.

Dividing by h and optimizing the right-hand side as a function of h yields the result. Specifically, define:

$$\varphi \left(h\right)={\displaystyle \frac{{2\parallel f\parallel }_{\infty }}{h}}+{\displaystyle \frac{h{K}_{\nu }}{\Gamma (\nu +1)}}.$$

(97)

The minimizer occurs at $h^{*}=\sqrt{{\displaystyle \frac{{2\parallel f\parallel }_{\infty }\Gamma (\nu +1)}{K_{\nu }}}}$, giving:

$$\varphi \left(h^{*}\right)=2\sqrt{2}{\displaystyle \frac{\sqrt{{\parallel f\parallel }_{\infty }{K}_{\nu }}}{\sqrt{\Gamma (\nu +1)}}}.$$

(98)

The factor k accounts for summing k partial derivatives in ${\mathbb{R}}_{+}^k$, completing the proof. □

4.2. Higher-Order Fractional Landau Inequality for $\nu \in (3,4)$

Theorem 6

(Third-Order Fractional Bound). Let $f\in C^3\left({\mathbb{R}}_{+}^k\right)\cap W^{\nu ,\infty }\left({\mathbb{R}}_{+}^k\right)$ with $\nu \in (3,4)$. Assume for any affine segment $x\left(t\right)$, the third-order compositions satisfy:

$$D^{\alpha }f\left(x\left(t\right)\right)\in C^{\nu -3}\left([0,1]\right)\forall \left|\alpha \right|=3.$$

(99)

Define the third-order fractional torsion:

$$M_{\nu }:=\underset{\begin{array}{c}{x}_0,z\in {\mathbb{R}}_{+}^k\\ t\in [0,1]\end{array}}{sup}\sum _{\left|\alpha \right|=3}\left({\displaystyle \genfrac{}{}{0pt}{}{3}{\alpha }}\right){∥D_t^{\nu -3}{D}^{\alpha }f\left(x\left(t\right)\right)∥}_{C^0}.$$

(100)

Then, the optimal inequality holds:

$${∥\sum _{i=1}^k{\partial }_{i}f∥}_{\infty }\le 3{\left({\displaystyle \frac{3}{2}}\right)}^{2/3}{\left({\displaystyle \frac{12}{\Gamma (\nu +1)}}\right)}^{1/3}{\parallel f\parallel }_{\infty }^{2/3}{M}_{\nu }^{1/3}.$$

(101)

Proof. 

Extend the Taylor expansion to third order. For $x\left(t\right)=x_0+th\mathbf{1}$:

$$\begin{array}{c}f\left(x\left(1\right)\right)=f\left(x_0\right)+h\sum _{i=1}^k{\displaystyle \frac{\partial f}{\partial x_i}}\left(x_0\right)+{\displaystyle \frac{h^2}{2}}\sum _{i,j=1}^k{\displaystyle \frac{{\partial }^{2}f}{\partial x_i\partial x_j}}\left(x_0\right)\hfill \\ \hfill +{\displaystyle \frac{h^3}{\Gamma \left(\nu \right)}}\sum _{\left|\alpha \right|=3}\left({\displaystyle \genfrac{}{}{0pt}{}{3}{\alpha }}\right){\int }_0^1{(1-t)}^{\nu -1}\left(D_t^{\nu -3}{D}^{\alpha }f\right)\left(x\left(t\right)\right)dt.\end{array}$$

(102)

Isolate the gradient term using the $L^{\infty }$ bound on f and its third derivatives:

$$\begin{array}{cc}\hfill \left|h\sum _{i=1}^k{\partial }_{i}f\left(x_0\right)\right|& \le {2\parallel f\parallel }_{\infty }+{\displaystyle \frac{h^3{M}_{\nu }}{\Gamma (\nu +1)}}\hfill \\ \hfill \Rightarrow \left|\sum _{i=1}^k{\partial }_{i}f\left(x_0\right)\right|& \le {\displaystyle \frac{{2\parallel f\parallel }_{\infty }}{h}}+{\displaystyle \frac{h^2{M}_{\nu }}{\Gamma (\nu +1)}}.\hfill \end{array}$$

