| Title:
|
Boundedness of para-product operators on spaces of homogeneous type (English) |
| Author:
|
Xiao, Yayuan |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
67 |
| Issue:
|
1 |
| Year:
|
2017 |
| Pages:
|
235-252 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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\le 1$, 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. (English) |
| Keyword:
|
boundedness |
| Keyword:
|
Calderón-Zygmund singular integral operator |
| Keyword:
|
para-product |
| Keyword:
|
spaces of homogeneous type |
| MSC:
|
42B25 |
| MSC:
|
42B30 |
| idZBL:
|
Zbl 06738515 |
| idMR:
|
MR3633009 |
| DOI:
|
10.21136/CMJ.2017.0536-15 |
| . |
| Date available:
|
2017-03-13T12:10:32Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/146051 |
| . |
| Reference:
|
[1] Christ, M.: A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral.Colloq. Math. 60/61 (1990), 601-628. Zbl 0758.42009, MR 1096400, 10.4064/cm-60-61-2-601-628 |
| Reference:
|
[2] Coifman, R. R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogènes. Etude de certaines intégrales singulières.Lecture Notes in Mathematics 242, Springer, Berlin French (1971). Zbl 0224.43006, MR 0499948, 10.1007/BFb0058946 |
| Reference:
|
[3] Coifman, R. R., Weiss, G.: Extensions of Hardy spaces and their use in analysis.Bull. Am. Math. Soc. 83 (1977), 569-645. Zbl 0358.30023, MR 0447954, 10.1090/S0002-9904-1977-14325-5 |
| Reference:
|
[4] David, G., Journé, J.-L., Semmes, S.: Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation.Rev. Mat. Iberoam. 1 (1985), 1-56. Zbl 0604.42014, MR 0850408, 10.4171/RMI/17 |
| Reference:
|
[5] Deng, D., Han, Y.: Harmonic Analysis on Spaces of Homogeneous Type.Lecture Notes in Mathematics 1966, Springer, Berlin (2009). Zbl 1158.43002, MR 2467074, 10.1007/978-3-540-88745-4 |
| Reference:
|
[6] Fefferman, C., Stein, E. M.: $H^p$ spaces of several variables.Acta Math. 129 (1972), 137-193. Zbl 0257.46078, MR 0447953, 10.1007/BF02392215 |
| Reference:
|
[7] Frazier, M., Jawerth, B.: A discrete transform and decompositions of distribution spaces.J. Funct. Anal. 93 (1990), 34-170. Zbl 0716.46031, MR 1070037, 10.1016/0022-1236(90)90137-A |
| Reference:
|
[8] Han, Y.: Calderón-type reproducing formula and the $Tb$ theorem.Rev. Mat. Iberoam. 10 (1994), 51-91. Zbl 0797.42009, MR 1271757, 10.4171/RMI/145 |
| Reference:
|
[9] Han, Y.: Discrete Calderón-type reproducing formula.Acta Math. Sin., Engl. Ser. 16 (2000), 277-294. Zbl 0978.42010, MR 1778708, 10.1007/s101140050021 |
| Reference:
|
[10] Han, Y., Sawyer, E. T.: Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces.Mem. Am. Math. Soc. 110 (1994), no. 530, 126 pages. Zbl 0806.42013, MR 1214968, 10.1090/memo/0530 |
| Reference:
|
[11] Macías, R. A., Segovia, C.: Lipschitz functions on spaces of homogeneous type.Adv. Math. 33 (1979), 257-270. Zbl 0431.46018, MR 0546295, 10.1016/0001-8708(79)90012-4 |
| Reference:
|
[12] Meyer, Y., Coifman, R.: Wavelets: Calderón-Zygmund and Multilinear Operators.Cambridge Studies in Advanced Mathematics 48, Cambridge University Press, Cambridge (1997). Zbl 0916.42023, MR 1456993 |
| Reference:
|
[13] Sawyer, E., Wheeden, R. L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces.Am. J. Math. 114 (1992), 813-874. Zbl 0783.42011, MR 1175693, 10.2307/2374799 |
| . |
