TY - JOUR

T1 - An improvement to the John-Nirenberg inequality for functions in critical Sobolev spaces

AU - Martínez, Ángel D.

AU - Spector, Daniel

N1 - Funding Information:
This work was initiated while the first named author was visiting the Nonlinear Analysis Unit in the Okinawa Institute of Science and Technology Graduate University. He warmly thanks OIST for the invitation and hospitality. The first named author is supported by the National Science Foundation under Grant No. DMS-1638352.
Publisher Copyright:
© 2021 Ángel D. Martínez and Daniel Spector, published by De Gruyter.

PY - 2021/1/1

Y1 - 2021/1/1

N2 - 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: |Iαf(x)| > t}) ≤ Ce−ctq for all ||f||LN/α,q(Ω) ≤ 1 and any β ∈ (0, N], where Ω ⊂ RN, H∞β 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).

AB - 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: |Iαf(x)| > t}) ≤ Ce−ctq for all ||f||LN/α,q(Ω) ≤ 1 and any β ∈ (0, N], where Ω ⊂ RN, H∞β 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).

KW - Critical Sobolev Embedding

KW - Hausdorff Content

KW - Riesz Potentials

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U2 - 10.1515/anona-2020-0157

DO - 10.1515/anona-2020-0157

M3 - Article

AN - SCOPUS:85098952340

VL - 10

SP - 877

EP - 894

JO - Advances in Nonlinear Analysis

JF - Advances in Nonlinear Analysis

SN - 2191-9496

IS - 1

ER -