Uniform rectifiability implies Varopoulos extensions
Hofmann, S., & Tapiola, O. (2021). Uniform rectifiability implies Varopoulos extensions. Advances in Mathematics, 390, Article 107961. https://doi.org/10.1016/j.aim.2021.107961
Published inAdvances in Mathematics
© 2021 the Authors
We construct extensions of Varopolous type for functions f∈BMO(E), for any uniformly rectifiable set E of codimension one. More precisely, let Ω⊂Rn+1 be an open set satisfying the corkscrew condition, with an n-dimensional uniformly rectifiable boundary ∂Ω, and let ≔σ≔Hn⌊∂Ω denote the surface measure on ∂Ω. We show that if f∈BMO(∂Ω,dσ) with compact support on ∂Ω, then there exists a smooth function V in Ω such that |∇V(Y)|dY is a Carleson measure with Carleson norm controlled by the BMO norm of f, and such that V converges in some non-tangential sense to f almost everywhere with respect to σ. Our results should be compared to recent geometric characterizations of Lp-solvability and of BMO-solvability of the Dirichlet problem, by Azzam, the first author, Martell, Mourgoglou and Tolsa and by the first author and Le, respectively. In combination, this latter pair of results shows that one can construct, for all f∈Cc(∂Ω), a harmonic extension u, with |∇u(Y)|2dist(Y,∂Ω)dY a Carleson measure with Carleson norm controlled by the BMO norm of f, only in the presence of an appropriate quantitative connectivity condition. ...
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Additional information about fundingS.H. was supported by NSF grant DMS-1664047. O.T. was partially supported by Emil Aaltosen Säätiö through Foundations' Post Doc Pool grant.
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