Quantitative Absolute Continuity of Harmonic Measure and the Dirichlet Problem: A Survey of Recent Progress
|
摘要
It is a well-known folklore result that quantitative, scale invariant absolute continuity(more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω ? R~(n+1) with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in L~p (?Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with continuous boundary data, can be solved, one may seek to characterize the open sets for which L~psolvability holds, thus allowing for singular boundary data.It has been known for some time that absolute continuity of harmonic measure is closely tied to rectifiability properties of ?Ω, but also that rectifiability alone is not sufficient to guarantee absolute continuity. In this note, we survey recent progress in this area, culminating in a geometric characterization of the weak-A∞ property, and hence of solvability of the L~p Dirichlet problem for some finite p. This characterization, obtained under rather optimal background hypotheses, follows from a combination of the present author's joint work with Martell, and the work of Azzam, Mourgoglou and Tolsa.
It is a well-known folklore result that quantitative, scale invariant absolute continuity(more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω ? R~(n+1) with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in L~p (?Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with continuous boundary data, can be solved, one may seek to characterize the open sets for which L~psolvability holds, thus allowing for singular boundary data.It has been known for some time that absolute continuity of harmonic measure is closely tied to rectifiability properties of ?Ω, but also that rectifiability alone is not sufficient to guarantee absolute continuity. In this note, we survey recent progress in this area, culminating in a geometric characterization of the weak-A∞ property, and hence of solvability of the L~p Dirichlet problem for some finite p. This characterization, obtained under rather optimal background hypotheses, follows from a combination of the present author's joint work with Martell, and the work of Azzam, Mourgoglou and Tolsa.
It is a well-known folklore result that quantitative, scale invariant absolute continuity(more precisely, the weak-A∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω ? R~(n+1) with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in L~p (?Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with continuous boundary data, can be solved, one may seek to characterize the open sets for which L~psolvability holds, thus allowing for singular boundary data.It has been known for some time that absolute continuity of harmonic measure is closely tied to rectifiability properties of ?Ω, but also that rectifiability alone is not sufficient to guarantee absolute continuity. In this note, we survey recent progress in this area, culminating in a geometric characterization of the weak-A∞ property, and hence of solvability of the L~p Dirichlet problem for some finite p. This characterization, obtained under rather optimal background hypotheses, follows from a combination of the present author's joint work with Martell, and the work of Azzam, Mourgoglou and Tolsa.
引文
[1]Aikawa,H.:Boundary Harnack principle and Martin boundary for a uniform domain.J.Math.Soc.Japan,53(1),119145.6(2001)