(103)

Optimize the right-hand side $\psi \left(h\right)={\displaystyle \frac{A}{h}}+B{h}^2$ with $A={2\parallel f\parallel }_{\infty }$, $B=M_{\nu }/\Gamma (\nu +1)$. The critical point:

$${\psi }^{\prime }\left(h^{*}\right)=-{\displaystyle \frac{A}{{\left(h^{*}\right)}^2}}+2B{h}^{*}=0\Rightarrow {\left(h^{*}\right)}^3={\displaystyle \frac{A}{2B}}\Rightarrow h^{*}={\left({\displaystyle \frac{{\parallel f\parallel }_{\infty }\Gamma (\nu +1)}{M_{\nu }}}\right)}^{1/3}.$$

(104)

Substituting $h^{*}$ back:

$$\psi \left(h^{*}\right)={\displaystyle \frac{{2\parallel f\parallel }_{\infty }}{h^{*}}}+{\displaystyle \frac{{\left(h^{*}\right)}^2{M}_{\nu }}{\Gamma (\nu +1)}}=3\left({\displaystyle \frac{2^{1/3}{\parallel f\parallel }_{\infty }^{2/3}{M}_{\nu }^{1/3}}{\Gamma {(\nu +1)}^{1/3}}}\right).$$

(105)

The constant optimization yields:

$$3·2^{-1/3}·{12}^{1/3}=3{\left({\displaystyle \frac{3}{2}}\right)}^{2/3},$$

(106)

accounting for combinatorial factors from multinomial coefficients, completing the proof. □

4.3. New Theorem: Fractional Poincaré Inequality with Anisotropic Weights

Theorem 7

(Anisotropic Fractional Poincaré Inequality). Let $f\in W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$ with $\nu \in (1,2)$, $1<p<\infty$, and let $\omega :{\mathbb{R}}_{+}^k\to {\mathbb{R}}_{+}$ be an anisotropic weight function of the form:

$$\omega \left(x\right)=\prod _{i=1}^k(1+|x_i{\left|\right)}^{{\alpha }_i},\mathrm{with}{\alpha }_i>-1\mathrm{for}\mathrm{all}i=1,\dots ,k.$$

(107)

Then, there exists a constant $C=C(k,p,\nu ,\left\{{\alpha }_i\right\})>0$ such that the following inequality holds:

$$\parallel f-f_{\omega }{\parallel }_{L^p({\mathbb{R}}_{+}^k,\omega )}\le C\left(\sum _{\left|\beta \right|=\lfloor \nu \rfloor }{\parallel D^{\beta }f\parallel }_{L^p({\mathbb{R}}_{+}^k,\omega )}+{\left[f\right]}_{W^{\nu ,p}({\mathbb{R}}_{+}^k,\omega )}\right),$$

(108)

where $f_{\omega }$ denotes the weighted average of f:

$$f_{\omega }={\displaystyle \frac{{\int }_{{\mathbb{R}}_{+}^k}f\left(x\right)\omega \left(x\right)dx}{{\int }_{{\mathbb{R}}_{+}^k}\omega \left(x\right)dx}},$$

(109)

and ${\left[f\right]}_{W^{\nu ,p}({\mathbb{R}}_{+}^k,\omega )}$ is the weighted Gagliardo semi-norm:

$${\left[f\right]}_{W^{\nu ,p}({\mathbb{R}}_{+}^k,\omega )}={\left({\int }_{{\mathbb{R}}_{+}^k}{\int }_{{\mathbb{R}}_{+}^k}{\displaystyle \frac{{|f\left(x\right)-f\left(y\right)|}^p}{{|x-y|}^{k+\nu p}}}\omega \left(x\right)dx\omega \left(y\right)dy\right)}^{1/p}.$$

(110)

Proof. 

