Sharp Paley-Titchmarsh inequalities in Orlicz spaces

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.