Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates

Cavero, Juan; Hofmann, Steve; Martell, José María; Toro, Tatiana

doi:10.1090/tran/8148

Perturbations of elliptic operators in 1-sided chord-arc domains. Part II: Non-symmetric operators and Carleson measure estimates
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by Juan Cavero, Steve Hofmann, José María Martell and Tatiana Toro PDF

Trans. Amer. Math. Soc. 373 (2020), 7901-7935 Request permission

Abstract:

We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and path-connectedness) and which have an Ahlfors regular boundary, a result of Kenig-Kirchheim-Pipher-Toro, in which Carleson measure estimates for bounded solutions of the equation $Lu=-\operatorname {div}(A\nabla u) = 0$ with $A$ being a real (not necessarily symmetric) uniformly elliptic matrix imply that the corresponding elliptic measure belongs to the Muckenhoupt $A_\infty$ class with respect to surface measure on the boundary. We present two applications of this result. In the first one we extend a perturbation result recently proved by Cavero-Hofmann-Martell presenting a simpler proof and allowing non-symmetric coefficients. Second, we prove that if an operator $L$ as above has locally Lipschitz coefficients satisfying certain Carleson measure condition, then $\omega _L\in A_\infty$ if and only if $\omega _{L^\top }\in A_\infty$. As a consequence, we can remove one of the main assumptions in the non-symmetric case of a result of Hofmann-Martell-Toro and show that if the coefficients satisfy a slightly stronger Carleson measure condition the membership of the elliptic measure associated with $L$ to the class $A_\infty$ yields that the domain is indeed a chord-arc domain.

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
Jonas Azzam, Steve Hofmann, José María Martell, Kaj Nyström, and Tatiana Toro, A new characterization of chord-arc domains, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 4, 967–981. MR 3626548, DOI 10.4171/JEMS/685
Christopher J. Bishop and Peter W. Jones, Harmonic measure and arclength, Ann. of Math. (2) 132 (1990), no. 3, 511–547. MR 1078268, DOI 10.2307/1971428
J. Bourgain, On the Hausdorff dimension of harmonic measure in higher dimension, Invent. Math. 87 (1987), no. 3, 477–483. MR 874032, DOI 10.1007/BF01389238
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
Lennart Carleson and John Garnett, Interpolating sequences and separation properties, J. Analyse Math., 28(1):273–299, 1975.
Juan Cavero, Steve Hofmann, and José María Martell, Perturbations of elliptic operators in 1-sided chord-arc domains. Part I: Small and large perturbation for symmetric operators, Trans. Amer. Math. Soc. 371 (2019), no. 4, 2797–2835. MR 3896098, DOI 10.1090/tran/7536
Michael Christ, A $T(b)$ theorem with remarks on analytic capacity and the Cauchy integral, Colloq. Math. 60/61 (1990), no. 2, 601–628. MR 1096400, DOI 10.4064/cm-60-61-2-601-628
Björn E. J. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 (1977), no. 3, 275–288. MR 466593, DOI 10.1007/BF00280445
G. David and D. Jerison, Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals, Indiana Univ. Math. J. 39 (1990), no. 3, 831–845. MR 1078740, DOI 10.1512/iumj.1990.39.39040
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
Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061, DOI 10.1090/surv/038
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
Juha Heinonen, Tero Kilpeläinen, and Olli Martio, Nonlinear potential theory of degenerate elliptic equations, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1993. Oxford Science Publications. MR 1207810
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, Phi Le, José María Martell, and Kaj Nyström, The weak-$A_\infty$ property of harmonic and $p$-harmonic measures implies uniform rectifiability, Anal. PDE 10 (2017), no. 3, 513–558. MR 3641879, DOI 10.2140/apde.2017.10.513
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
Steve Hofmann and José María Martell, $A_\infty$ estimates via extrapolation of Carleson measures and applications to divergence form elliptic operators, Trans. Amer. Math. Soc. 364 (2012), no. 1, 65–101. MR 2833577, DOI 10.1090/S0002-9947-2011-05397-3
Steve Hofmann and José María Martell, Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in $L^p$, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 3, 577–654 (English, with English and French summaries). MR 3239100, DOI 10.24033/asens.2223
Steve Hofmann, Dorina Mitrea, Marius Mitrea, and Andrew J. Morris, $L^p$-square function estimates on spaces of homogeneous type and on uniformly rectifiable sets, Mem. Amer. Math. Soc. 245 (2017), no. 1159, v+108. MR 3589162, DOI 10.1090/memo/1159
Steve Hofmann, José María Martell, and Tatiana Toro, Elliptic operators on non-smooth domains work in progress, 2014.
Steve Hofmann, José María Martell, and Tatiana Toro, $A_\infty$ implies NTA for a class of variable coefficient elliptic operators, J. Differential Equations 263 (2017), no. 10, 6147–6188. MR 3693172, DOI 10.1016/j.jde.2017.06.028
Steve Hofmann, José María Martell, and Ignacio Uriarte-Tuero, Uniform rectifiability and harmonic measure, II: Poisson kernels in $L^p$ imply uniform rectifiability, Duke Math. J. 163 (2014), no. 8, 1601–1654. MR 3210969, DOI 10.1215/00127094-2713809
David S. Jerison and Carlos E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 (1982), no. 1, 80–147. MR 676988, DOI 10.1016/0001-8708(82)90055-X
C. Kenig, B. Kirchheim, J. Pipher, and T. Toro, Square functions and the $A_\infty$ property of elliptic measures, J. Geom. Anal. 26 (2016), no. 3, 2383–2410. MR 3511480, DOI 10.1007/s12220-015-9630-6
C. Kenig, H. Koch, J. Pipher, and T. Toro, A new approach to absolute continuity of elliptic measure, with applications to non-symmetric equations, Adv. Math. 153 (2000), no. 2, 231–298. MR 1770930, DOI 10.1006/aima.1999.1899
Mikhail A. Lavrentiev, Boundary problems in the theory of univalent functions, (Russian). Math Sb., 43:815–846, 1936.
John L. Lewis and Margaret A. M. Murray, Regularity properties of commutators and layer potentials associated to the heat equation, Trans. Amer. Math. Soc. 328 (1991), no. 2, 815–842. MR 1020043, DOI 10.1090/S0002-9947-1991-1020043-6
Emmanouil Milakis, Jill Pipher, and Tatiana Toro, Harmonic analysis on chord arc domains, J. Geom. Anal. 23 (2013), no. 4, 2091–2157. MR 3107693, DOI 10.1007/s12220-012-9322-4
Emmanouil Milakis, Jill Pipher, and Tatiana Toro, Perturbations of elliptic operators in chord arc domains, Harmonic analysis and partial differential equations, Contemp. Math., vol. 612, Amer. Math. Soc., Providence, RI, 2014, pp. 143–161. MR 3204862, DOI 10.1090/conm/612/12229
Frigyes Riesz and Marcel Riesz, Uber die randwerte einer analytischen funktion, Comptes Rendus du Quatrième Congrès des Mathématiciens Scandinaves, 1916, pp, 27–44.
Stephen Semmes, Analysis vs. geometry on a class of rectifiable hypersurfaces in $\textbf {R}^n$, Indiana Univ. Math. J. 39 (1990), no. 4, 1005–1035. MR 1087183, DOI 10.1512/iumj.1990.39.39048
Zihui Zhao, BMO solvability and $A_{\infty }$ condition of the elliptic measures in uniform domains, J. Geom. Anal. 28 (2018), no. 2, 866–908. MR 3790485, DOI 10.1007/s12220-017-9845-9