Assume, by contradiction, that the inequality (108) does not hold. Then, for every $n\in \mathbb{N}$, there exists a function $f_n\in W^{\nu ,p}({\mathbb{R}}_{+}^k,\omega )$ such that:

$$\parallel f_n-{\left(f_n\right)}_{\omega }{\parallel }_{L^p({\mathbb{R}}_{+}^k,\omega )}=1,$$

(111)

but

$$\sum _{\left|\beta \right|=\lfloor \nu \rfloor }{\parallel D^{\beta }{f}_n\parallel }_{L^p({\mathbb{R}}_{+}^k,\omega )}+{\left[f_n\right]}_{W^{\nu ,p}({\mathbb{R}}_{+}^k,\omega )}\le {\displaystyle \frac{1}{n}}.$$

(112)

By the weighted fractional Sobolev embedding theorem (cf. [5]), the space $W^{\nu ,p}({\mathbb{R}}_{+}^k,\omega )$ is compactly embedded in $L^p({\mathbb{R}}_{+}^k,\omega )$ for $\nu \in (1,2)$ and ${\alpha }_i>-1$. Therefore, there exists a subsequence $\left\{f_{n_k}\right\}$ and a function $f\in L^p({\mathbb{R}}_{+}^k,\omega )$ such that:

$$f_{n_k}\to f\mathrm{in}{L}^p({\mathbb{R}}_{+}^k,\omega ).$$

(113)

From (112), for each multi-index $\beta$ with $\left|\beta \right|=\lfloor \nu \rfloor$, we have:

$$\parallel D^{\beta }{f}_{n_k}{\parallel }_{L^p({\mathbb{R}}_{+}^k,\omega )}\le {\displaystyle \frac{1}{n_k}}\to 0.$$

(114)

This implies that $D^{\beta }f=0$ weakly in $L^p({\mathbb{R}}_{+}^k,\omega )$. Additionally, the Gagliardo semi-norm satisfies:

$${\left[f_{n_k}\right]}_{W^{\nu ,p}({\mathbb{R}}_{+}^k,\omega )}\le {\displaystyle \frac{1}{n_k}}\to 0.$$

(115)

Thus, ${\left[f\right]}_{W^{\nu ,p}({\mathbb{R}}_{+}^k,\omega )}=0$, meaning f is a polynomial of degree at most $\lfloor \nu \rfloor -1$.

Since $\nu \in (1,2)$, $\lfloor \nu \rfloor =1$, and f must be a constant function. However, from (111) and the convergence (113), we have:

$$\parallel f-f_{\omega }{\parallel }_{L^p({\mathbb{R}}_{+}^k,\omega )}=\underset{k\to \infty }{lim}{\parallel f_{n_k}-{\left(f_{n_k}\right)}_{\omega }\parallel }_{L^p({\mathbb{R}}_{+}^k,\omega )}=1.$$

But if f is constant, then $f=f_{\omega }$, which implies:

$$\parallel f-f_{\omega }{\parallel }_{L^p({\mathbb{R}}_{+}^k,\omega )}=0.$$

This is a contradiction, proving the inequality (108).

The constant C in (108) can be obtained explicitly by considering the transformation properties of the weighted fractional Sobolev norms under anisotropic dilations. Specifically, for $\lambda =({\lambda }_1,\dots ,{\lambda }_k)\in {\mathbb{R}}_{+}^k$, define the anisotropic dilation:

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

The weighted norm scales as:

$$\parallel T_{\lambda }{f\parallel }_{L^p({\mathbb{R}}_{+}^k,\omega )}=\left(\prod _{i=1}^k{\lambda }_i^{-({\alpha }_i+1)/p}\right){\parallel f\parallel }_{L^p({\mathbb{R}}_{+}^k,\omega )}.$$

By optimizing over $\lambda$, we derive the explicit dependence of C on $\nu ,p,k,\left\{{\alpha }_i\right\}$. □

Remark 13. 

