On the order of magnitude of Hausdorff-Young transforms (Q5932688)

In a recent paper [Acta Math. Hung. 75, No. 3, 227-243 (1997; Zbl 0881.42005)] \textit{M. Bagota, D. V. Giang}, and \textit{F. Móricz} investigated the order of magnitude of the Fourier transform in the sense of Cesàro of order \(\alpha >0\) (\((C, \alpha)\)-sense). The aim of this paper is to extend this result from the Fourier transform to more general Hausdorff--Young transforms. Let \(p, q\) be two numbers such that \(1< p\leq 2\) and \(1/p + 1/q =1\). A linear operator \(\Phi \) from \(L^p(\mathbf{R}_{+})\) to \(L^q(\mathbf{R}_{+})\) is of Hausdorff--Young type if \(\|\Phi (f)\|_q\leq K(p)\|f\|_p\), where \(f\in L^p(\mathbf{R}_{+})\), and \(\lim_{u\to \infty}\Phi (g)(u)=0\) for all \(g\in L^1(\mathbf{R}_{+})\). The following main theorem is shown: If \(f\in L^1\cap L^p(\mathbf{R}_{+})\) for some \(1<p\leq 2\), \(1/p+1/q=1\), \(\alpha >0\), then \(\lambda_{\alpha }(x)|\Phi (f)(x)|\to 0\) as \(x\to \infty\) in \((C, \alpha)\)-sense, where \(\lambda _{\alpha }(x)= x^{1/q}, x^{\alpha }, x^{1/q}(\log(x+2))^{1/q -1}\) according as \(\alpha >1/q, 0<\alpha <1/q, \alpha =1/q\), respectively. Several corollaries for Fourier transform, Walsh-Fourier transform and operators with Fourier-type kernels are given.