[2]Aikawa,H.:Potential theoretic characterizations of nonsmooth domains.Bull.London Math.Soc.,36,469-482(2004)
[3]Aikawa,H.,Hirata,K.:Doubling conditions for harmonic measure in John domains.Ann.Inst.Fourier(Grenoble),58(2),429-445(2008)
[4]Akman,M.,Azzam,J.,Mourgoglou,M.:Absolute continuity of harmonic measure for domains with lower regular boundaries,preprint,arXiv:1605.07291
[5]Akman,M.,Bortz,S.,Hofmann,S.,et al.:Rectifiability,interior approximation and Harmonic Measure,preprint,arXiv:1601.08251
[6]Alt,H.,Caffarelli,L.:Existence and regularity for a minimum problem with free boundary.J.Reine Angew.Math.,325,105-144(1981)
[7]Alt,H.W.,Caffarelli,L.A.,Friedman,A.:Variational problems with two phases and their free boundaries.Trans.Amer.Math.Soc.,282(2),431-461(1984)
[8]Azzam,J.:Semi-uniform domains and a characterization of the A∞property for harmonic measure,preprint,arXiv:1711.03088
[9]Azzam,J.,Garnett,J.,Mourgoglou,M.,et al.:Uniform rectifiability,elliptic measure,square functions,and-approximability via an ACF monotonicity formula.Preprint 2016.arXiv:1612.02650
[10]Azzam,J.,Hofmann,S.,Martell,J.M.,et al.:Volberg rectifiability of harmonic measure.GAFA,26,703-728(2016)
[11]Azzam,J.,Mourgoglou,M.,Tolsa,X.:Singular sets for harmonic measure on locally flat domains with locally finite surface measure.Int.Math.Res.Not.IMRN,2017(12),3751-3773(2017)
[12]Azzam,J.,Mourgoglou,M.,Tolsa,X.:Harmonic measure and quantitative connectivity:geometric characterization of the Lp-solvability of the Dirichlet problem.Part II,preprint,arXiv:1803.07975
[13]Badger,M.:Null sets of harmonic measure on NTA domains:Lipschitz approximation revisited.Math.Z.,270(1-2),241-262(2012)
[14]Bennewitz,B.,Lewis,J.L.:On weak reverse H?lder inequalities for nondoubling harmonic measures.Complex Var.Theory Appl.,49(7-9),571-582(2004)
[15]Bishop,C.,Jones,P.:Harmonic measure and arclength.Ann.of Math.(2),132,511-547(1990)
[16]Bortz,S.,Hofmann,S.:Quantitative Fatou Theorems and uniform rectifiability,preprint,arXiv:1801.01371,to appear in Potential Analysis
[17]Bourgain,J.:On the Hausdorff dimension of harmonic measure in higher dimensions.Invent.Math.,87,477-483(1987)
[18]Christ,M.:A T(b)theorem with remarks on analytic capacity and the Cauchy integral.Colloq.Math.,LX/LXI,601-628(1990)
[19]Coifman,R.,Fefferman,C.:Weighted norm inequalities for maximal functions and singular integrals.Studia Math.,51,241-250(1974)
[20]Dahlberg,B.:On estimates for harmonic measure.Arch.Rat.Mech.Analysis,65,272-288(1977)
[21]David,G.:Morceaux de graphes lipschitziens et int′egrales singulières sur une surface.(French)[Pieces of Lipschitz graphs and singular integrals on a surface].Rev.Mat.Iberoamericana,4(1),73-114(1988)
[22]David,G.,Jerison,D.:Lipschitz approximation to hypersurfaces,harmonic measure,and singular integrals.Indiana Univ.Math.J.,39(3),831-845(1990)
[23]David,G.,Semmes,S.:Singular integrals and rectifiable sets in Rn:Beyond Lipschitz graphs.Asterisque,193,(1991)
[24]David,G.,Semmes,S.:Analysis of and on Uniformly Rectifiable Sets,Mathematical Monographs and Surveys.38,AMS,Providence,RI,1993
[25]Garnett,J.,Mourgoglou,M.,Tolsa,X.:Uniform rectifiability from Carleson measure estimates andε-approximability of bounded harmonic functions.Duke Math.J.,167(8),1473-1524(2018)
[26]Heinonen,J.,Kilpel?inen,T.,Martio.O.:Nonlinear potential theory of degenerate elliptic equations.Dover(Rev.ed.),2006
