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## Studia Mathematica

1998 | 131 | 2 | 115-135
Tytuł artykułu

### $B^q$ for parabolic measures

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
If Ω is a Lip(1,1/2) domain, μ a doubling measure on $∂_{p}Ω, ∂/∂t - L_{i}$, i = 0,1, are two parabolic-type operators with coefficients bounded and measurable, 2 ≤ q < ∞, then the associated measures $ω_{0}$, $ω_{1}$ have the property that $ω_{0} ∈ B^{q}(μ)$ implies $ω_{1}$ is absolutely continuous with respect to $ω_{0}$ whenever a certain Carleson-type condition holds on the difference function of the coefficients of $L_{1}$ and $L_{0}$. Also $ω_{0} ∈ B^{q}(μ)$ implies $ω_{1} ∈ B^{q}(μ)$ whenever both measures are center-doubling measures. This is B. Dahlberg's result for elliptic measures extended to parabolic-type measures on time-varying domains. The method of proof is that of Fefferman, Kenig and Pipher.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
115-135
Opis fizyczny
Daty
wydano
1998
otrzymano
1996-12-05
poprawiono
1997-12-29
Twórcy
autor
• Department of Mathematical Sciences, MSC 3MB, New Mexico State University, P.O. Box 30001, Las Cruces, New Mexico 88003-8001, U.S.A., csweezy@nmsu.edu
Bibliografia
• [A] D. G. Aronson, Non-negative solutions of linear parabolic equations, Ann. Scuola Norm. Sup. Pisa 22 (1968), 607-694.
• [RB] R. Brown, Area integral estimates for caloric functions, Trans. Amer. Math. Soc. 315 (1989), 565-589.
• [CW] S. Chanillo and R. Wheeden, Harnack's inequality and mean-value inequalities for solutions of degenerate elliptic equations, Comm. Partial Differential Equations 11 (1986), 1111-1134.
• [CF] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250.
• [D] B. E. J. Dahlberg, On the absolute continuity of elliptic measures, Amer. J. Math. 108 (1986), 1119-1138.
• [DJK] B. E. J. Dahlberg, D. Jerison and C. Kenig, Area integral estimates for elliptic differential operators with non-smooth coefficients, Ark. Mat. 22 (1984), 97-108.
• [Do] J. Doob, Classical Potential Theory and its Probabilistic Counterpart, Springer, 1984.
• [FGS] E. Fabes, N. Garofalo, and S. Salsa, A backward Harnack inequality and Fatou theorem for nonnegative solutions of parabolic equations, Illinois J. Math. 30 (1986), 536-565.
• [FGSII] E. Fabes, N. Garofalo, and S. Salsa, Comparison theorems for temperatures in non-cylindrical domains, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 77 (1984), 1-12.
• [FKP] R. Fefferman, C. Kenig and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. 134 (1991), 65-124.
• [GS] M. Giaquinta and M. Struwe, On the partial regularity of weak solutions of nonlinear parabolic systems, Math. Z. 179 (1982), 437-451.
• [GW] C. Gutiérrez and R. Wheeden, Harnack's inequality for degenerate parabolic equations, Comm. Partial Differential Equations 16 (1991), 745-770.
• [YH] Y. Heurteaux, Inégalités de Harnack à la frontière pour des opérateurs paraboliques, C. R. Acad. Sci. Paris 308 (1989), 401-404, 441-444.
• [HL] S. Hofmann and J. Lewis, $L^2$ solvability and representation by caloric layer potentials in time-varying domains, Ann. of Math. 144 (1996), 349-420.
• [JK] D. Jerison and C. Kenig, Boundary behavior of harmonic functions in non-tangentially accessible domains, Adv. Math. 46 (1982), 80-147.
• [K] J. T. Kemper, Temperatures in several variables; kernel functions, representations and parabolic boundary values, Trans. Amer. Math. Soc. 167 (1972), 243-262.
• [CK] C. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Regional Conf. Ser. in Math. 83, Amer. Math. Soc., 1994.
• [LSU] O. Ladyzhenskaya, V. Solonnikov, and N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, Transl. Math. Monographs, Amer. Math. Soc., 1968.
• [N] K. Nystrom, The Dirichlet problem for second order parabolic operators, Indiana Univ. Math. J. 46 (1997), 183-245.
• [S] S. Salsa, Some properties of non-negative solutions of parabolic differential operators, Ann. Mat. Pura Appl. 128 (1981), 193-206.
• [CS] C. Sweezy, Absolute continuity for elliptic-caloric measures, Studia Math. 120 (1996), 95-112.
Typ dokumentu
Bibliografia
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