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. ...
Publication in research information system
MetadataShow full item record
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.
Showing items with similar title or keywords.
Hofmann, Steve; Tapiola, Olli (Centre Mersenne; l'Institut Fourier,, 2020)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 ...
Bortz, Simon; Tapiola, Olli (American Mathematical Society, 2019)
Coulhon, Thierry; Jiang, Renjin; Koskela, Pekka; Sikora, Adam (Elsevier, 2020)Let (X,d,μ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a “carré du champ”. Assume that (X,d,μ,E) supports a scale-invariant L2-Poincaré inequality. In this article, we study the following ...
Guaranteed and computable error bounds for approximations constructed by an iterative decoupling of the Biot problem Kumar, Kundan; Kyas, Svetlana; Nordbotten, Jan Martin; Repin, Sergey (Elsevier, 2021)The paper is concerned with guaranteed a posteriori error estimates for a class of evolutionary problems related to poroelastic media governed by the quasi-static linear Biot equations. The system is decoupled by employing ...
Guaranteed error bounds and local indicators for adaptive solvers using stabilised space-time IgA approximations to parabolic problems Langer, Ulrich; Matculevich, Svetlana; Repin, Sergey (Elsevier, 2019)The paper is concerned with space–time IgA approximations to parabolic initial–boundary value problems. We deduce guaranteed and fully computable error bounds adapted to special features of such type of approximations and ...