Fatou theorems for parabolic equations
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- by Caroline Sweezy
- Proc. Amer. Math. Soc. 124 (1996), 2343-2355
- DOI: https://doi.org/10.1090/S0002-9939-96-03687-8
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Abstract:
For elliptic parabolic operators with time dependent coefficients, bounded and measurable, the absolute continuity of the two caloric measures plus a Fatou theorem are shown to hold on the parabolic boundary of a smooth cylinder given a Carleson-type condition on the coefficients of the operators, and assuming one of the measures is a center doubling measure. Given a stronger Carleson condition, and no doubling assumption, another kind of Fatou theorem result holds. The method of proof follows that of Fefferman, Kenig and Pipher.References
- D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 22 (1968), 607–694. MR 435594
- P. Erdös and T. Grünwald, On polynomials with only real roots, Ann. of Math. (2) 40 (1939), 537–548. MR 7, DOI 10.2307/1968938
- 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
- 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
- Björn E. J. Dahlberg, David S. Jerison, and Carlos E. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients, Ark. Mat. 22 (1984), no. 1, 97–108. MR 735881, DOI 10.1007/BF02384374
- J. L. Doob, Classical potential theory and its probabilistic counterpart, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR 731258, DOI 10.1007/978-1-4612-5208-5
- Neil A. Eklund, Existence and representation of solutions of parabolic equations, Proc. Amer. Math. Soc. 47 (1975), 137–142. MR 361442, DOI 10.1090/S0002-9939-1975-0361442-1
- Eugene B. Fabes, Nicola Garofalo, and Sandro Salsa, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. Math. 30 (1986), no. 4, 536–565. MR 857210
- R. A. Fefferman, C. E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. (2) 134 (1991), no. 1, 65–124. MR 1114608, DOI 10.2307/2944333
- Yanick Heurteaux, Inégalités de Harnack à la frontière pour des opérateurs paraboliques, C. R. Acad. Sci. Paris Sér. I Math. 308 (1989), no. 13, 401–404 (French, with English summary). MR 992517
- Carlos E. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems, CBMS Regional Conference Series in Mathematics, vol. 83, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1994. MR 1282720, DOI 10.1090/cbms/083
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- C. Sweezy, Absolute continuity for elliptic-caloric measures, Studia Math. (to appear).
Bibliographic Information
- Caroline Sweezy
- Affiliation: Department of Mathematical Sciences, New Mexico State University, Las Cruces, New Mexico 88003
- Email: csweezy@nmsu.edu
- Received by editor(s): May 18, 1994
- Received by editor(s) in revised form: December 7, 1994
- Communicated by: J. Marshall Ash
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 2343-2355
- MSC (1991): Primary 35K20, 42K25
- DOI: https://doi.org/10.1090/S0002-9939-96-03687-8
- MathSciNet review: 1363188