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- W2912388832 abstract "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." @default.
- W2912388832 created "2019-02-21" @default.
- W2912388832 creator A5075033673 @default.
- W2912388832 date "2019-02-06" @default.
- W2912388832 modified "2023-10-15" @default.
- W2912388832 title "On a problem of Pichorides" @default.
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