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Weighted Carleson Measure Spaces Associated with Different Homogeneities

Published online by Cambridge University Press:  24 December 2018

Xinfeng Wu
Xinfeng Wu*
Affiliation:
Department of Mathematics, China University of Mining ℘ Technology (Beijing), Beijing 100083, China. e-mail: [email protected]
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Abstract

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In this paper, we introduce weighted Carleson measure spaces associated with different homogeneities and prove that these spaces are the dual spaces of weighted Hardy spaces studied in a forthcoming paper. As an application, we establish the boundedness of composition of two Calderón–Zygmund operators with different homogeneities on the weighted Carleson measure spaces; this, in particular, provides the weighted endpoint estimates for the operators studied by Phong–Stein.

Keywords

42B2042B35composition of operatorsweighted Carleson measure spacesduality
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 66 , Issue 6 , 01 December 2014 , pp. 1382 - 1412
Copyright
Copyright © Canadian Mathematical Society 2014

References

[CF1] Chang, S.-Y. A. and Fefferman, R., A continuous verion of duality of H1with BMO on the bi-disc. Ann. of Math. 112(1980), no. 1, 179–201.http://dx.doi.org/10.2307/1971324 Google Scholar
[CF2] Chang, S.-Y. A. and Fefferman, R., Some recent developments in Fourier analysis and Hp-theory on product domains. Bull. Amer. Math. Soc. (N.S.) 12(1985), no. 1, 1–43.http://dx.doi.org/10.1090/S0273-0979-1985-15291-7 Google Scholar
[DHLW] Ding, Y., Han, Y., Lu, G., and Wu, X., Boundedness of singular integrals on multiparameter weighted Hardy spaces Hp w(Rn✗ Rm). Potential Anal. 37(2012), no. 1, 31–56. http://dx.doi.org/10.1007/s11118-011-9244-y Google Scholar
[FS1] Fefferman, R. and Stein, E. M., Singular integrals on product spaces. Adv. in Math. 45(1982),no. 2, 117–143.http://dx.doi.org/10.1016/S0001-8708(82)80001-7 Google Scholar
[FL] Ferguson, S. and Lacey, M. T., A characterization of product BMO by commutators. Acta Math. 189(2002), no. 2, 143–160.http://dx.doi.org/10.1007/BF02392840 Google Scholar
[FS2] Folland, G. B. and Stein, E. M., Hardy spaces on homogeneous groups. Mathematical Notes, 28, Princeton University Press, Princeton, NJ; University of Tokyo Press, Tokyo, 1982.Google Scholar
[FJ] Frazier, M. and Jawerth, B., A discrete transform and decompositions of distribution spaces. J. Func. Anal. 93(1990), no. 1, 34–170.http://dx.doi.org/10.1016/0022-1236(90)90137-A Google Scholar
[FJW] Frazier, M., Jawerth, B., and Weiss, G., Littlewood-Paley theory and the study of function spaces. CBMS Regional Conference Series in Mathematics, 79, American Mathematical Society, Providence, RI, 1991.Google Scholar
[Ga] Garcia-Cuerva, J., Weighted Hardy spaces. In: Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll.,Williamstown, Mass., 1978), Part 1, Proc. Sympos. Pure Math., 35, American Mathematical Society, Providence, RI, 1979, pp. 253–261.Google Scholar
[GR] Garcia-Cuerva, J. and Rubio de Francia, J., Weighted norm inequalities and related topics. North-Holland Mathematics Studies, 116, Mathematical Notes, 104, North-Holland, Amsterdam, 1985.Google Scholar
[Ha] Han, Y., Discrete Calderön-type reproducing formula. Acta Math. Sin. (Engl. Ser.) 16(2000), no. 2, 277–294.http://dx.doi.org/10.1007/s101140000037 Google Scholar
[HLLRS] Han, Y., Lin, C.-C., Lu, G., Ruan, Z., and Sawyer, E., Hardy spaces associated with different homogeneities and boundedness of composition operators. Rev. Mat. Iberoamericana, to appear.Google Scholar
[Jo] Journé, J.-L., Calderón-Zygmund operators on product spaces. Rev. Mat. Iberomaricana 1(1985), no. 3, 55–91.http://dx.doi.org/10.4171/RMI/15 Google Scholar
[Kr] Krug, D., A weighted version of the atomic decomposition for Hp (bi-half space). Indiana Univ. Math. J. 37(1988), no. 2, 277–300.http://dx.doi.org/10.1512/iumj.1988.37.37014 Google Scholar
[KT] Krug, D. and Torchinsky, A. , A weighted version of Journé's Lemma. Rev. Mat. Iberoamericana 10(1994), no. 2, 363–378.http://dx.doi.org/10.4171/RMI/155 Google Scholar
[LPPW] Lacey, M., Petermichl, S., Pipher, J., and Wick, B., Multiparameter Riesz commutators. Amer. J. Math. 131(2009), no. 3, 731–769.http://dx.doi.org/10.1353/ajm.0.0059 Google Scholar
[LLL] Lee, M.-Y., Lin, C.-C., and Lin, Y.-C., A wavelet characterization for the dual of weighted Hardyspaces. Proc. Amer. Math. Soc. 137(2009), no. 12, 4219–4225.http://dx.doi.org/10.1090/S0002-9939-09-10044-8 Google Scholar
[MR] Madych, W. and Rivière, N., Multipliers of the Hölder classes. J. Functional Analysis 21(1976), no. 4, 369–379.http://dx.doi.org/10.1016/0022-1236(76)90032-X Google Scholar
[PS] Phong, D. H. and Stein, E. M. , Some further classes of pseudodifferential and singular-integral operators arising in boundary value problems. I. Composition of operators. Amer. J. Math. 104(1982), no. 1, 141–172.http://dx.doi.org/10.2307/2374071 Google Scholar
[St] Stein, E. M., Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton Mathematical Series, 43, Monographs in Harmonic Analysis, III, Princeton University Press. Princeton, NJ, 1993.Google Scholar
[ST] Strömberg, J.-O. and Torchinsky, A., Weighted Hardy spaces. Lecture Notes in Mathematics, 1381, Springer-Verlag, Berlin, 1989.Google Scholar
[WW] Wainger, S. and Weiss, G., Harmonic analysis in Euclidean spaces. I. Proceedings of Symp. in Pure Math., 35, American Mathematical Society, Providence, RI, 1979.Google Scholar
[Wu] Wu, X., Weighted norm inequalities for composition of operators associated with different homogeneities. Submitted,http://lxy.cumtb.edu.cn/1.pdf.Google Scholar