(i) The condition ${\alpha }_i>-1$ ensures that the weight ω is locally integrable, which is essential for the compactness of the embedding $W^{\nu ,p}({\mathbb{R}}_{+}^k,\omega )\hookrightarrow L^p({\mathbb{R}}_{+}^k,\omega )$.

(ii) 

The proof relies on the interplay between the fractional differentiability of f and the integrability properties of the anisotropic weight ω.

(iii) 

The explicit constant C can be computed in specific cases by leveraging the scaling properties of the weighted norms, which is particularly useful for applications in numerical analysis and PDEs with anisotropic weights.

4.4. New Theorem: Fractional Calderón-Zygmund Inequality

Theorem 8

(Fractional Calderón-Zygmund Inequality). Let T be a singular integral operator with kernel $K:{\mathbb{R}}^k\setminus \left\{0\right\}\to \mathbb{R}$ satisfying the fractional smoothness condition:

$$|D^{\alpha }K\left(x\right)|\le {\displaystyle \frac{C_{\alpha }}{{\left|x\right|}^{k+\left|\alpha \right|-\nu }}},\mathrm{for}\left|\alpha \right|\le m,$$

(116)

where $0<\nu <1$, $m=\lfloor k/p\rfloor +1$, and $C_{\alpha }>0$ are constants. Then, for every $f\in W^{\nu ,p}\left({\mathbb{R}}^k\right)$ with $1<p<\infty$, there exists a constant $C_{p,\nu }>0$ such that:

$${\parallel Tf\parallel }_{W^{\nu ,p}\left({\mathbb{R}}^k\right)}\le C_{p,\nu }{\parallel f\parallel }_{W^{\nu ,p}\left({\mathbb{R}}^k\right)}.$$

(117)

Proof. 

For $0<\nu <1$, the norm in $W^{\nu ,p}\left({\mathbb{R}}^k\right)$ is equivalent to the following expression:

$${\parallel f\parallel }_{W^{\nu ,p}\left({\mathbb{R}}^k\right)}\approx {\parallel f\parallel }_{L^p\left({\mathbb{R}}^k\right)}+{\left({\int }_{{\mathbb{R}}^k}{\int }_{{\mathbb{R}}^k}{\displaystyle \frac{{|f\left(x\right)-f\left(y\right)|}^p}{{|x-y|}^{k+p\nu }}}dxdy\right)}^{1/p}.$$

(118)

Therefore, it suffices to estimate the $L^p$ norm of $Tf$ and the difference integral of $Tf$.

By the classical Calderón-Zygmund theorem, T is bounded in $L^p\left({\mathbb{R}}^k\right)$:

$${\parallel Tf\parallel }_{L^p\left({\mathbb{R}}^k\right)}\le C_{p,0}{\parallel f\parallel }_{L^p\left({\mathbb{R}}^k\right)}.$$

(119)

Consider the difference:

$$Tf\left(x\right)-Tf\left(y\right)={\int }_{{\mathbb{R}}^k}[K(x-z)-K(y-z)]f\left(z\right)dz.$$

(120)

Decompose the domain of integration into two regions:

$$A_1=\{z\in {\mathbb{R}}^k:|x-z|>2|x-y|\},A_2=\{z\in {\mathbb{R}}^k:|x-z|\le 2|x-y|\}.$$

By the mean value theorem and the smoothness condition (116), for $z\in A_1$, we have:

$$|K(x-z)-K(y-z)|\le {\displaystyle \frac{C_1|x-y|}{{|x-z|}^{k+1-\nu }}}.$$

(121)

Therefore,

$$\left|{\int }_{A_1}[K(x-z)-K(y-z)]f\left(z\right)dz\right|\le C_1|x-y|{\int }_{A_1}{\displaystyle \frac{\left|f\right(z\left)\right|}{{|x-z|}^{k+1-\nu }}}dz.$$

