Quantitative Weighted Estimates for Rubio de Francia’s Littlewood–Paley Square Function
Abstract: We consider the Rubio de Francia's Littlewood-Paley square function associated with an arbitrary family of intervals in R with finite overlapping. Quantitative weighted estimates are obtained for this operator. The linear dependence on the characteristic of the weight [w] A p/2 turns out to be sharp for 3 ≤ p < ∞, whereas the sharpness in the range 2 < p < 3 remains as an open question. The results arise as a consequence of a sparse domination shown for these operators, obtained by suitably adapting the ideas … Show more
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Cited by 5 publications
References 39 publications
Let $$S^{(\Lambda )}$$ 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.
Let $$S^{(\Lambda )}$$ 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.
We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual $$\ell ^1$$ ℓ 1 -sum in the sparse operator is replaced by an $$\ell ^r$$ ℓ r -sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the $$A_2$$ A 2 -theorem for vector-valued Calderón–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.
In this paper we obtain quantitative weighted L p -inequalities for some operators involving Bessel convolutions. We consider maximal operators, Littlewood-Paley functions and variational operators. We obtain L p (w)-operator norms in terms of the Ap-characteristic of the weight w. In order to do this we show that the operators under consideration are dominated by a suitable family of sparse operators in the space of homogeneous type ((0, ∞), | • |, x 2λ dx).x d dx on (0, ∞). h λ , # λ and λ τ x , x ∈ (0, ∞), are connected with ∆ λ . For every f, g ∈ S(0, ∞), the Schwartz space on (0, ∞), we have that
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