Uniform rectifiability and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>ε</mml:mi></mml:math>-approximability of harmonic functions in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup></mml:math>

Hofmann, Steve; Tapiola, Olli

doi:10.5802/aif.3359

“…These papers show that this estimate is also equivalent to an approximation property of harmonic functions. Related results were proven by Hofmann and Tapiola [HT20] and Bortz and Tapiola [BT19]. 10.5.…”

Section: Bounded Analytic Functions and The Cauchy Transformsupporting

confidence: 54%

Rectifiability; a survey

Mattila¹

2021

Preprint

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“…The concept of ε-approximability in L p for p ∈ (1, ∞) was introduced by Hytönen and Rosén in [HR18] who showed that the dyadic average extension operator as well as any weak solution to certain elliptic PDE's in R n+1 + are ε-approximable in L p for every ε ∈ (0, 1) and p ∈ (1, ∞). The second part of that work was extended by Hofmann and Tapiola in [HT17] to harmonic functions in Ω = R n+1 \ E where E ⊂ R n+1 is a uniformly n-rectifiable set. The converse direction was shown by Bortz and Tapiola in [BT19].…”

Section: Andmentioning

confidence: 99%

“…Then we can invoke Theorem 1.2 to obtain the desired extension of g, which, in the previous proofs was attained via PDE methods and in particular with the help of harmonic measure producing a bounded harmonic function. For instance, in co-dimension 1, Hofmann and Tapiola [HT17] had to assume uniform rectifiability of the boundary in order to use ε-approximability of bounded harmonic functions. We are able to overcome this obstacle thanks to Theorem 1.1, which is valid in AR(s) domains (such domains could have fractal boundaries-e.g.…”

Section: Andmentioning

confidence: 99%

“…Concerning the higher co-dimensional version of our theorems (when s < n), it is worth noting that in [HT17], the extension of b can be done in such setting. The main difference is that the Carleson functional is defined the same way as in the case s = n without having the weight δ Ω (•) s−n in the integral.…”

Section: Andmentioning

confidence: 99%

“…Extensions of this kind were first constructed by Varopoulos [Var77], [Var78] in the upper half-space R n+1 + for boundary functions in BMO, and by Hytönen and Rosén [HR18] for boundary functions in L p for p ∈ (1, ∞). Hofmann and Tapiola [HT17] showed that in corkscrew domains with uniformly n-rectifiable boundary (in the sense of David and S. Semmes [DS1], [DS2]), one can also extend BMO functions with the desired bounds. Recently, the first named author with Tolsa [MT22] constructed an almost harmonic extension of functions in the Hajłasz Sobolev space Ẇ 1,p (∂Ω), which is the correct analogue of the L p version of Varopoulos extension for one "smoothness level" up.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Varopoulos' extensions of boundary functions in $L^p$ and BMO in domains with Ahlfors-regular boundaries

Mourgoglou¹,

Zacharopoulos²

2023

Preprint

0

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Let Ω ⊂ R n+1 , n ≥ 1, be an open set with s-Ahlfors regular boundary ∂Ω, for some s ∈ (0, n], such that either s = n and Ω is a corkscrew domain with the pointwise John condition, or s < n and Ω = R n+1 \E, for some s-Ahlfors regular set E ⊂ R n+1 . In this paper we construct Varopoulos' type extensions of L p and BMO boundary functions. In particular, we show that a) if f ∈ L p (∂Ω), 1 < p ≤ ∞, there exists F ∈ C ∞ (Ω) such that the Carleson functional of dist(x, Ω c ) s−n ∇F (x) and the non-tangential maximal function of F are in L p (∂Ω) with norms controlled by the L p -norm of f , andwith norm controlled by the BMO-norm of f , and F → f in some non-tangential sense H s | ∂Ω -almost everywhere. If, in addition, the boundary functions are Lipschitz with compact support, then both F and F can be modified so that they are also Lipschitz on Ω and converge to the boundary data continuously. In the latter case, the pointwise John condition when s = n is only needed for the BMO extension.

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The $$A_\infty $$ Condition, $$\varepsilon $$-Approximators, and Varopoulos Extensions in Uniform Domains

Bortz,

Poggi,

Tapiola

et al. 2024

J Geom Anal

1

0

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Suppose that $$\Omega \subset {\mathbb {R}}^{n+1}$$ Ω ⊂ R n + 1 , $$n\ge 1$$ n ≥ 1 , is a uniform domain with n-Ahlfors regular boundary and L is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in $$\Omega $$ Ω . We show that the corresponding elliptic measure $$\omega _L$$ ω L is quantitatively absolutely continuous with respect to surface measure of $$\partial \Omega $$ ∂ Ω in the sense that $$\omega _L \in A_\infty (\sigma )$$ ω L ∈ A ∞ ( σ ) if and only if any bounded solution u to $$Lu = 0$$ L u = 0 in $$\Omega $$ Ω is $$\varepsilon $$ ε -approximable for any $$\varepsilon \in (0,1)$$ ε ∈ ( 0 , 1 ) . By $$\varepsilon $$ ε -approximability of u we mean that there exists a function $$\Phi = \Phi ^\varepsilon $$ Φ = Φ ε such that $$\Vert u-\Phi \Vert _{L^\infty (\Omega )} \le \varepsilon \Vert u\Vert _{L^\infty (\Omega )}$$ ‖ u - Φ ‖ L ∞ ( Ω ) ≤ ε ‖ u ‖ L ∞ ( Ω ) and the measure $${\widetilde{\mu }}_\Phi $$ μ ~ Φ with $$d{\widetilde{\mu }} = |\nabla \Phi (Y)| \, dY$$ d μ ~ = | ∇ Φ ( Y ) | d Y is a Carleson measure with $$L^\infty $$ L ∞ control over the Carleson norm. As a consequence of this approximability result, we show that boundary $${{\,\textrm{BMO}\,}}$$ BMO functions with compact support can have Varopoulos-type extensions even in some sets with unrectifiable boundaries, that is, smooth extensions that converge non-tangentially back to the original data and that satisfy $$L^1$$ L 1 -type Carleson measure estimates with $${{\,\textrm{BMO}\,}}$$ BMO control over the Carleson norm. Our result complements the recent work of Hofmann and the third named author who showed the existence of these types of extensions in the presence of a quantitative rectifiability hypothesis.

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

Abstract: Les Annales de l'institut Fourier sont membres du Centre Mersenne pour l'édition scienti que ouverte www.centre-mersenne.org

Cited by 6 publications

References 17 publications

Rectifiability; a survey

Rectifiability; a survey

Varopoulos' extensions of boundary functions in $L^p$ and BMO in domains with Ahlfors-regular boundaries

The $$A_\infty $$ Condition, $$\varepsilon $$-Approximators, and Varopoulos Extensions in Uniform Domains

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