Let $S^{ (\Lambda)}$ denote the classical Littlewood-Paley square function formed with respect to a lacunary sequence $\Lambda$ of positive integers. Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of the behaviour of the operator norm of $S^{ (\Lambda)}$ from the analytic Hardy space $H^p_A (\mathbb{ T})$ to $L^p (\mathbb{ T})$ and of the behaviour of the $L^p (\mathbb{ T}) \rightarrow L^p (\mathbb{ T})$ operator norm of $S^{ (\Lambda)}$ ($1 < p < 2$) in terms of the ratio of the lacunary sequence $\Lambda$. Namely, if $\rho_{ \Lambda}$ denotes the ratio of $\Lambda$, then we prove that $$ \sup_{ \substack{ \| f \|_{ L^p (\mathbb{ T})} = 1 \\ f \in H^p_A (\mathbb{ T}) } } \big\| S^{ (\Lambda)} (f) \big\|_{ L^p (\mathbb{ T})} \lesssim \frac{ 1}{ p-1} (\rho_{ \Lambda} - 1 )^{ -1/2} \quad (1<p<2)$$ and $$ \big\| S^{ (\Lambda)} \big\|_{ L^p (\mathbb{ T}) \rightarrow L^p (\mathbb{ T})} \lesssim \frac{ 1}{ (p-1)^{ 3/2}} (\rho_{ \Lambda} - 1 )^{ -1/2} \quad (1<p<2)$$ and that the exponents $r=1/2$ in $(\rho_{ \Lambda} - 1 )^{ -1/2} $ cannot be improved in general. Variants in higher dimensions and in the Euclidean setting are also obtained.
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