Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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[Lp] ESTIMATES FOR A ROUGH MAXIMAL OPERATOR ON PRODUCT SPACES

  • Published : 2005.05.01
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Abstract

We establish appropriate $L^p$ estimates for a class of maximal operators $S_{\Omega}^{(\gamma)}$ on the product space $R^n\;\times\;R^m\;when\;\Omega$ lacks regularity and $1\;\le\;\gamma\;\le\;2.\;Also,\;when\;\gamma\;=\;2$, we prove the $L^p\;(2\;{\le}\;P\;<\;\infty)\;boundedness\;of\;S_{\Omega}^{(\gamma)}\;whenever\;\Omega$ is a function in a certain block space $B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ (for some q > 1). Moreover, we show that the condition $\Omega\;{\in}\;B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ is nearly optimal in the sense that the operator $S_{\Omega}^{(2)}$ may fail to be bounded on $L^2$ if the condition $\Omega\;{\in}\;B_q^{(0,0)}(S^{n-1}\;\times\;S^{m-1})$ is replaced by the weaker conditions $\Omega\;{\in}\;B_q^{(0,\varepsilon)}(S^{n-1}\;\times\;S^{m-1})\;for\;any\;-1\;<\;\varepsilon\;<\;0.$

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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 https://doi.org/10.14492/hokmj/1285851910
  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 https://doi.org/10.14492/hokmj/1350911903
  3. J. Bourgain, Average in the plane over convex curves and maximal operators, J. Anal. Math. 47 (1986), 69-85 https://doi.org/10.1007/BF02792533
  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 https://doi.org/10.1007/BF01388746
  6. J. Duoandikoetxea, Multiple singular integrals and maximal functions along hypersurfaces, Ann. Ins. Fourier (Grenoble), 36 (1986), 185-206 https://doi.org/10.5802/aif.1073
  7. D. Fan, Y. Pan and D. Yang, A weighted norm inequality for rough singular integrals, Tohoku Math. J. 51 (1999), 141-161 https://doi.org/10.2748/tmj/1178224808
  8. R. Fefferman, Singular integrals on product domains, Bull. Amer. Math. Soc. 4 (1981), 195-201 https://doi.org/10.1090/S0273-0979-1981-14883-7
  9. R. Fefferman and E. M. Stein, Singular integrals on product spaces, Adv. in Math., 45 (1982), 117-143 https://doi.org/10.1016/S0001-8708(82)80001-7
  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 https://doi.org/10.14492/hokmj/1380892533
  11. M. Keitoku and E. Sato, Block spaces on the unit sphere in $R^{n}$, Proc. Amer. Math. Soc. 119 (1993), 453-455 https://doi.org/10.2307/2159928
  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 https://doi.org/10.1512/iumj.1985.34.34028
  14. E. M. Stein, Maximal functions: spherical means, Proc. Natl. Acad. Sci. USA 73 (1976), 2174-2175 https://doi.org/10.1073/pnas.73.7.2174
  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