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Some remarks on Campanato Theorem and the Anisotropic Bessel Spaces

H. Hajaiej, R. Leitao

TL;DR

This work develops an anisotropic regularity theory for fractional operators by proving a Campanato-type embedding in anisotropic spaces and linking it to Hölder continuity under the regime $c_{\beta}=n$. It defines and analyzes the anisotropic Besov/Bessel spaces $W_{\beta}^{\alpha,q}$ and $H_{r_{\beta}}^{\alpha,q}$, establishing the embedding $H_{r_{\beta}}^{\alpha,q}(\mathbb{R}^{n}) \hookrightarrow C^{\theta}_{\|\cdot\|_{\beta}}(\mathbb{R}^{n})$ with $\theta=\alpha-\frac{c_{\beta}}{q}$ and clarifying their relation to $W_{\beta}^{\alpha,q}$. Building on this, the paper develops a viscosity-solution framework and barrier methods to construct anisotropic fundamental solutions for the nonlocal operator $\Delta^{\beta,\alpha}$, obtaining existence, uniqueness (up to constants), and detailed asymptotics and regularity. The results yield Liouville-type theorems and provide a robust toolkit for analyzing anisotropic nonlocal PDEs, with implications for anisotropic diffusion models and related PDEs.

Abstract

In this paper, we establish an anisotropic version of Campanato Theorem and show that the anisotropic Bessel spaces are continuously embedded in the spaces of Holder continuous functions. As an application of this embedding, we build fundamental solutions for a class of anisotropic fractional Laplacian operators.

Some remarks on Campanato Theorem and the Anisotropic Bessel Spaces

TL;DR

This work develops an anisotropic regularity theory for fractional operators by proving a Campanato-type embedding in anisotropic spaces and linking it to Hölder continuity under the regime . It defines and analyzes the anisotropic Besov/Bessel spaces and , establishing the embedding with and clarifying their relation to . Building on this, the paper develops a viscosity-solution framework and barrier methods to construct anisotropic fundamental solutions for the nonlocal operator , obtaining existence, uniqueness (up to constants), and detailed asymptotics and regularity. The results yield Liouville-type theorems and provide a robust toolkit for analyzing anisotropic nonlocal PDEs, with implications for anisotropic diffusion models and related PDEs.

Abstract

In this paper, we establish an anisotropic version of Campanato Theorem and show that the anisotropic Bessel spaces are continuously embedded in the spaces of Holder continuous functions. As an application of this embedding, we build fundamental solutions for a class of anisotropic fractional Laplacian operators.

Paper Structure

This paper contains 8 sections, 26 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Preliminaries:
  3. Anisotropic topology and elements of the Campanato theory
  4. Fundamental solution: ingredients
  5. Viscosity solutions and $L^{p}$-estimates
  6. $-\gamma$-homogeneous solutions
  7. Proof of Theorem \ref{['main theorem']}
  8. Existence of Fundamental solutions and consequences

Key Result

Theorem 1.1

Let $\beta = (b_{1}, \dots, b_{n}),$ and $b_{\max}=\max_{1\leq i\leq n}b_i.$ If $0 < \alpha < 2/b_{\max}$ and $\alpha q > c_{\beta}$, then $W_{\beta}^{\alpha, q}(\mathbb{R}^{n})$ is continuously embedded in $C^{\theta}_{\Vert \cdot \Vert_{\beta}}\left( \mathbb{R}^{n}\right)$ for In particular, if $c_{\beta}=n$, $H_{r_{\beta}}^{\alpha, q}(\mathbb{R}^{n})$ is continuously embedded in $C^{\theta}_{\

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • ...and 33 more