Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 235-252, 2017


Boundedness of para-product operators on spaces of homogeneous type

Yayuan Xiao

Received October 8, 2015.  First published February 24, 2017.

Abstract:  We obtain the boundedness of Calderón-Zygmund singular integral operators $T$ of non-convolution type on Hardy spaces $H^p(\mathcal X)$ for $ 1/{(1+\epsilon)}<p\le1$, where ${\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss (1971), and $\epsilon$ is the regularity exponent of the kernel of the singular integral operator $T$. Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón's identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast to what was used in the literature.
Keywords:  boundedness; Calderón-Zygmund singular integral operator; para-product; spaces of homogeneous type
Classification MSC:  42B25, 42B30
DOI:  10.21136/CMJ.2017.0536-15


References:
[1] M. Christ: A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 60/61 (1990), 601-628. MR 1096400 | Zbl 0758.42009
[2] R. R. Coifman, G. Weiss: Analyse harmonique non-commutative sur certains espaces homogènes. Etude de certaines intégrales singulières. Lecture Notes in Mathematics 242, Springer, Berlin French (1971). DOI 10.1007/BFb0058946 | MR 0499948 | Zbl 0224.43006
[3] R. R. Coifman, G. Weiss: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83 (1977), 569-645. DOI 10.1090/S0002-9904-1977-14325-5 | MR 0447954 | Zbl 0358.30023
[4] G. David, J.-L. Journé, S. Semmes: Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation. Rev. Mat. Iberoam. 1 (1985), 1-56. DOI 10.4171/RMI/17 | MR 0850408 | Zbl 0604.42014
[5] D. Deng, Y. Han: Harmonic Analysis on Spaces of Homogeneous Type. Lecture Notes in Mathematics 1966, Springer, Berlin (2009). DOI 10.1007/978-3-540-88745-4 | MR 2467074 | Zbl 1158.43002
[6] C. Fefferman, E. M. Stein: $H^p$ spaces of several variables. Acta Math. 129 (1972), 137-193. DOI 10.1007/BF02392215 | MR 0447953 | Zbl 0257.46078
[7] M. Frazier, B. Jawerth: A discrete transform and decompositions of distribution spaces. J. Funct. Anal. 93 (1990), 34-170. DOI 10.1016/0022-1236(90)90137-A | MR 1070037 | Zbl 0716.46031
[8] Y. Han: Calderón-type reproducing formula and the $Tb$ theorem. Rev. Mat. Iberoam. 10 (1994), 51-91. DOI 10.4171/RMI/145 | MR 1271757 | Zbl 0797.42009
[9] Y. Han: Discrete Calderón-type reproducing formula. Acta Math. Sin., Engl. Ser. 16 (2000), 277-294. DOI 10.1007/s101140050021 | MR 1778708 | Zbl 0978.42010
[10] Y. Han, E. T. Sawyer: Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces. Mem. Am. Math. Soc. 110 (1994), no. 530, 126 pages. DOI 10.1090/memo/0530 | MR 1214968 | Zbl 0806.42013
[11] R. A. Macías, C. Segovia: Lipschitz functions on spaces of homogeneous type. Adv. Math. 33 (1979), 257-270. DOI 10.1016/0001-8708(79)90012-4 | MR 0546295 | Zbl 0431.46018
[12] Y. Meyer, R. Coifman: Wavelets: Calderón-Zygmund and Multilinear Operators. Cambridge Studies in Advanced Mathematics 48, Cambridge University Press, Cambridge (1997). MR 1456993 | Zbl 0916.42023
[13] E. Sawyer, R. L. Wheeden: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114 (1992), 813-874. DOI 10.2307/2374799 | MR 1175693 | Zbl 0783.42011

Affiliations:   Yayuan Xiao, Department of Mathematics, Ball State University, 2000 W University Ave, Muncie 47306, Indianapolis, USA, e-mail: yxiao3@bsu.edu

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