Startseite Mathematik Harmonic measure and Riesz transform in uniform and general domains
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Harmonic measure and Riesz transform in uniform and general domains

  • Mihalis Mourgoglou EMAIL logo und Xavier Tolsa
Veröffentlicht/Copyright: 17. Oktober 2017

Abstract

Let Ωn+1 be open and let μ be some measure supported on Ω such that μ(B(x,r))Crn for all xn+1, r>0. We show that if the harmonic measure in Ω satisfies some scale invariant A-type conditions with respect to μ, then the n-dimensional Riesz transform

μf(x)=x-y|x-y|n+1f(y)𝑑μ(y)

is bounded in L2(μ). We do not assume any doubling condition on μ. We also consider the particular case when Ω is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the harmonic measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.

Funding statement: The authors were supported by the ERC grant 320501 of the European Research Council (FP7/2007-2013). X.T. was also partially supported by MTM-2013-44304-P, MTM-2016-77635-P, MDM-2014-044 (MICINN, Spain), and by Marie Curie ITN MAnET (FP7-607647).

Acknowledgements

We would like to thank Jonas Azzam for very helpful discussions in connection with this paper.

References

[1] H. Aikawa, Boundary Harnack principle and Martin boundary for a uniform domain, J. Math. Soc. Japan 53 (2001), no. 1, 119–145. 10.2969/jmsj/05310119Suche in Google Scholar

[2] H. Aikawa, Equivalence between the boundary Harnack principle and the Carleson estimate, Math. Scand. 103 (2008), no. 1, 61–76. 10.7146/math.scand.a-15069Suche in Google Scholar

[3] H. Aikawa and K. Hirata, Doubling conditions for harmonic measure in John domains, Ann. Inst. Fourier (Grenoble) 58 (2008), no. 2, 429–445. 10.5802/aif.2357Suche in Google Scholar

[4] J. Azzam, S. Hofmann, J. M. Martell, S. Mayboroda, M. Mourgoglou, X. Tolsa and A. Volberg, Rectifiability of harmonic measure, Geom. Funct. Anal. 26 (2016), no. 3, 703–728. 10.1007/s00039-016-0371-xSuche in Google Scholar

[5] J. Azzam, S. Hofmann, J. M. Martell, K. Nyström and T. Toro, A new characterization of chord-arc domains, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 4, 967–981. 10.4171/JEMS/685Suche in Google Scholar

[6] J. Azzam, M. Mourgoglou and X. Tolsa, Rectifiability of harmonic measure in domains with porous boundaries, preprint (2015), https://arxiv.org/abs/1505.06088. Suche in Google Scholar

[7] J. Azzam and X. Tolsa, Characterization of n-rectifiability in terms of Jones’ square function: Part II, Geom. Funct. Anal. 25 (2015), no. 5, 1371–1412. 10.1007/s00039-015-0334-7Suche in Google Scholar

[8] S. Bortz and S. Hofmann, Harmonic measure and approximation of uniformly rectifiable sets, Rev. Mat. Iberoam. 33 (2017), no. 1, 351–373. 10.4171/RMI/940Suche in Google Scholar

[9] J. Bourgain, On the Hausdorff dimension of harmonic measure in higher dimension, Invent. Math. 87 (1987), no. 3, 477–483. 10.1007/BF01389238Suche in Google Scholar

[10] G. David and D. Jerison, Lipschitz approximation to hypersurfaces, harmonic measure, and singular integrals, Indiana Univ. Math. J. 39 (1990), no. 3, 831–845. 10.1512/iumj.1990.39.39040Suche in Google Scholar

[11] G. David and P. Mattila, Removable sets for Lipschitz harmonic functions in the plane, Rev. Mat. Iberoam. 16 (2000), no. 1, 137–215. 10.4171/RMI/272Suche in Google Scholar

[12] G. David and S. Semmes, Analysis of and on uniformly rectifiable sets, Math. Surveys Monogr. 38, American Mathematical Society, Providence 1993. 10.1090/surv/038Suche in Google Scholar

[13] L. L. Helms, Potential theory, 2nd ed., Universitext, Springer, London 2014. 10.1007/978-1-4471-6422-7Suche in Google Scholar

[14] S. Hofmann, Non-degeneracy of harmonic measure plus ADR implies corkscrew, private communication (2015). Suche in Google Scholar

[15] S. Hofmann and J. M. Martell, Uniform rectifiability and harmonic measure I: Uniform rectifiability implies Poisson kernels in Lp, Ann. Sci. Éc. Norm. Supér. (4) 47 (2014), no. 3, 577–654. 10.24033/asens.2223Suche in Google Scholar

[16] S. Hofmann and J. M. Martell, Uniform rectifiability and harmonic measure IV: Ahlfors regularity plus Poisson kernels in Lp implies uniform rectifiability, preprint (2015), https://arxiv.org/abs/1505.06499. Suche in Google Scholar

[17] S. Hofmann, J. M. Martell and S. Mayboroda, Uniform rectifiability and harmonic measure III: Riesz transform bounds imply uniform rectifiability of boundaries of 1-sided NTA domains, Int. Math. Res. Not. IMRN 2014 (2014), no. 10, 2702–2729. 10.1093/imrn/rnt002Suche in Google Scholar

[18] S. Hofmann, J. M. Martell and I. Uriarte-Tuero, Uniform rectifiability and harmonic measure II: Poisson kernels in Lp imply uniform rectifiability, Duke Math. J. 163 (2014), no. 8, 1601–1654. 10.1215/00127094-2713809Suche in Google Scholar

[19] D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. Math. 46 (1982), no. 1, 80–147. 10.1016/0001-8708(82)90055-XSuche in Google Scholar

[20] F. Nazarov, X. Tolsa and A. Volberg, On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1, Acta Math. 213 (2014), no. 2, 237–321. 10.1007/s11511-014-0120-7Suche in Google Scholar

[21] F. Nazarov, X. Tolsa and A. Volberg, The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions, Publ. Mat. 58 (2014), no. 2, 517–532. 10.5565/PUBLMAT_58214_26Suche in Google Scholar

[22] F. Nazarov, S. Treil and A. Volberg, The Tb-theorem on non-homogeneous spaces, Acta Math. 190 (2003), no. 2, 151–239. 10.1007/BF02392690Suche in Google Scholar

[23] X. Tolsa, Bilipschitz maps, analytic capacity, and the Cauchy integral, Ann. of Math. (2) 162 (2005), no. 3, 1243–1304. 10.4007/annals.2005.162.1243Suche in Google Scholar

[24] X. Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous Calderón–Zygmund theory, Progr. Math. 307, Birkhäuser, Basel 2014. 10.1007/978-3-319-00596-6Suche in Google Scholar

[25] X. Tolsa, Rectifiable measures, square functions involving densities, and the Cauchy transform, Mem. Amer. Math. Soc. 245 (2017), no. 1158. 10.1090/memo/1158Suche in Google Scholar

[26] A. Volberg, Calderón–Zygmund capacities and operators on nonhomogeneous spaces, CBMS Reg. Conf. Ser. Math. 100, American Mathematical Society, Providence 2003. 10.1090/cbms/100Suche in Google Scholar

Received: 2017-01-07
Revised: 2017-07-27
Published Online: 2017-10-17
Published in Print: 2020-01-01

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