Abstract
We establish a connection between the absolute continuity of elliptic measure associated with a second order divergence form operator with bounded measurable coefficients with the solvability of an end-point BMO Dirichlet problem. We show that these two notions are equivalent. As a consequence we obtain an end-point perturbation result, i.e., the solvability of the BMO Dirichlet problem implies L p solvability for all p>p 0.
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Caffarelli, L., Fabes, E., Mortola, S., Salsa, S.: Boundary behavior of non-negative solutions of elliptic operators in divergence form. Indiana Univ. Math. J. 30, 621–640 (1981)
Coifman, R., Fefferman, C.: Weighted norm inequalities for maximal functions and singular integrals. Studia Math. 51, 241–250 (1974)
Coifman, R., Rochberg, R., Weiss, G.: Factorization theorems for Hardy spaces in several variables. Ann. Math. 103, 611–635 (1976)
Dahlberg, B.E.J.: On estimates for harmonic measure. Arch. Ration. Mech. Anal. 65, 272–288 (1977)
Dahlberg, B.E.J.: Weighted norm inequalities for the Luisin area integral and the nontangential maximal function for functions harmonic in a Lipschitz domain. Studia Math. 67(3), 297–314 (1980)
Dahlberg, B.E.J.: Approximation of harmonic functions. Ann. Inst. Fourier 30, 97–107 (1980) (Grenoble)
Dahlberg, B.E.J., Jerison, D., Kenig, C.: Area integral estimates for elliptic differential operators with nonsmooth coefficients. Ark. Mat. 22, 97–108 (1984)
Dindos, M.: Hardy spaces and potential theory for C 1 domains in Riemannian manifolds. Mem. Am. Math. Soc. 191(894) (2008)
Fabes, E., Neri, U.: Dirichlet problem in Lipschitz domains with BMO data. Proc. Am. Math. Soc. 78, 33–39 (1980)
Fefferman, C., Stein, E.M.: H p spaces of several variables. Acta Math. 129, 137–193 (1972)
Fefferman, R., Kenig, C., Pipher, J.: The theory of weights and the Dirichlet problem for elliptic equations. Ann. Math, Ser. 2 131(1), 65–121 (1991)
Garnett, J.B.: Bounded Analytic Functions. Academic Press, New York (1981)
Hofmann, S., Lewis, J.: The Dirichlet problem for parabolic operators with singular drift terms. Mem. Am. Math. Soc. 719 (2001)
Jerison, D., Kenig, C.: The Dirichlet problem in nonsmooth domains. Ann. Math., Ser. 2 113(2), 367–382 (1981)
Journé, J.-L., Jones, P.: On weak convergence in ℋ1(R d). Proc. Am. Math. Soc. 120, 137–138 (1994)
Kenig, C.: Harmonic analysis techniques for second order elliptic boundary value problems. CBMS Regional Conference Series, No. 83 (1992)
Kenig, C., Pipher, J.: The Neumann problem for elliptic equations with non-smooth coefficients. Invent. Math. 113, 447–509 (1993)
Kenig, C., Pipher, J.: The Dirichlet problem for elliptic equations with drift terms. Publ. Mat. 45(1), 199–217 (2001)
Kenig, C., Koch, H., Pipher, J., Toro, T.: A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations. Adv. Math. 153(2), 231–298 (2000)
Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)
Sarason, D.: Functions of vanishing mean oscillations. Trans. Am. Math. Soc. 207, 391–405 (1975)
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Communicated by Marco Peloso.
Research of M. Dindos was supported by EPRC grant EP/F014589/1-253000.
Research of C. Kenig and J. Pipher was supported by NSF.
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Dindos, M., Kenig, C. & Pipher, J. BMO Solvability and the A ∞ Condition for Elliptic Operators. J Geom Anal 21, 78–95 (2011). https://doi.org/10.1007/s12220-010-9142-3
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DOI: https://doi.org/10.1007/s12220-010-9142-3