Sharp asymptotic estimates for a class of Littlewood-Paley operators

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 $\| S_{\mathcal{I}_{E_2}} \|_{L^p (\mathbb{R}) \rightarrow L^p (\mathbb{R})} \sim (p-1)^{-2} \quad (p \rightarrow 1^+) ,$ where $S_{\mathcal{I}_{E_2}}$ denotes the classical Littlewood-Paley operator formed with respect to the second order lacunary set $E_2 = \{ \pm ( 2^k - 2^l ) : k,l \in \mathbb{Z} \text{ with } k>l \}$.

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