Two-weight inequalities for the Hilbert transform of monotone functions
type: Item
instance of: scholarly article
title: Two-weight inequalities for the Hilbert transform of monotone functions
zbMATH Open document ID: 1253.26034
DOI: 10.1134/S1064562411020359
published in: Doklady Mathematics
publication date: 2012-01-17T00:00:00Z
review text: The authors characterize the weighted inequality \[ \left( \int_{-\infty }^{\infty }\left| \mathcal{H} f(x)\right| ^{q}\left| x\right| ^{\alpha }dx\right) ^{\frac{1}{q} }\leq C\left( \int_{-\infty }^{\infty }\left| f(x)\right| ^{p}\left| x\right| ^{\beta }dx\right) ^{\frac{1}{p}} \] restricted to the cone of odd or even monotone functions, where \(\mathcal{H}\) is the Hilbert transform and \(0<p\leq q<\infty\). Furthermore, the discrete analogue of the above result is also established. Several consequences of the above results are examined and discussed.
reviewed by: James A. Oguntuase
Mathematics Subject Classification ID: 26D10
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 5998378
zbMATH Keywords: Hilbert transform
zbMATH Keywords: weight
zbMATH Keywords: monotone functions
author: V. D. Stepanov
author: Sergey Tikhonov
MaRDI profile type: Publication
cites work: Some more theorems concerning Fourier series and Fourier power series
cites work: Some Theorems on Odd and Even Functions
cites work: Weighted Paley-Wiener theorem on the Hilbert transform
cites work: Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture
cites work: Inequalities with Weights for Discrete Hilbert Transforms✝
description: scientific article; zbMATH DE number 5998378