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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.

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B30 \(H^p\)-spaces
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