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

We apply the discrete version of Calderón’s reproducing formula and Littlewood–Paley theory with weights to establish the H w → H w (0 < p < ∞) and H w → Lw (0 < p ≤ 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 ∈ A∞. The bounds will be expressed in terms of the Aq constant of w if q > qw = inf{s : w ∈ As}. Our results can be regarded as a natural extension of the results about the growth of the Ap constant of singular integral operators on classical weighted Lebesgue spaces Lw in Hytonen et al. (arXiv:1006.2530, 2010; arXiv:0911.0713, 2009), Lerner (Ill. J. Math. 52:653–666, 2008; Proc. Am. Math. Soc. 136(8):2829–2833, 2008), Lerner et al. (Int. Math. Res. Notes 2008:rnm 126, 2008; Math. Res. Lett. 16:149–156, 2009), Lacey et al. (arXiv:0905.3839v2, 2009; arXiv:0906.1941, 2009), Petermichl (Am. J. Math. 129(5):1355–1375, 2007; Proc. Am. Math. Soc. 136(4):1237–1249, 2008), 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.