Abstract
In this paper we study the asymptotic behavior of functions defined on domains of a multidimensional real or complex space when the point tends to the boundary in the approach region with different orders of tangency. The main results are related to the boundary behavior of functions from Hardy-Sobolev spaces in a multidimensional complex ball and of solutions to elliptic boundary-value problems in a Lipschitz domain of a real Euclidean space. The methods used are based on two-weighted estimates for tangential maximal functions in an abstract ball. The boundary of this ball is a space equipped with measure and quasimetric.
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References
A. Nagel, W. Rudin, and J. Shapiro, “Tangential boundary behavior of functions in Dirichlet-type spaces,”Ann. of Math.,116, No. 2, 331–360 (1982).
A. Nagel and E. M. Stein, “On certain maximal functions and approach regions,”Adv. Math.,54, No. 1, 83–106 (1984).
P. Ahern and A. Nagel, “StrongL p-estimates for maximal functions with respect to singular measure with applications to exceptional sets,”Duke Math. J.,53, No. 2, 359–393 (1986).
V. G. Krotov, “Estimates for maximal operators associated with boundary behavior, and their applications,”Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.],190, 117–138 (1989).
V. G. Krotov, “Boundary behavior of functions from Hardy-type spaces,”Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv] 54, No. 1, 957–974 (1990).
J. Sueiro, “Tangential boundary limits and exceptional sets for harmonic functions in Dirichlet-type spaces,”J. Math. Ann.,286, No. 4, 661–678 (1990).
P. Cifuentes, J. Dorronsoro, and J. Sueiro, “Boundary tangential convergence in spaces of homogeneous type,”Trans. Amer. Math. Soc.,332, No. 1, 331–350 (1992).
C. Cascante and J. M. Ortega, “Tangential-exceptional sets for Hardy-Sobolev spaces,”Illinois J. Math.,39, No. 1, 68–85 (1995).
G. Verhota, “The Dirichlet problem for the polyharmonic equation in Liphschitz domains,”Indiana Univ. Math. J.,39, No. 3, 671–702 (1990).
V. G. Krotov, “Weighted inequalities related to the boundary behavior of functions from multidimensional Hardy-Sobolev classes and of the solutions of elliptic boundary-value problems” in:Boundary-Value Problems, Special Functions and Fractional Calculus. Proc. of the Internat. Conference Dedicated to the 90th Anniversary of Academician F. D. Gakhov [in Russian], 16–20 February, 1996, Minsk (1996), pp. 172–177.
R. R. Coifman and G. Weiss, “Extensions of Hardy spaces and their use in analysis,”Bull. Amer. Math. Soc.,83, No. 4, 569–645 (1977).
W. Rudin, Function Theory in the Unit Ball of ℂn, Springer-Verlag, Heidelberg (1981).
V. G. Krotov, “The differential properties on the boundary of functions holomorphic in the unit ball of ℂn,”Mat. Zametki [Math. Notes],45, No. 2, 51–59 (1989).
J. Pipher and G. C. Verhota, “Dilatation invariant estimates and the boundary Gårding inequality for the higher order elliptic operators,”Ann. of Math.,142, No. 1, 1–36 (1995).
P. Ahern and W. Cohn, “Exceptional sets for Hardy-Sobolev functions,”Indiana Univ. Math. J.,38, No. 2, 417–452 (1989).
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Translated fromMatematicheskie Zametki, Vol. 68, No. 2, pp. 230–248, August, 2000.
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Krotov, V.G. Tangential boundary behavior of functions of several variables. Math Notes 68, 201–216 (2000). https://doi.org/10.1007/BF02675346
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DOI: https://doi.org/10.1007/BF02675346