0000000000674576
AUTHOR
Steve Hofmann
showing 2 related works from this author
Uniform rectifiability and ε-approximability of harmonic functions in Lp
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 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. peerReviewed
Uniform rectifiability implies Varopoulos extensions
2020
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 $\sigma := \mathcal{H}^n\lfloor_{\partial \Omega}$ denote the surface measure on $\partial \Omega$. We show that if $f \in \text{BMO}(\partial \Omega,d\sigma)$ 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 th…