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Fractional weighted Sobolev spaces associated to the Riesz fractional gradient

Guillermo García-Sáez

Abstract

In this work, we introduce a new family of functions spaces, the weighted fractional Sobolev spaces $X^{s,p}_{0,w}(Ω)$, where $w$ is a weight in the Muckenhoupt class $A_p$. This space is a natural extension of the fractional Sobolev spaces $H^{s,p}_0$, obtained by means of the Riesz fractional gradient $D^s$, to the setting of the weighted Lebesgue spaces $L^p_w$. As it happened in the unweighted space, the spaces $X^{s,p}_{0,w}(Ω)$ coincide with the weighted version of the Bessel potential space. We obtaien several structural properties for these spaces, as well as continuous and compact embeddings. We conclude with the study of a family of degenerate fractional elliptic partial differential equations.

Fractional weighted Sobolev spaces associated to the Riesz fractional gradient

Abstract

In this work, we introduce a new family of functions spaces, the weighted fractional Sobolev spaces , where is a weight in the Muckenhoupt class . This space is a natural extension of the fractional Sobolev spaces , obtained by means of the Riesz fractional gradient , to the setting of the weighted Lebesgue spaces . As it happened in the unweighted space, the spaces coincide with the weighted version of the Bessel potential space. We obtaien several structural properties for these spaces, as well as continuous and compact embeddings. We conclude with the study of a family of degenerate fractional elliptic partial differential equations.

Paper Structure

This paper contains 13 sections, 27 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Outline of the work
  3. Preliminaries
  4. Interpolation theory
  5. Weighted spaces
  6. Calderón-Zymgund operators and multiplier theory
  7. Nonlocal calculus
  8. Weighted Bessel potential spaces
  9. Fractional weighted spaces on bounded domains
  10. Interpolation on bounded domains and the fractional-weighted setting
  11. Continuous embeddings and interpolation inequalities
  12. Compactness results
  13. A Fractional nonlinear Degenerate Elliptic problem

Key Result

Theorem 2.1

Let $(E_0,E_1)$ a compatible couple of Banach spaces, and $\theta\in [0,1]$. Then, we have

Theorems & Definitions (32)

  • Theorem 2.1: Properties of Complex interpolation spaces
  • Definition 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Definition 3.1
  • Definition 3.2
  • ...and 22 more