$A_\infty$ estimates via extrapolation of Carleson measures and applications to divergence form elliptic operators
HTML articles powered by AMS MathViewer
- 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
Additional Information
- Steve Hofmann
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 251819
- ORCID: 0000-0003-1110-6970
- Email: hofmanns@missouri.edu
- José María Martell
- Affiliation: Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM, Consejo Superior de Investigaciones Científicas, C/ Nicolás Cabrera, 13-15, E-28049 Madrid, Spain
- MR Author ID: 671782
- ORCID: 0000-0001-6788-4769
- Email: chema.martell@icmat.es
- Received by editor(s): November 23, 2009
- Published electronically: August 2, 2011
- Additional Notes: The first author was supported by NSF grants DMS-0245401 and DMS-0801079.
The second author was supported by MEC Grant MTM2010-16518 and by CSIC PIE 200850I015. This work has been possible thanks to the support and hospitality of the University of Missouri-Columbia (USA), the Universidad Autónoma de Madrid (Spain), the Centre de Recerca Matemàtica (Spain), the Consejo Superior de Investigaciones Científicas (Spain), and the BIRS Centre in Banff (Canada). Both authors would like to express their gratitude to these institutions. - © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 65-101
- MSC (2010): Primary 42B99, 42B25, 35J25
- DOI: https://doi.org/10.1090/S0002-9947-2011-05397-3
- MathSciNet review: 2833577