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Harmonic measure and Riesz transform in uniform and general domains

  • Mihalis Mourgoglou EMAIL logo and Xavier Tolsa

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.

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

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