On $L^p$ bounds for the Littlewood-Paley $g_\alpha$ function with fractional poisson kernel

Abstract

Let \(0<\alpha\leq1\), and let \(g_\alpha\) be the Littlewood-Paley function associated with the fractional Poisson kernel, which reduces to the classical \(g\)-function when \(\alpha=1\). We establish dimension-free \(L^p\) bounds for \(g_\alpha\) for every \(1<p<\infty\) and identify its exact \(L^2\) norm as \(\sqrt{\alpha/2}\). We further determine the precise asymptotic behavior of the weak-type \((1,1)\) constant as \(n\to\infty\), yielding quantitative upper and lower bounds in terms of the dimension.