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Nonuniform Sobolev Spaces

Ting Chen, Loukas Grafakos, Wenchang Sun

Abstract

We study nonuniform Sobolev spaces, i.e., spaces of functions whose partial derivatives lie in possibly different Lebesgue spaces. Although standard proofs do not apply, we show that nonuniform Sobolev spaces share similar properties as the classical ones. These spaces arise naturally in the study of certain PDEs. For instance, we illustrate that nonuniform fractional Sobolev spaces are useful in the study of local estimates for solutions of heat equations and the convergence of Schrödinger operators. In this work we extend recent advances on local energy estimates for solutions of heat equations and the convergence of Schrödinger operators to nonuniform fractional Sobolev spaces.

Nonuniform Sobolev Spaces

Abstract

We study nonuniform Sobolev spaces, i.e., spaces of functions whose partial derivatives lie in possibly different Lebesgue spaces. Although standard proofs do not apply, we show that nonuniform Sobolev spaces share similar properties as the classical ones. These spaces arise naturally in the study of certain PDEs. For instance, we illustrate that nonuniform fractional Sobolev spaces are useful in the study of local estimates for solutions of heat equations and the convergence of Schrödinger operators. In this work we extend recent advances on local energy estimates for solutions of heat equations and the convergence of Schrödinger operators to nonuniform fractional Sobolev spaces.
Paper Structure (8 sections, 20 theorems, 276 equations)

This paper contains 8 sections, 20 theorems, 276 equations.

Table of Contents

  1. Introduction
  2. Embedding Inequalities for Nonuniform Sobolev Spaces
  3. Nonuniform Fractional Sobolev Spaces
  4. Embedding Theorem for Nonuniform Fractional Sobolev Spaces
  5. Density of Compactly Supported Infinitely Differentiable Functions
  6. Applications
  7. Local estimates for solutions of heat equations
  8. Convergence of Schrödinger Operators

Key Result

Lemma 2.1

For any $k\ge 1$ and $\vec{p}\in [1,\infty)^{k+1}$, the space $C^{\infty}\cap W_k^{\vec{p}}(\mathbb{R}^N)$ is dense in $W_k^{\vec{p}}(\mathbb{R}^N)$.

Theorems & Definitions (44)

  • Definition 1.1
  • Definition 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Corollary 2.4
  • Remark 2.5
  • ...and 34 more