[27]Hofmann,S.,Le,P.:BMO solvability and absolute continuity of harmonic measure.J.Geom.Anal,https://doi.org/10.1007/s12220-017-9959-0
[28]Hofmann,S.,Le,P.,Martell,J.M.:The weak-A∞property of harmonic and p-harmonic measures implies uniform rectifiability.Anal.PDE.,10(3),513-558(2017)
[29]Hofmann,S.,Martell,J.M.:Uniform rectifiability and harmonic measure IV:Ahlfors regularity plus Poisson kernels in Lpimplies uniform rectifiability,preprint,arXiv:1505.06499
[30]Hofmann,S.,Martell,J.M.:Harmonic measure and quantitative connectivity:geometric characterization of the Lp-solvability of the Dirichlet problem,Part I,preprint,arXiv:1712.03696v3
[31]Hofmann,S.,Martell,J.M.,Mayboroda,S.:Uniform rectifiability,Carleson measure estimates,and approximation of harmonic functions.Duke Math.J.,165(12),2331-2389(2016)
[32]Hofmann,S.,Martell,J.M.,Mayboroda,S.,et al.:Uniform rectifiability and elliptic operators with small Carleson norm,preprint,arXiv:1710.06157
[33]Hofmann,S.,Martell,J.M.,Toro,T.:A∞implies NTA for a class of variable coefficient elliptic operators.J.Differential Equations,263(10),6147-6188(2017)
[34]Jerison,D.:Regularity of the Poisson kernel and free boundary problems.Colloquium Mathematicum,LX/LXI,547-567(1990)
[35]Jerison,D.,Kenig,C.:Boundary behavior of harmonic functions in nontangentially accessible domains.Adv.in Math.,46(1),80-147(1982)
[36]Kenig,C.:Harmonic analysis techniques for second order elliptic boundary value problems,CBMS Regional Conference Series in Mathematics,83.Published for the Conference Board of the Mathematical Sciences,Washington,DC;by the American Mathematical Society,Providence,RI,1994
[37]Kenig,C.,Toro,T.:Poisson kernel characterizations of Reifenberg flat chord arc domains.Ann.Sci.′Ecole Norm.Sup.(4),36(3),323-401(2003)
[38]Lavrentiev,M.:Boundary problems in the theory of univalent functions(Russian).Math Sb.,43,815-846(1936),AMS Transl.Series 2,32,1-35(1963)
[39]Lewis,J.L.,Vogel,A.:Symmetry theorems and uniform rectifiability.Boundary Value Problems,2007,(2007),article ID 030190,59 pages
[40]Mattila,P.,Melnikov,M.,Verdera,J.:The Cauchy integral,analytic capacity,and uniform rectifiability.Ann.of Math.(2),144(1),127-136(1996)
[41]Mourgoglou,M.,Tolsa,X.:Harmonic measure and Riesz transform in uniform and general domains.J.Reine Angew.Math.,to appear
[42]Muckenhoupt,B.:Weighted norm inequalities for the Hardy maximal function.Trans.Amer.Math.Soc.,165,207-226(1972)
[43]Nazarov,F.,Tolsa,X.,Volberg,A.:On the uniform rectifiability of ad-regular measures with bounded Riesz transform operator:The case of codimension 1.Acta Math.,213(2),237-321(2014)
[44]Riesz,F.and Riesz,M.:¨Uber die randwerte einer analtischen funktion,Compte Rendues du Quatrième Congrès des Math′ematiciens Scandinaves,Stockholm 1916,Almqvists and Wilksels,Uppsala,1920
[45]Semmes,S.:A criterion for the boundedness of singular integrals on hypersurfaces.Trans.Amer.Math.Soc.,311,501-513(1989)
1)In fact,even the latter condition may be relaxed somewhat;we refer the reader to[22]for details.
2)“Capacity Density Condition”:a quantitative version of Wiener regularity.
3)which is itself an endpoint version of the results of[6]and of[34]
4)This idea is also exploited in[37].
5)The alert reader may object that in Theorem 2,one assumes an interior Corkscrew condition;in fact,the latter condition holds automatically in domains(with ADR boundaries)whose harmonic measure is doubling,and of course,A∞entails doubling.
