eb6c9a3d-e8ce-45d7-b8f9-e27fc293a71220220312084948326naun:naunmdt@crossref.orgMDT DepositInternational Journal of Pure Mathematics2313-057110.46300/91019http://www.naun.org/cms.action?id=698531220223122022910.46300/91019.2022.9https://npublications.com/journals/puremath/2022.phpGabriellaZeccaDepartment of Mathematics, University of Naples ’Federico II’ Naples, ItalyWe establish a connection between the solvability of end-point BMO and Lp Dirichlet problems for a second order divergence form elliptic operator (not necessarily symmetric) with bounded measurable coefficients. In particular, we give a lower bound for the exponent p > 1 in terms of the BMO-constant of L (see Definition 8).3122022312202231348https://npublications.com/journals/puremath/2022/a162019-008(2022).pdf10.46300/91019.2022.9.8https://npublications.com/journals/puremath/2022/a162019-008(2022).pdf10.1512/iumj.1981.30.30049L. Caffarelli, E. Fabes, S. Mortola and S. Salsa, “Boundary behavior of non-negative solutions of elliptic operators in divergence form”, Ind. U. Math. J., 30, (1981) 621-640; 10.4064/sm-51-3-241-250Coifmann, R. Fefferman, “Weighted norm inequalities for maximal functions and singular integrals”, Studia Math., 51 (1974), 241-250. B. E. J. Dahlberg, “On estimates of harmonic measure”, Arch. Rat. Mech. Anal., 65, (1977), 272-288; B. E. J. Dahlberg, “On the Poisson integral for Lipschitz and C1 domains”, Studia Math., 66, (1979), 7-24; 10.1007/s12220-010-9142-3M. Dindos, C. Kenig, J.Pipher “BMO Solvability and the A∞ Condition for Elliptic Operator ”. J. Geom. Anal. (2011) 21 78-95; 10.2307/2944333R. Fefferman, C. Kenig and J. Pipher, “The Theory of Weights and the Dirichlet Problem for Elliptic Equations” Annals of Math., 134, (1991), 65-124; 10.1016/s0304-0208(08)73086-xJ. Garcia-Cuerva, J. L. Rubio de Francia, “Weighted norm inqualities and related topics”, North- Holland Math. Stud., vol. 116, North-Holland, Amsterdam, (1985); C. E. Kenig, “Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems”, Conference Board of the Mathematical Sciences, Amer. Math. Soc. 83, (1991); 10.1006/aima.1999.1899C. E. Kenig, H. Koch, J. Pipher and T. Toro “A new approach to absolute continuity of the elliptic measure, with applications to non-symmetric equations” Adv. Math. 153, 2 (2000) 231-298; 10.1090/s0002-9947-1972-0293384-6B. Muckenhoupt, “Weighted norm inqualities for the Hardy maximal function”, Trans. Amer. Math. Soc., 165, (1972), 207-226; 10.1007/bf03017196C. Sbordone, “Sharp embeddings for classes of weights and applications”, Rend Accad. Naz. delle Scienze detta dei XL, Memorie di Matematica e Applicazioni 1230 XXIX, fasc. 1, (2005), 339-354; 10.1007/s00041-009-9075-zC. Sbordone, G.Zecca “ The Lp -solvability of the Dirichlet problem for planar elliptic equations, sharp results.” J. Fourier Anal. Appl. 15 (2009), no. 6, 871-903. G. Zecca, “On the Dirichlet problem with Orlicz boundary data”, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 10 (2007), no. 3, 661–679; G. Zecca, “The unsolvability of the Dirichlet problem with L(logL) α boundary data”, Rend. Acc. Sc. Fis. Mat. Napoli, 72, (2005) 71-80.