$\varepsilon $-approximability of harmonic functions in $L^p$ implies uniform rectifiability

Abstract

Suppose that Ω ⊂ R n+1 , n ≥ 2, is an open set satisfying the corkscrew condition with an n-dimensional ADR boundary, ∂Ω. In this note, we show that if harmonic functions are ε-approximable in L p for any p > n/(n − 1), then ∂Ω is uniformly rectifiable. Combining our results with those in [HT] gives us a new characterization of uniform rectifiability which complements the recent results in [HMM], [GMT] and [AGMT].

Uniform rectifiability and ε-approximability of harmonic functions in L p

“…Although the L p version of ε-approximability seems like the weakest one of all the properties, it is equivalent with the other properties in the codimension 1 ADR context provided that p is large enough. This follows from the recent results of S. Bortz and the second author [4]. Hence, combining our results with the results in [4,13,18] gives us the following characterization theorem: Theorem 1.6.…”

“…This follows from the recent results of S. Bortz and the second author [4]. Hence, combining our results with the results in [4,13,18] gives us the following characterization theorem: Theorem 1.6. -Suppose that E ⊂ R n+1 is an n-dimensional ADR set and let Ω := R n+1 \ E. The following conditions are equivalent:…”

Uniform rectifiability and $\varepsilon$-approximability of harmonic functions in $L^p$

“…Although the L p version of ε-approximability seems like the weakest one of all the properties, it is equivalent with the other properties in the codimension 1 ADR context provided that p is large enough. This follows from the recent results of S. Bortz and the second author [BT19]. Hence, combining our results with the results in [HMM16], [GMT18] and [BT19] gives us the following characterization theorem: Theorem 1.6.…”

“…This follows from the recent results of S. Bortz and the second author [BT19]. Hence, combining our results with the results in [HMM16], [GMT18] and [BT19] gives us the following characterization theorem: Theorem 1.6. Suppose that E ⊂ R n+1 is an n-dimensional ADR set and let Ω := R n+1 \ E. The following conditions are equivalent:…”

Uniform rectifiability implies Varopoulos extensions

“…The direction UR implies ε-approximability appears in [HMM16], and the converse is proved in [GMT18]. (see also [HT17] and [BT19] for pointwise and L p versions of this result). For other characterizations of UR with respect to properties of harmonic functions or solutions to other elliptic PDE, see [HMM16, HMM19, GMT18, HT17, BT19, AGMT16].…”