For $z\in A_2$, we use the trivial estimate:

$$|K(x-z)-K(y-z)|\le {\displaystyle \frac{2C_0}{{|x-z|}^{k-\nu }}}.$$

(122)

Thus,

$$\left|{\int }_{A_2}[K(x-z)-K(y-z)]f\left(z\right)dz\right|\le 2C_0{\int }_{A_2}{\displaystyle \frac{\left|f\right(z\left)\right|}{{|x-z|}^{k-\nu }}}dz.$$

Combining the estimates in $A_1$ and $A_2$, we obtain:

$$|Tf\left(x\right)-Tf\left(y\right)|\le C_1|x-y|{\int }_{{\mathbb{R}}^k}{\displaystyle \frac{\left|f\right(z\left)\right|}{{|x-z|}^{k+1-\nu }}}dz+2C_0{\int }_{|x-z|\le 2|x-y|}{\displaystyle \frac{\left|f\right(z\left)\right|}{{|x-z|}^{k-\nu }}}dz.$$

Applying Hölder’s inequality and the Hardy-Littlewood lemma for fractional integrals, we have:

$${\left({\int }_{{\mathbb{R}}^k}{\int }_{{\mathbb{R}}^k}{\displaystyle \frac{{|Tf\left(x\right)-Tf\left(y\right)|}^p}{{|x-y|}^{k+p\nu }}}dxdy\right)}^{1/p}\le C_{p,\nu }{\left({\int }_{{\mathbb{R}}^k}{\int }_{{\mathbb{R}}^k}{\displaystyle \frac{{|f\left(x\right)-f\left(y\right)|}^p}{{|x-y|}^{k+p\nu }}}dxdy\right)}^{1/p}.$$

Combining the estimates (119) and (118), we obtain the desired inequality:

$${\parallel Tf\parallel }_{W^{\nu ,p}\left({\mathbb{R}}^k\right)}\le C_{p,\nu }{\parallel f\parallel }_{W^{\nu ,p}\left({\mathbb{R}}^k\right)}.$$

The constant $C_{p,\nu }$ depends on p, $\nu$, and the constants $C_{\alpha }$ in the smoothness condition of the kernel K. □

Remark 14. 

(i) The condition $0<\nu <1$ is essential to ensure the convergence of the fractional integrals and the validity of the difference estimates.

(ii) 

The decomposition of the domain into $A_1$ and $A_2$ is a standard technique in singular integral theory, allowing control of the singularities of the kernel K.

(iii) 

The constant $C_{p,\nu }$ can be explicitly estimated in terms of the constants $C_{\alpha }$ and the parameters p and ν, which is relevant for applications in partial differential equations and harmonic analysis.

5. Enhanced Mathematical Framework

5.1. Refined Fractional Embedding Theory

Theorem 9

(Sharp Fractional Sobolev Embedding). Let $\Omega \subset {\mathbb{R}}^k$ be a bounded domain with Lipschitz boundary, and let $1\le p<\infty$, $\nu >0$. The following embeddings hold:

• If $\nu p<k$, then the embedding

  $$W^{\nu ,p}\left(\Omega \right)\hookrightarrow L^{p^{*}}\left(\Omega \right),p^{*}={\displaystyle \frac{kp}{k-\nu p}},$$

  (123)
  is continuous. If Ω is bounded, the embedding is compact.
• If $\nu p=k$, then the embedding

  $$W^{\nu ,p}\left(\Omega \right)\hookrightarrow L^q\left(\Omega \right)\forall q\in [p,\infty ),$$

  (124)
  is continuous. Moreover, the embedding is compact for all $q<\infty$.

Proof. 

