Boundary tangential convergence on spaces of homogeneous type

Cifuentes, Patricio; Dorronsoro, José R.; Sueiro, Juan

doi:10.1090/S0002-9947-1992-1069747-0

Boundary tangential convergence on spaces of homogeneous type
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by Patricio Cifuentes, José R. Dorronsoro and Juan Sueiro PDF

Trans. Amer. Math. Soc. 332 (1992), 331-350 Request permission

Abstract:

We study tangential convergence of convolutions with approximate identities of functions defined on a homogeneous type space and having a certain regularity. Our results contain those already known for the Euclidean case and give new ones for stratified nilpotent Lie groups and for solutions of the Dirichlet problem on Lipschitz domains.

References

Patrick Ahern and William Cohn, Exceptional sets for Hardy Sobolev functions, $p>1$, Indiana Univ. Math. J. 38 (1989), no. 2, 417–453. MR 997390, DOI 10.1512/iumj.1989.38.38020
Patrick Ahern and Alexander Nagel, Strong $L^p$ estimates for maximal functions with respect to singular measures; with applications to exceptional sets, Duke Math. J. 53 (1986), no. 2, 359–393. MR 850541, DOI 10.1215/S0012-7094-86-05323-8
L. Caffarelli, E. Fabes, S. Mortola, and S. Salsa, Boundary behavior of nonnegative solutions of elliptic operators in divergence form, Indiana Univ. Math. J. 30 (1981), no. 4, 621–640. MR 620271, DOI 10.1512/iumj.1981.30.30049
Alberto-P. Calderón and Ridgway Scott, Sobolev type inequalities for $p>0$, Studia Math. 62 (1978), no. 1, 75–92. MR 487419, DOI 10.4064/sm-62-1-75-92
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
Ronald R. Coifman and Guido Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Mathematics, Vol. 242, Springer-Verlag, Berlin-New York, 1971 (French). Étude de certaines intégrales singulières. MR 0499948
Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, DOI 10.1007/BF00280445
Björn E. J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), no. 5, 1119–1138. MR 859772, DOI 10.2307/2374598
Ronald A. DeVore and Robert C. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc. 47 (1984), no. 293, viii+115. MR 727820, DOI 10.1090/memo/0293
José R. Dorronsoro, Poisson integrals of regular functions, Trans. Amer. Math. Soc. 297 (1986), no. 2, 669–685. MR 854092, DOI 10.1090/S0002-9947-1986-0854092-3
Eugene B. Fabes, David S. Jerison, and Carlos E. Kenig, Necessary and sufficient conditions for absolute continuity of elliptic-harmonic measure, Ann. of Math. (2) 119 (1984), no. 1, 121–141. MR 736563, DOI 10.2307/2006966
G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups, Ark. Mat. 13 (1975), no. 2, 161–207. MR 494315, DOI 10.1007/BF02386204
G. B. Folland, Lipschitz classes and Poisson integrals on stratified groups, Studia Math. 66 (1979), no. 1, 37–55. MR 562450, DOI 10.4064/sm-66-1-37-55
G. B. Folland and E. M. Stein, Estimates for the $\bar \partial _{b}$ complex and analysis on the Heisenberg group, Comm. Pure Appl. Math. 27 (1974), 429–522. MR 367477, DOI 10.1002/cpa.3160270403
G. B. Folland and Elias M. Stein, Hardy spaces on homogeneous groups, Mathematical Notes, vol. 28, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1982. MR 657581

Doctoral dissertation

Lars Inge Hedberg, On certain convolution inequalities, Proc. Amer. Math. Soc. 36 (1972), 505–510. MR 312232, DOI 10.1090/S0002-9939-1972-0312232-4
Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
Peter W. Jones, A geometric localization theorem, Adv. in Math. 46 (1982), no. 1, 71–79. MR 676987, DOI 10.1016/0001-8708(82)90054-8
A. Korányi and S. Vági, Singular integrals on homogeneous spaces and some problems of classical analysis, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 25 (1971), 575–648 (1972). MR 463513
Alexander Nagel, Walter Rudin, and Joel H. Shapiro, Tangential boundary behavior of functions in Dirichlet-type spaces, Ann. of Math. (2) 116 (1982), no. 2, 331–360. MR 672838, DOI 10.2307/2007064
Alexander Nagel and Elias M. Stein, On certain maximal functions and approach regions, Adv. in Math. 54 (1984), no. 1, 83–106. MR 761764, DOI 10.1016/0001-8708(84)90038-0
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Juan Sueiro, Tangential boundary limits and exceptional sets for holomorphic functions in Dirichlet-type spaces, Math. Ann. 286 (1990), no. 4, 661–678. MR 1045395, DOI 10.1007/BF01453595
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 0107776