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- W3011999095 abstract "We construct extensions of Varopolous type for functions $f in text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More precisely, let $Omega subset mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $partial Omega$, and let $sigma := mathcal{H}^nlfloor_{partial Omega}$ denote the surface measure on $partial Omega$. We show that if $f in text{BMO}(partial Omega,dsigma)$ with compact support on $partial Omega$, then there exists a smooth function $V$ in $Omega$ such that $|nabla V(Y)| , dY$ 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 $sigma$. Our results should be compared to recent geometric characterizations of $L^p$-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 $f in C_c(partial Omega)$, a harmonic extension $u$, with $|nabla u(Y)|^2 text{dist}(Y,partial Omega) , dY $ a Carleson measure controlled by the BMO norm of $f$, only in the presence of an appropriate quantitative connectivity condition." @default.
- W3011999095 created "2020-03-23" @default.
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- W3011999095 date "2020-03-17" @default.
- W3011999095 modified "2023-09-27" @default.
- W3011999095 title "Uniform rectifiability implies Varopoulos extensions" @default.
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