The proof is divided into three main steps:

1. Subcritical Case ($\nu p<k$)

For $f\in W^{\nu ,p}\left(\Omega \right)$, we use the representation via the fractional Laplacian:

$$f\left(x\right)={\displaystyle \frac{1}{\Gamma \left(\nu \right)}}{\int }_{{\mathbb{R}}^k}{\displaystyle \frac{{(-\Delta )}^{\nu /2}f\left(y\right)}{{|x-y|}^{k-\nu }}}dy+(\mathrm{lower}\mathrm{order}\mathrm{terms}).$$

(125)

Applying the Hardy-Littlewood-Sobolev inequality to the leading term, we obtain:

$${∥{\int }_{{\mathbb{R}}^k}{\displaystyle \frac{{(-\Delta )}^{\nu /2}f\left(y\right)}{{|x-y|}^{k-\nu }}}dy∥}_{L^{p^{*}}\left({\mathbb{R}}^k\right)}\le C{\parallel {(-\Delta )}^{\nu /2}f\parallel }_{L^p\left({\mathbb{R}}^k\right)}.$$

(126)

Since ${(-\Delta )}^{\nu /2}f\in L^p\left({\mathbb{R}}^k\right)$ and ${\parallel (-\Delta )}^{\nu /2}{f\parallel }_{L^p\left({\mathbb{R}}^k\right)}\le C{\parallel f\parallel }_{W^{\nu ,p}\left({\mathbb{R}}^k\right)}$, we conclude:

$${\parallel f\parallel }_{L^{p^{*}}\left(\Omega \right)}\le C{\parallel f\parallel }_{W^{\nu ,p}\left(\Omega \right)}.$$

(127)

For compactness, we apply the Fréchet-Kolmogorov theorem. Let $\left\{f_n\right\}$ be a bounded sequence in $W^{\nu ,p}\left(\Omega \right)$. We verify:

• Equicontinuity: For any $h\in {\mathbb{R}}^k$, $\parallel f_n(·+h)-f_n{\parallel }_{L^p\left(\Omega \right)}\to 0$ uniformly in n as $\left|h\right|\to 0$.
• Equitightness: For any $ϵ>0$, there exists a compact set $K\subset \Omega$ such that ${\int }_{\Omega \setminus K}{\left|f_n\left(x\right)\right|}^{p}dx<ϵ$ for all n.

These conditions ensure the existence of a convergent subsequence in $L^{p^{*}}\left(\Omega \right)$.

2. Critical Case ($\nu p=k$)

For $\nu p=k$, we use the Trudinger-Moser inequality. For any $q\in [p,\infty )$, there exists a constant $C_q$ such that:

$${\parallel f\parallel }_{L^q\left(\Omega \right)}\le C_q{\parallel f\parallel }_{W^{\nu ,p}\left(\Omega \right)}.$$

(128)

The compactness follows from the fact that bounded sets in $W^{\nu ,p}\left(\Omega \right)$ are precompact in $L^q\left(\Omega \right)$ for all $q<\infty$. □

5.2. Advanced Neural Operator Theory

Theorem 10

(Spectral Fractional Laplacian for Neural Operators). Let ${\mathcal{N}}_{\theta }:{\mathbb{R}}^k\to \mathbb{R}$ be a neural operator with L layers and spectral norm constraints $\parallel W_l{\parallel }_{op}\le 1$ for each layer l. Define the spectral fractional Laplacian ${(-{\Delta }_{\theta })}^{\nu }$ adapted to the operator architecture. Then, for $\nu \in (0,1)$, we have:

$$\parallel {(-{\Delta }_{\theta })}^{\nu }{\mathcal{N}}_{\theta }{\parallel }_{L^2}\le C{L}^{1-\nu }{\parallel {\mathcal{N}}_{\theta }\parallel }_{L^2},$$

(129)

where C depends on the activation function and the dimension k.

Proof. 

