Sharp asymptotic estimates for a class of Littlewood-Paley operators

Abstract

It is well-known that Littlewood-Paley operators formed with respect to lacunary sets of finite order are bounded on $L^p(\mathbb{R})$ for all $1<p<\infty$. In this note it is shown that $$\Vert S_{{\mathcal{I}}_{E_2}}{\Vert }_{L^p(\mathbb{R})\to L^p(\mathbb{R})}\sim (p-1{)}^{-2}(p\to 1^{+}),$$ where $S_{{\mathcal{I}}_{E_2}}$ denotes the classical Littlewood-Paley operator formed with respect to the second order lacunary set $E_2=\{\pm (2^k-2^l):k,l\in \mathbb{Z}\text{ with }k>l\}$.

11 pages