Bounds of Singular Integrals on Weighted Hardy Spaces and Discrete Littlewood–Paley Analysis

Abstract

1355-1375, 2007 Proc. Am. Math. Soc. 136(4):1237-1249), and Petermichl and Volberg (Duke Math. J. 112(2):281-305, 2002. Our main result is stated in Theorem 1.1. Our method avoids the atomic decomposition which was usually used in proving boundedness of singular integral operators on Hardy spaces.

Mixed estimates for singular integrals on weighted Hardy spaces

“…More recently, Lu and Zhu [22] apply a discrete version of Calderón's reproducing formula and Littlewood-Paley theory with weights to establish…”

“…The authors in [22,Theorem 1.1] show the boundedness of some sort of singular integrals in H p w (R n ) for w ∈ A ∞ . They claim that the bounds obtained in their article are quantitative but they fail in giving the explicit constant depending on the weight when switching from one H p w -norm to another (see [22,Proposition 2.1 and Corollary 2.1]); there is actually a dependence on the A ∞ constant of the weight w. Therefore, a difficulty in obtaining quantitative bounds for norms of operators in this setting is that one needs to be very careful when interchanging the norms in H p w (R n ) since there will be constants relating them that might depend on the weight. When one is just concerned about the boundedness of an operator, the constant can be disregarded.…”

“…In this article we shall precise the quantitative bounds obtained in [22]. While they provide bounds for the norm of the singular integral operator, they are not completely quantitative and accurate.…”

“…Afterwards, we consider singular integrals of convolution type with more general kernels that include the homogeneous type operators mentioned above. We follow the ideas by Lu and Zhu [22] and use the mixed bound A q − A ∞ for the vector-valued Hardy-Littlewood maximal operator obtained in [2], which is an improvement of the bound for this operator used by the authors. Moreover, we take special care of the equivalence between H p w (R n ) norms in order to get the correct dependence on the weight w.…”

A new approach to norm inequalities on weighted and variable Hardy spaces

“…Weighted Hardy space estimates for singular integrals (actually for the more general class of multipliers) were proved by Strömberg and Torchinsky [35]. Theorem 1.1 was proved by Lu and Zhu [25] for singular integrals with C ∞ kernels; see the bibliography of their paper for earlier results. Then T : H p(·) → H p(·) .…”