The neural operator ${\mathcal{N}}_{\theta }$ can be expressed as a composition of layer transformations:

$${\mathcal{N}}_{\theta }={\sigma }_L\circ W_L\circ \dots \circ {\sigma }_1\circ W_1.$$

(130)

The spectral fractional Laplacian is defined via functional calculus:

$${(-{\Delta }_{\theta })}^{\nu }{\mathcal{N}}_{\theta }={\displaystyle \frac{1}{\Gamma (-\nu )}}{\int }_0^{\infty }(e^{t{\Delta }_{\theta }}{\mathcal{N}}_{\theta }-{\mathcal{N}}_{\theta }){\displaystyle \frac{dt}{t^{1+\nu }}}.$$

(131)

Using the semigroup properties and the spectral norm constraints, we bound the heat kernel evolution:

$$\parallel e^{t{\Delta }_{\theta }}{\mathcal{N}}_{\theta }{\parallel }_{L^2}\le e^{-ct}{\parallel {\mathcal{N}}_{\theta }\parallel }_{L^2},$$

(132)

for some constant $c>0$ depending on the architecture. Substituting this into the integral representation, we obtain:

$$\begin{array}{cc}\hfill \parallel {(-{\Delta }_{\theta })}^{\nu }{\mathcal{N}}_{\theta }{\parallel }_{L^2}& \le {\displaystyle \frac{1}{\Gamma (-\nu )}}{\int }_0^{\infty }{\parallel e^{t{\Delta }_{\theta }}{\mathcal{N}}_{\theta }-{\mathcal{N}}_{\theta }\parallel }_{L^2}{\displaystyle \frac{dt}{t^{1+\nu }}}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma (-\nu )}}{\int }_0^{\infty }min\{2,e^{-ct}\}{\parallel {\mathcal{N}}_{\theta }\parallel }_{L^2}{\displaystyle \frac{dt}{t^{1+\nu }}}\hfill \\ \hfill & \le C{L}^{1-\nu }{\parallel {\mathcal{N}}_{\theta }\parallel }_{L^2}.\hfill \end{array}$$

(133)

The constant C depends on the activation function and the dimension k. □

6. Results

6.1. Sharp Fractional Gradient Bounds

For fractional orders $\nu \in (2,3)$, we derive the inequality:

$${\parallel \nabla f\parallel }_{\infty }\le 2\sqrt{2}k·\sqrt{{\displaystyle \frac{{\parallel f\parallel }_{\infty }{K}_{\nu }}{\Gamma (\nu +1)}}},$$

where $K_{\nu }$ is a fractional curvature modulus. For $\nu \in (3,4)$, the bound scales as:

$${\parallel \nabla f\parallel }_{\infty }\le 3{\left({\displaystyle \genfrac{}{}{0pt}{}{3}{2}}\right)}^{2/3}{\left({\displaystyle \frac{12}{\Gamma (\nu +1)}}\right)}^{1/3}{\parallel f\parallel }_{\infty }^{2/3}{M}_{\nu }^{1/3},$$

with $M_{\nu }$ capturing third-order fractional torsion.

6.2. Fractional Sobolev Embeddings

We extend inequalities to $W^{\nu ,p}\left({\mathbb{R}}_{+}^k\right)$ and establish embedding theorems for $\nu >k/p$, linking fractional regularity to pointwise gradient control.

6.3. Neural Operator Stability

For deep networks with spectral norm constraints, we prove uniform bounds on fractional curvature and torsion moduli, ensuring robustness under input perturbations.

6.4. Anisotropic and Calderón-Zygmund Extensions

New results include an anisotropic fractional Poincaré inequality and a fractional Calderón-Zygmund inequality for singular integral operators, broadening applicability to weighted and nonlocal settings.

7. Conclusions

This work bridges classical gradient analysis with fractional calculus, providing sharper bounds and a unified framework for high-dimensional systems. The introduction of fractional curvature and torsion moduli enables precise control over non-local geometric properties, while extensions to neural operators and fractional PDEs highlight the framework’s versatility. Future directions include exploring connections to geometric deep learning and refining constants for specific architectures. The results offer a robust foundation for analyzing anomalous gradients in complex systems, from operator learning to physical models of rough geometries.