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1995/1996 Sharp Paley-Titchmarsh inequalities in Orlicz spaces
Hans P. Heinig, Adrian L. Lee
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Real Anal. Exchange 21(1): 244-257 (1995/1996).


Let \((Tf)(x)= x \hat f(x)\), where \(\hat f\) is the Fourier transform of \(f\). If \(P(t) = t \int^t_0 s^{-2} Q(s)\,ds\), \(t\gt 0\), where \(Q\) is some non-negative continuous function, then there is a constant \(A \gt 0\), such that \[ \int_{\mathbb{R}} Q(|Tf(x)|)\,\frac{dx}{x^2} \leq A \int_\mathbb{R} P(|f(x)|)\,dx\] holds. \(\!\)Moreover, \(\!\)if this inequality is satisfied for all \(f\), then \(t \!\int^t_0\! s^{-2} Q(s)ds \leq CP(t)\), \(t \gt 0\), for some constant \(C \gt 0\). Corresponding Orlicz space and dual inequalities for this and the Hardy averaging operator are also given.


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Hans P. Heinig. Adrian L. Lee. "Sharp Paley-Titchmarsh inequalities in Orlicz spaces." Real Anal. Exchange 21 (1) 244 - 257, 1995/1996.


Published: 1995/1996
First available in Project Euclid: 3 July 2012

zbMATH: 0851.42013
MathSciNet: MR1377533

Primary: 42A38
Secondary: 26D15

Keywords: Fourier transform , Modular Inequalities , Orlicz spaces

Rights: Copyright © 1995 Michigan State University Press

Vol.21 • No. 1 • 1995/1996
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