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Nonlocal gradients: Fundamental theorem of calculus, Poincaré inequalities and embeddings

José Carlos Bellido, Carlos Mora-Corral, Hidde Schönberger

Abstract

We address the study of nonlocal gradients defined through general radial kernels $ρ$. Our investigation focuses on the properties of the associated function spaces, which depend on the characteristics of the kernel function. Specifically, even with minimal assumptions on $ρ$, we establish Poincaré inequalities and compact embeddings into Lebesgue spaces. Additionally, we present a fundamental theorem of calculus that enables us to recover a function from its nonlocal gradient through a convolution. This is used to demonstrate embeddings into Orlicz spaces and spaces of continuous functions that mirror the well-known Sobolev and Morrey inequalities for classical gradients. Finally, we establish conditions for inclusions and equality of spaces associated to different kernels.

Nonlocal gradients: Fundamental theorem of calculus, Poincaré inequalities and embeddings

Abstract

We address the study of nonlocal gradients defined through general radial kernels . Our investigation focuses on the properties of the associated function spaces, which depend on the characteristics of the kernel function. Specifically, even with minimal assumptions on , we establish Poincaré inequalities and compact embeddings into Lebesgue spaces. Additionally, we present a fundamental theorem of calculus that enables us to recover a function from its nonlocal gradient through a convolution. This is used to demonstrate embeddings into Orlicz spaces and spaces of continuous functions that mirror the well-known Sobolev and Morrey inequalities for classical gradients. Finally, we establish conditions for inclusions and equality of spaces associated to different kernels.
Paper Structure (15 sections, 29 theorems, 232 equations)

This paper contains 15 sections, 29 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. First properties of $\mathcal{G}_{\rho}$
  3. Function spaces
  4. Definition and first properties
  5. Equivalence of spaces with different kernels
  6. Poincaré inequalities and compact embeddings
  7. Positivity of $\widehat{Q}_\rho$
  8. Poincaré inequality and Compactness in $L^2$
  9. Poincaré inequality and Compactness in $L^p$
  10. Fundamental theorem of calculus
  11. Embeddings
  12. Embeddings into Orlicz spaces
  13. Embeddings into spaces of continuous functions
  14. Compact embeddings
  15. Inclusion between spaces for different kernels

Key Result

Lemma 2.3

Let $u \in C^{\infty}_c (\mathbb{R}^{n})$. Then $\mathcal{G}_{\rho} u \in L^1 (\mathbb{R}^{n},\mathbb{R}^{n}) \cap L^{\infty} (\mathbb{R}^{n},\mathbb{R}^{n})$ and the integral eq:Grux is absolutely convergent for each $x \in \mathbb{R}^{n}$.

Theorems & Definitions (70)

  • Example 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 60 more