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Bounds of singular integrals on weighted Hardy spaces and discrete Littlewood-Paley analysis. (English) Zbl 1408.42009
Summary: We apply the discrete version of Calderón’s reproducing formula and Littlewood-Paley theory with weights to establish the $$H^{p}_{w} \to H^{p}_{w} (0<p<\infty)$$ and $$H^{p}_{w}\to L^{p}_{w} (0<p\leq 1)$$ boundedness for singular integral operators and derive some explicit bounds for the operator norms of singular integrals acting on these weighted Hardy spaces when we only assume $$w\in A_\infty$$. The bounds will be expressed in terms of the $$A_q$$ constant of $$w$$ if $$q>q_w =\inf\{s:w\in A_s\}$$. Our results can be regarded as a natural extension of the results about the growth of the $$A_p$$ constant of singular integral operators on classical weighted Lebesgue spaces $$L^{p}_{w}$$ in [T. P. Hytönen et al., “Weak and strong type $$A_p$$ estimates for Calderón-Zygmund operators”, Preprint, arXiv:1006.2530; “Weak and strong-type estimates for Haar shift operators: sharp power on the $$A_p$$ characteristic”, Preprint, arXiv:0911.0713; A. K. Lerner [Ill. J. Math. 52, No. 2, 653–666 (2008; Zbl 1177.42016); Proc. Am. Math. Soc. 136, No. 8, 2829–2833 (2008; Zbl 1273.42019); Int. Math. Res. Not. 2008, Article ID rnm161, 11 p. (2008; Zbl 1237.42012); Math. Res. Lett. 16, No. 1, 149–156 (2009; Zbl 1169.42006); M. T. Lacey et al., J. Funct. Anal. 259, No. 5, 1073–1097 (2010; Zbl 1196.42014); Math. Ann. 348, No. 1, 127–141 (2010; Zbl 1210.42017); S. Petermichl, Am. J. Math. 129, No. 5, 1355–1375 (2007; Zbl 1139.44002); Proc. Am. Math. Soc. 136, No. 4, 1237–1249 (2008; Zbl 1142.42005); with A. Volberg, Duke Math. J. 112, No. 2, 281–305 (2002; Zbl 1025.30018)]. 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.

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B30 $$H^p$$-spaces
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