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Pointwise estimates for rough operators with applications to Sobolev inequalities

Cong Hoang, Kabe Moen, Carlos Pérez

Abstract

We investigate Sobolev inequalities for several rough operators. We prove that several operators satisfy a pointwise bound by the Riesz potential applied to the gradient. From this inequality, we derive several new Sobolev-type inequalities with an operator on the left-hand side.

Pointwise estimates for rough operators with applications to Sobolev inequalities

Abstract

We investigate Sobolev inequalities for several rough operators. We prove that several operators satisfy a pointwise bound by the Riesz potential applied to the gradient. From this inequality, we derive several new Sobolev-type inequalities with an operator on the left-hand side.
Paper Structure (9 sections, 16 theorems, 184 equations)

This paper contains 9 sections, 16 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. The main operators and the main result.
  4. Proofs of our main results
  5. One weight consequences
  6. Weighted estimates from previously known sharp bounds
  7. New bounds
  8. Two weight consequences
  9. Endpoint results

Key Result

Theorem 1.1

If $f\in C_c^\infty(\mathbb R^n)$, then and hence

Theorems & Definitions (25)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Lemma 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 15 more