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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 \lt p \lt \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 \gt l \} $. Variants in the periodic setting and for certain lacunary sets of order $N$ are also presented.