An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces

Ángel D. Martínez, Daniel Spector*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

It is known that functions in a Sobolev space with critical exponent embed into the space of functions of bounded mean oscillation, and therefore satisfy the John-Nirenberg inequality and a corresponding exponential integrability estimate. While these inequalities are optimal for general functions of bounded mean oscillation, the main result of this paper is an improvement for functions in a class of critical Sobolev spaces. Precisely, we prove the inequality Hβ ({x ∈ Ω: |Iαf(x)| > t}) ≤ Ce−ctq for all ||f||LN/α,q(Ω) ≤ 1 and any β ∈ (0, N], where Ω ⊂ RN, Hβ is the Hausdorff content, LN/α,q(Ω) is a Lorentz space with q ∈ (1, ∞], q' = q/(q − 1) is the Hölder conjugate to q, and Iαf denotes the Riesz potential of f of order α ∈ (0, N).

Original languageEnglish
Pages (from-to)877-894
Number of pages18
JournalAdvances in Nonlinear Analysis
Volume10
Issue number1
DOIs
Publication statusPublished - 2021 Jan 1
Externally publishedYes

Keywords

  • Critical Sobolev Embedding
  • Hausdorff Content
  • Riesz Potentials

ASJC Scopus subject areas

  • Analysis

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