## 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_{α}f(x)| > t}) ≤ Ce^{−ctq} 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_{α}f denotes the Riesz potential of f of order α ∈ (0, N).

Original language | English |
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Pages (from-to) | 877-894 |

Number of pages | 18 |

Journal | Advances in Nonlinear Analysis |

Volume | 10 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2021 Jan 1 |

Externally published | Yes |

## Keywords

- Critical Sobolev Embedding
- Hausdorff Content
- Riesz Potentials

## ASJC Scopus subject areas

- Analysis