𝐴_{∞} estimates via extrapolation of Carleson measures and applications to divergence form elliptic operators

Hofmann, Steve; Martell, José María

doi:10.1090/S0002-9947-2011-05397-3

$A_\infty$ estimates via extrapolation of Carleson measures and applications to divergence form elliptic operators
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by Steve Hofmann and José María Martell PDF

Trans. Amer. Math. Soc. 364 (2012), 65-101 Request permission

Abstract:

We revisit the “extrapolation method” for Carleson measures, introduced by Lewis and Murray (1995), to prove $A_\infty$ estimates for certain caloric measures, and we present a purely real variable version of the method suitable for establishing $A_\infty$ estimates. To illustrate the use of this technique, we then reprove a well-known result of Fefferman, Kenig, and Pipher (1991).

References

Pascal Auscher, Steve Hofmann, John L. Lewis, and Philippe Tchamitchian, Extrapolation of Carleson measures and the analyticity of Kato’s square-root operators, Acta Math. 187 (2001), no. 2, 161–190. MR 1879847, DOI 10.1007/BF02392615
P. Auscher, S. Hofmann, C. Muscalu, T. Tao, and C. Thiele, Carleson measures, trees, extrapolation, and $T(b)$ theorems, Publ. Mat. 46 (2002), no. 2, 257–325. MR 1934198, DOI 10.5565/PUBLMAT_{4}6202_{0}1
Lennart Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547–559. MR 141789, DOI 10.2307/1970375
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, 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
G. David and S. Semmes, Singular integrals and rectifiable sets in $\textbf {R}^n$: Beyond Lipschitz graphs, Astérisque 193 (1991), 152 (English, with French summary). MR 1113517
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
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
Steve Hofmann and John L. Lewis, The Dirichlet problem for parabolic operators with singular drift terms, Mem. Amer. Math. Soc. 151 (2001), no. 719, viii+113. MR 1828387, DOI 10.1090/memo/0719
Steve Hofmann and José María Martell, A note on $A_\infty$ estimates via extrapolation of Carleson measures, The AMSI-ANU Workshop on Spectral Theory and Harmonic Analysis, Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 44, Austral. Nat. Univ., Canberra, 2010, pp. 143–166. MR 2655385
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
John L. Lewis and Margaret A. M. Murray, The method of layer potentials for the heat equation in time-varying domains, Mem. Amer. Math. Soc. 114 (1995), no. 545, viii+157. MR 1323804, DOI 10.1090/memo/0545