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

TL;DR

This paper improves the John-Nirenberg inequality for functions in critical Sobolev spaces, providing sharper exponential integrability estimates for Riesz potentials within Lorentz spaces.

Contribution

It establishes a refined inequality involving Hausdorff content for functions in critical Sobolev spaces, enhancing the classical John-Nirenberg inequality with sharper bounds.

Findings

Proves a new exponential decay inequality for Riesz potentials.

Extends the John-Nirenberg inequality to Lorentz space settings.

Provides sharper bounds for functions in critical Sobolev spaces.

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 \[\mathcal{H}^{\beta}_{\infty}(\{x\in \Omega:|I_\alpha f(x)|>t\})\leq Ce^{-ct^{q'}}\] for all $\|f\|_{L^{N/\alpha,q}(\Omega)}\leq 1$ and any $\beta \in (0,N]$, where $\Omega \subset \mathbb{R}^N$, $\mathcal{H}^{\beta}_{\infty}$ is the Hausdorff content, $L^{N/\alpha,q}(\Omega)$ is a Lorentz space with $q \in (1,\infty]$, $q'=q/(q-1)$ is the H\"older conjugate to $q$, and $I_\alpha f$ denotes the Riesz potential of $f$ of order $\alpha \in (0,N)$.