  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 \} $.
