On a Problem of Pichorides

Abstract

AbstractLet $$S^{(\Lambda )}$$ S ( Λ ) denote the classical Littlewood–Paley operator formed with respect to a lacunary sequence $$\Lambda $$ Λ of positive integers. Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of the behaviour of the operator norm of $$S^{(\Lambda )}$$ S ( Λ ) from the analytic Hardy space $$H^p_A (\mathbb {T})$$ H A p ( T ) to $$L^p (\mathbb {T})$$ L p ( T ) and of the behaviour of the $$L^p (\mathbb {T}) \rightarrow L^p (\mathbb {T})$$ L p ( T ) → L p ( T ) operator norm of $$S^{(\Lambda )}$$ S ( Λ ) ($$1< p < 2$$ 1 < p < 2 ) in terms of the ratio of the lacunary sequence $$\Lambda $$ Λ . Namely, if $$\rho _{\Lambda }$$ ρ Λ denotes the ratio of $$\Lambda $$ Λ , then we prove that $$\begin{aligned} \sup _{\begin{array}{c} \Vert f \Vert _{L^p (\mathbb {T})} = 1 \\ f \in H^p_A (\mathbb {T}) \end{array}} \big \Vert S^{(\Lambda )} (f) \big \Vert _{L^p (\mathbb {T})} \lesssim \frac{1}{p-1} (\rho _{\Lambda } - 1)^{-1/2} \quad (1< p < 2) \end{aligned}$$ sup ‖ f ‖ L p ( T ) = 1 f ∈ H A p ( T ) ‖ S ( Λ ) ( f ) ‖ L p ( T ) ≲ 1 p - 1 ( ρ Λ - 1 ) - 1 / 2 ( 1 < p < 2 ) and $$\begin{aligned} \big \Vert S^{(\Lambda )} \big \Vert _{L^p (\mathbb {T}) \rightarrow L^p (\mathbb {T})} \lesssim \frac{1}{(p-1)^{3/2}} (\rho _{\Lambda } - 1)^{-1/2} \quad (1<p < 2) \end{aligned}$$ ‖ S ( Λ ) ‖ L p ( T ) → L p ( T ) ≲ 1 ( p - 1 ) 3 / 2 ( ρ Λ - 1 ) - 1 / 2 ( 1 < p < 2 ) and that these results are optimal as $$p \rightarrow 1^+$$ p → 1 + . Variants in higher dimensions and in the Euclidean setting are also obtained.