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

created by spector on 19 Jul 2020

[BibTeX]

preprint

Inserted: 19 jul 2020
Last Updated: 19 jul 2020

Year: 2020

ArXiv: 2007.04576 PDF

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 $\alpha \in (0,N)$.

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