On a problem of Pichorides

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.