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

2020/07/09 by Martínez, Ángel D., Spector, Daniel · 1 citation
#Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2007.04576

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 H^β_∞(\x∈ Ω:|I_αf(x)|gt;t\)≤ Ce^-ct^q' for all ‖f‖_L^N/α,q(Ω)≤ 1 and any β∈ (0,N], where Ω⊂ ℝ^N, H^β_∞ is the Hausdorff content, L^N/α,q(Ω) is a Lorentz space with q ∈ (1,∞], q'=q/(q-1) is the Hölder conjugate to q, and I_αf denotes the Riesz potential of f of order α∈ (0,N).

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