Uniform rectifiability and ε-approximability of harmonic functions in Lp
Hofmann, S., & Tapiola, O. (2020). Uniform rectifiability and ε-approximability of harmonic functions in Lp. Annales de l'Institut Fourier, 70(4), 1595-1638. https://doi.org/10.5802/aif.3359
Published inAnnales de l'Institut Fourier
DisciplineMatematiikkaAnalyysin ja dynamiikan tutkimuksen huippuyksikköMathematicsAnalysis and Dynamics Research (Centre of Excellence)
© 2020 Association des Annales de l’institut Fourier
Suppose that E⊂Rn+1 is a uniformly rectifiable set of codimension 1. We show that every harmonic function is ε-approximable in Lp(Ω) for every p∈(1,∞), where Ω:=Rn+1∖E. Together with results of many authors this shows that pointwise, L∞ and Lp type ε-approximability properties of harmonic functions are all equivalent and they characterize uniform rectifiability for codimension 1 Ahlfors–David regular sets. Our results and techniques are generalizations of recent works of T. Hytönen and A. Rosén and the first author, J. M. Martell and S. Mayboroda.
PublisherCentre Mersenne; l'Institut Fourier,
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Related funder(s)Academy of Finland
Funding program(s)Centre of Excellence, AoF
Additional information about fundingS.H. was supported by NSF grant DMS-1664047. O.T. was supported by Emil Aaltosen Säätiö through Foundations’ Post Doc Pool grant. In the previous stages of this work, he was supported by the European Union through T. Hytönen’s ERC Starting Grant 278558 “Analytic-probabilistic methods for borderline singular integrals” and the Finnish Centre of Excellence in Analysis and Dynamics Research.
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