×

zbMATH — the first resource for mathematics

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
PDF BibTeX Cite
Full Text: DOI
References:
[1] Andersen, K.F., John, R.T.: Weighted inequalities for vector-valued maximal function and singular integrals. Stud. Math. 69, 19–31 (1980) · Zbl 0448.42016
[2] Bownik, M., Li, B., Yang, D., Zhou, Y.: Weighted anisotropic Hardy spaces and their applications in boundedness of sublinear operators. Indiana Univ. Math. J. 57(7), 3065–3100 (2008) · Zbl 1161.42014
[3] Buckley, S.: Estimates for operator norms on weighted spaces and reverse Jensen inequalities. Trans. Am. Math. Soc. 340, 253–272 (1993) · Zbl 0795.42011
[4] Ding, Y., Han, Y., Lu, G., Wu, X.: Boundedness of singular integrals on multi-parameter weighted Hardy spaces $H\^{p}_{w}(\(\backslash\)mathbf{R}\^{n} \(\backslash\)times \(\backslash\)mathbf{R}\^{m})$ . Preprint (2008)
[5] Dragicević, O., Grafakos, L., Pereyra, M., Petermichl, S.: Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces. Publ. Math. 49(1), 73–91 (2005) · Zbl 1081.42007
[6] Fefferman, C., Stein, E.M.: H p spaces of several variables. Acta Math. 129, 137–194 (1972) · Zbl 0257.46078
[7] Garcia-Cuerva, J.: Weighted Hardy spaces. Diss. Math. 162, 1–63 (1979) · Zbl 0417.30026
[8] Garcia-Cuerva, J., Martell, J.: Wavelet characterization of weighted spaces. J. Geom. Anal. 11(2), 241–264 (2001) · Zbl 0995.42016
[9] Garcia-Cuerva, J., Rubio de Francia, J.: Weighted Norm Inequalities and Related Topics. North-Holland, Amsterdam (1985) · Zbl 0578.46046
[10] Grafakos, L.: Classical and Modern Fourier Analysis. Pearson/Prentice Hall, New York (2004) · Zbl 1148.42001
[11] Han, Y., Lu, G.: Discrete Littlewood–Paley–Stein theory and multi-parameter Hardy spaces associated with flag singular integrals. arXiv:0801.1701v1 (2008)
[12] Han, Y., Lu, G.: Some recent works on multiparameter Hardy space theory and discrete Littlewood–Paley analysis. In: Trends in Partial Differential Equations. Adv. Lect. Math. (ALM), vol. 10, pp. 99–191. Int. Press, Somerville (2009) · Zbl 1201.42016
[13] Hytonen, T., Lacey, M., Reguera, M.C., Sawyer, E., Uriarte-Tuero, I., Vagharshakyan, A.: Weak and strong type A p estimates for Calderón–Zygmund operators. arXiv:1006.2530
[14] Hytonen, T.P., Lacey, M., Reguera, M.C., Vagharshakyan, A.: Weak and strong-type estimates for Haar shift operators: sharp power on the A p characteristic. arXiv:0911.0713
[15] Lacey, M., Petermichl, S., Reguera, M.C.: Sharp A 2 inequality for Haar shift operators. arXiv:0906.1941 · Zbl 1210.42017
[16] Lacey, M., Moen, K., Pérez, C., Torres, R.: Sharp weighted inequalities for fractional operators. arXiv:0905.3839v2 [math.CA] (2009) · Zbl 1196.42014
[17] Lee, M.-Y., Lin, C.-C.: The molecular characterization of weighted Hardy spaces. J. Funct. Anal. 188, 442–460 (2002) · Zbl 0998.42013
[18] Lee, M.Y., Lin, C.C., Lin, Y.C., Dunyan, D.Y.: Boundedness of singular integral operators with variable kernels. J. Math. Anal. Appl. 348(2), 787–796 (2008) · Zbl 1159.42011
[19] Lerner, A.K.: On some sharp weighted norm inequalities for Littlewood–Paley operators. Ill. J. Math. 52, 653–666 (2008) · Zbl 1177.42016
[20] Lerner, A.K.: An elementary approach to several results on the Hardy–Littlewood maximal operators. Proc. Am. Math. Soc. 136(8), 2829–2833 (2008) · Zbl 1273.42019
[21] Lerner, A.K., Ombrosi, S., Pérez, C.: Sharp A 1 bounds for Calderón–Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden. Int. Math. Res. Notes 2008, article ID rnm 126 (2008), 11 pp. · Zbl 1237.42012
[22] Lerner, A.K., Ombrosi, S., Pérez, C.: A 1 bounds for Calderón–Zygmund operators related to a problem of Muckenhoupt and Wheeden. Math. Res. Lett. 16, 149–156 (2009) · Zbl 1169.42006
[23] Nagel, A., Ricci, F., Stein, E.M.: Singular integrals with flag kernels and analysis on quadratic CR manifolds. J. Funct. Anal. 181, 29–118 (2001) · Zbl 0974.22007
[24] Perez, C., Treil, S., Volberg, A.: On A2 conjecture and Corona decomposition of weights. arXiv:1006.2630
[25] Petermichl, S.: The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p -characteristic. Am. J. Math. 129(5), 1355–1375 (2007) · Zbl 1139.44002
[26] Petermichl, S.: The sharp weighted bound for the Riesz transforms. Proc. Am. Math. Soc. 136(4), 1237–1249 (2008) · Zbl 1142.42005
[27] Petermichl, S., Volberg, A.: Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular. Duke Math. J. 112(2), 281–305 (2002) · Zbl 1025.30018
[28] Ruan, Z.: Weighted Hardy spaces in three parameter case. J. Math. Anal. Appl. 367, 625–639 (2010) · Zbl 1198.42015
[29] Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993) · Zbl 0821.42001
[30] Strömberg, J.O., Torchinsky, A.: Weights, Sharp maximal functions and Hardy spaces. Bull. Am. Math. Soc. 3, 1053–1056 (1982) · Zbl 0452.43004
[31] Strömberg, J.O., Torchinsky, A.: Weighted Hardy Spaces. Lecture Notes in Math., vol. 1381. Springer, Berlin (1989) · Zbl 0676.42021
[32] Strömberg, J.O., Wheeden, R.L.: Relations between $H\^{p}_{u}$ and $L\^{p}_{u}$ in a product space. Trans. Am. Math. Soc. 315, 769–797 (1989) · Zbl 0689.42020
[33] Wu, S.: A wavelet characterization for weighted Hardy spaces. Rev. Mat. Iberoam. 8(3), 329–349 (1992) · Zbl 0769.42011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.