[2]Aikawa,H.:Potential theoretic characterizations of nonsmooth domains.Bull.London Math.Soc.,36,469-482(2004)
[3]Aikawa,H.,Hirata,K.:Doubling conditions for harmonic measure in John domains.Ann.Inst.Fourier(Grenoble),58(2),429-445(2008)
[4]Akman,M.,Azzam,J.,Mourgoglou,M.:Absolute continuity of harmonic measure for domains with lower regular boundaries,preprint,arXiv:1605.07291
[5]Akman,M.,Bortz,S.,Hofmann,S.,et al.:Rectifiability,interior approximation and Harmonic Measure,preprint,arXiv:1601.08251
[6]Alt,H.,Caffarelli,L.:Existence and regularity for a minimum problem with free boundary.J.Reine Angew.Math.,325,105-144(1981)
[7]Alt,H.W.,Caffarelli,L.A.,Friedman,A.:Variational problems with two phases and their free boundaries.Trans.Amer.Math.Soc.,282(2),431-461(1984)
[8]Azzam,J.:Semi-uniform domains and a characterization of the A∞property for harmonic measure,preprint,arXiv:1711.03088
[9]Azzam,J.,Garnett,J.,Mourgoglou,M.,et al.:Uniform rectifiability,elliptic measure,square functions,and-approximability via an ACF monotonicity formula.Preprint 2016.arXiv:1612.02650
[10]Azzam,J.,Hofmann,S.,Martell,J.M.,et al.:Volberg rectifiability of harmonic measure.GAFA,26,703-728(2016)
[11]Azzam,J.,Mourgoglou,M.,Tolsa,X.:Singular sets for harmonic measure on locally flat domains with locally finite surface measure.Int.Math.Res.Not.IMRN,2017(12),3751-3773(2017)
[12]Azzam,J.,Mourgoglou,M.,Tolsa,X.:Harmonic measure and quantitative connectivity:geometric characterization of the Lp-solvability of the Dirichlet problem.Part II,preprint,arXiv:1803.07975
[13]Badger,M.:Null sets of harmonic measure on NTA domains:Lipschitz approximation revisited.Math.Z.,270(1-2),241-262(2012)
[14]Bennewitz,B.,Lewis,J.L.:On weak reverse H?lder inequalities for nondoubling harmonic measures.Complex Var.Theory Appl.,49(7-9),571-582(2004)
[15]Bishop,C.,Jones,P.:Harmonic measure and arclength.Ann.of Math.(2),132,511-547(1990)
[16]Bortz,S.,Hofmann,S.:Quantitative Fatou Theorems and uniform rectifiability,preprint,arXiv:1801.01371,to appear in Potential Analysis
[17]Bourgain,J.:On the Hausdorff dimension of harmonic measure in higher dimensions.Invent.Math.,87,477-483(1987)
[18]Christ,M.:A T(b)theorem with remarks on analytic capacity and the Cauchy integral.Colloq.Math.,LX/LXI,601-628(1990)
[19]Coifman,R.,Fefferman,C.:Weighted norm inequalities for maximal functions and singular integrals.Studia Math.,51,241-250(1974)
[20]Dahlberg,B.:On estimates for harmonic measure.Arch.Rat.Mech.Analysis,65,272-288(1977)
[21]David,G.:Morceaux de graphes lipschitziens et int′egrales singulières sur une surface.(French)[Pieces of Lipschitz graphs and singular integrals on a surface].Rev.Mat.Iberoamericana,4(1),73-114(1988)
[22]David,G.,Jerison,D.:Lipschitz approximation to hypersurfaces,harmonic measure,and singular integrals.Indiana Univ.Math.J.,39(3),831-845(1990)
[23]David,G.,Semmes,S.:Singular integrals and rectifiable sets in Rn:Beyond Lipschitz graphs.Asterisque,193,(1991)
[24]David,G.,Semmes,S.:Analysis of and on Uniformly Rectifiable Sets,Mathematical Monographs and Surveys.38,AMS,Providence,RI,1993
[25]Garnett,J.,Mourgoglou,M.,Tolsa,X.:Uniform rectifiability from Carleson measure estimates andε-approximability of bounded harmonic functions.Duke Math.J.,167(8),1473-1524(2018)
[26]Heinonen,J.,Kilpel?inen,T.,Martio.O.:Nonlinear potential theory of degenerate elliptic equations.Dover(Rev.ed.),2006
