Rearrangement Estimates for Fourier Transforms in Lp and Hp in Terms of Moduli of Continuity

• Published in: Mathematische Nachrichten Vol. 228; no. 1; pp. 123 - 144
• Main Authors: García-Cuerva, José, Kolyada, Víctor I.
• Format: Journal Article
• Language: English
• Published: Berlin WILEY-VCH Verlag Berlin GmbH 01-08-2001; WILEY‐VCH Verlag Berlin GmbH
• Subjects: Fourier transforms; Hardy spaces; moduli of continuity; rearrangements
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/1522-2616(200108)228:1<123::AID-MANA123>3.0.CO;2-A

One of the main purposes of this paper is to obtain estimates for Fourier transforms of functions in Lp(ℝn) (1 ≤ p ≤ 2) in terms of their moduli of continuity. More precisely, we study the following problem: find sharp conditions on the modulus of continuity of a function f ∈ Lp(ℝn), under which the non‐increasing rearragement of $\hat f$, the Fourier transform of f, is integrable against a given non‐negative weight function ρ. We shall also study similar problems for the Fourier transforms of functions or distributions in the Hardy spaces Hp(ℝn) (0 < p ≤ 1, n ∈ ℝ).