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