Fractional integration of summable functions: Maz'ya's Φ-inequalities

2021/09/16 by Dmitriy Stolyarov, Stolyarov, Dmitriy · 1 citation
Mathematics · #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Approximation and Integration

paper · pdf · doi:10.48550/arxiv.2109.08014

openalex publication_date 2021/09/16 · openalex created_date 2023/03/08 · openalex updated_date 2026/07/28

Abstract

We study the inequalities of the type |∫_ℝ^d Φ(K*f)| \lesssim ‖f‖_L1(ℝ^d)^p, where the kernel K is homogeneous of order α- d and possibly vector-valued, the function Φ is positively p-homogeneous, and p = d/(d-α). Under mild regularity assumptions on K and Φ, we find necessary and sufficient conditions on these functions under which the inequality holds true with a uniform constant for all sufficiently regular functions f.

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