[Lp] ESTIMATES FOR A ROUGH MAXIMAL OPERATOR ON PRODUCT SPACES

Title & Authors
[Lp] ESTIMATES FOR A ROUGH MAXIMAL OPERATOR ON PRODUCT SPACES
AL-QASSEM HUSSAIN MOHAMMED;

Abstract
We establish appropriate $\small{L^p}$ estimates for a class of maximal operators $\small{S_{\Omega}^{(\gamma)}}$ on the product space $\small{R^n\;\times\;R^m\;when\;\Omega}$ lacks regularity and $1\;\le\;\gamma\;\le\;2.\;Also,\;when\;\gamma\; Keywords rough kernel;block space;singular integral;maximal operator;product domains; Language English Cited by References 1. A. Al-Salman, H. Al-Qassem, and Y. Pan, Singular integrals associated to homogeneous mappings with rough kernels, Hokkaido Math. J. 33 (2004), 551-569 2. H. Al-Qassem and Y. Pan,$L^p$boundedness for singular integrals with rough ker nels on product domains, Hokkaido Math. J. 31 (2002), 555-613 3. J. Bourgain, Average in the plane over convex curves and maximal operators, J. Anal. Math. 47 (1986), 69-85 4. L. K. Chen and H. Lin, A maximal operator related to a class of singular integrals, Illinois J. Math. 34 (1990), 120-126 5. J. Duoandikoetxea and J. L. Rubio de Francia, Maximal functions and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), 541- 561 6. J. Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Ins. Fourier (Grenoble), 36 (1986), 185-206 7. D. Fan, Y. Pan and D. Yang, A weighted norm inequality for rough singular integrals, Tohoku Math. J. 51 (1999), 141-161 8. R. Fefferman, Singular integrals on product domains, Bull. Amer. Math. Soc. 4 (1981), 195-201 9. R. Fefferman and E. M. Stein, Singular integrals on product spaces, Adv. in Math., 45 (1982), 117-143 10. Y. Jiang and S. Lu, A class of singular integral operators with rough kernels on product domains, Hokkaido Math. J. 24 (1995), 1-7 11. M. Keitoku and E. Sato, Block spaces on the unit sphere in$R^{n}$, Proc. Amer. Math. Soc. 119 (1993), 453-455 12. S. Lu, M. Taibleson, and G. Weiss, Spaces Generated by Blocks, Beijing Normal University Press, Beijing, 1989 13. Y. Meyer, M. Taibleson, and G. Weiss, Some functional analytic properties of the space$B_q\$ generated by blocks, Indiana Univ. Math. J. 34 (1985), no. 3, 493-515

14.
E. M. Stein, Maximal functions: spherical means, Proc. Natl. Acad. Sci. USA 73 (1976), 2174-2175

15.
E. M. Stein, Singular integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970

16.
M. H. Taibleson and G. Weiss, Certain function spaces associated with a.e. convergence of Fourier series, Univ. of Chicago Conf. in honor of Zygmund, Woodsworth, 1983