[27]Hofmann,S.,Le,P.:BMO solvability and absolute continuity of harmonic measure.J.Geom.Anal,https://doi.org/10.1007/s12220-017-9959-0
[28]Hofmann,S.,Le,P.,Martell,J.M.:The weak-A∞property of harmonic and p-harmonic measures implies uniform rectifiability.Anal.PDE.,10(3),513-558(2017)
[29]Hofmann,S.,Martell,J.M.:Uniform rectifiability and harmonic measure IV:Ahlfors regularity plus Poisson kernels in Lpimplies uniform rectifiability,preprint,arXiv:1505.06499
[30]Hofmann,S.,Martell,J.M.:Harmonic measure and quantitative connectivity:geometric characterization of the Lp-solvability of the Dirichlet problem,Part I,preprint,arXiv:1712.03696v3
[31]Hofmann,S.,Martell,J.M.,Mayboroda,S.:Uniform rectifiability,Carleson measure estimates,and approximation of harmonic functions.Duke Math.J.,165(12),2331-2389(2016)
[32]Hofmann,S.,Martell,J.M.,Mayboroda,S.,et al.:Uniform rectifiability and elliptic operators with small Carleson norm,preprint,arXiv:1710.06157
[33]Hofmann,S.,Martell,J.M.,Toro,T.:A∞implies NTA for a class of variable coefficient elliptic operators.J.Differential Equations,263(10),6147-6188(2017)
[34]Jerison,D.:Regularity of the Poisson kernel and free boundary problems.Colloquium Mathematicum,LX/LXI,547-567(1990)
[35]Jerison,D.,Kenig,C.:Boundary behavior of harmonic functions in nontangentially accessible domains.Adv.in Math.,46(1),80-147(1982)
[36]Kenig,C.:Harmonic analysis techniques for second order elliptic boundary value problems,CBMS Regional Conference Series in Mathematics,83.Published for the Conference Board of the Mathematical Sciences,Washington,DC;by the American Mathematical Society,Providence,RI,1994
[37]Kenig,C.,Toro,T.:Poisson kernel characterizations of Reifenberg flat chord arc domains.Ann.Sci.′Ecole Norm.Sup.(4),36(3),323-401(2003)
[38]Lavrentiev,M.:Boundary problems in the theory of univalent functions(Russian).Math Sb.,43,815-846(1936),AMS Transl.Series 2,32,1-35(1963)
[39]Lewis,J.L.,Vogel,A.:Symmetry theorems and uniform rectifiability.Boundary Value Problems,2007,(2007),article ID 030190,59 pages
[40]Mattila,P.,Melnikov,M.,Verdera,J.:The Cauchy integral,analytic capacity,and uniform rectifiability.Ann.of Math.(2),144(1),127-136(1996)
[41]Mourgoglou,M.,Tolsa,X.:Harmonic measure and Riesz transform in uniform and general domains.J.Reine Angew.Math.,to appear
[42]Muckenhoupt,B.:Weighted norm inequalities for the Hardy maximal function.Trans.Amer.Math.Soc.,165,207-226(1972)
[43]Nazarov,F.,Tolsa,X.,Volberg,A.:On the uniform rectifiability of ad-regular measures with bounded Riesz transform operator:The case of codimension 1.Acta Math.,213(2),237-321(2014)
[44]Riesz,F.and Riesz,M.:¨Uber die randwerte einer analtischen funktion,Compte Rendues du Quatrième Congrès des Math′ematiciens Scandinaves,Stockholm 1916,Almqvists and Wilksels,Uppsala,1920
[45]Semmes,S.:A criterion for the boundedness of singular integrals on hypersurfaces.Trans.Amer.Math.Soc.,311,501-513(1989)
1)In fact,even the latter condition may be relaxed somewhat;we refer the reader to[22]for details.
2)“Capacity Density Condition”:a quantitative version of Wiener regularity.
3)which is itself an endpoint version of the results of[6]and of[34]
4)This idea is also exploited in[37].
5)The alert reader may object that in Theorem 2,one assumes an interior Corkscrew condition;in fact,the latter condition holds automatically in domains(with ADR boundaries)whose harmonic measure is doubling,and of course,A∞entails doubling.