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Olli Tapiola

Publications and source records attributed to Olli Tapiola.

10 recordsLinked to original sources
arXiv2023

The $A_\infty$ condition, $\varepsilon$-approximators, and Varopoulos extensions in uniform domains

Suppose that $\Omega \subset\mathbb R^{n+1}$, $n\geq1$, is a uniform domain with $n$-Ahlfors regular boundary and $L$ is a (not necessarily symmetric) divergence form elliptic, real, bounded operator in $\Omega$. We show that the corresponding elliptic measure $\omega_L$ is quantitatively absolutely continuous with respect to surface measure of $\partial\Omega$ in the sense that $\omega_L \in A_\infty(\sigma)$ if and only if any bounded solution $u$ to $Lu = 0$ in $\Omega$ is $\varepsilon$-approximable for any $\varepsilon \in (0,1)$. By $\varepsilon$-approximability of $u$ we mean that there exists a function $\Phi = \Phi^\varepsilon$ such that $\|u-\Phi\|_{L^\infty(\Omega)} \le \varepsilon\|u\|_{L^\infty(\Omega)}$ and the measure $\widetilde{\mu}_\Phi$ with $d\widetilde{\mu} = |\nabla \Phi(Y)| \, dY$ is a Carleson measure with $L^\infty$ control over the Carleson norm. As a consequence of this approximability result, we show that boundary $\operatorname{BMO}$ 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 $L^1$-type Carleson measure estimates with $\operatorname{BMO}$ 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.

math.AP↗
arXiv2022

Connectivity conditions and boundary Poincar\'e inequalities

Inspired by recent work of Mourgoglou and the second named author, and earlier work of Hofmann, Mitrea and Taylor, we consider connections between the local John condition, the Harnack chain condition and weak boundary Poincar\'e inequalities in open sets $\Omega \subset \mathbb{R}^{n+1}$, with codimension $1$ Ahlfors--David regular boundaries. First, we prove that if $\Omega$ satisfies both the local John condition and the exterior corkscrew condition, then $\Omega$ also satisfies the Harnack chain condition (and hence, is a chord-arc domain). Second, we show that if $\Omega$ is a $2$-sided chord-arc domain, then the boundary $\partial \Omega$ supports a Heinonen--Koskela type weak $1$-Poincar\'e inequality. We also construct an example of a set $\Omega \subset \mathbb{R}^{n+1}$ such that the boundary $\partial \Omega$ is Ahlfors--David regular and supports a weak boundary $1$-Poincar\'e inequality but $\Omega$ is not a chord-arc domain. Our proofs utilize significant advances in particularly harmonic measure, uniform rectifiability and metric Poincar\'e theories.

math.AP↗
arXiv2020

Uniform rectifiability implies Varopoulos extensions

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 $Ω\subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\partial Ω$, and let $σ:= \mathcal{H}^n\lfloor_{\partial Ω}$ denote the surface measure on $\partial Ω$. We show that if $f \in \text{BMO}(\partial Ω,dσ)$ with compact support on $\partial Ω$, then there exists a smooth function $V$ in $Ω$ 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 $σ$. 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 $f \in C_c(\partial Ω)$, a harmonic extension $u$, with $|\nabla u(Y)|^2 \text{dist}(Y,\partial Ω) \, dY $ a Carleson measure controlled by the BMO norm of $f$, only in the presence of an appropriate quantitative connectivity condition.

math.AP↗
arXiv2019

$C_p$ estimates for rough homogeneous singular integrals and sparse forms

We consider Coifman--Fefferman inequalities for rough homogeneous singular integrals $T_Ω$ and $C_p$ weights. It was recently shown by Li-Pérez-Rivera-Ríos-Roncal that $$ \|T_Ω\|_{L^p(w)} \le C_{p,T,w} \|Mf\|_{L^p(w)} $$ for every $0< p < \infty$ and every $w \in A_\infty$. Our first goal is to generalize this result for every $w \in C_q$ where $q > \max\{1,p\}$ without using extrapolation theory. Although the bounds we prove are new even in a qualitative sense, we also give the quantitative bound with respect to the $C_q$ characteristic. Our techniques rely on recent advances in sparse domination theory and we actually prove most of our estimates for sparse forms. Our second goal is to continue the structural analysis of $C_p$ classes. We consider some weak self-improving properties of $C_p$ weights and weak and dyadic $C_p$ classes. We also revisit and generalize a counterexample by Kahanpää and Mejlbro who showed that $C_p \setminus \bigcup_{q > p} C_q \neq \emptyset$. We combine their construction with techniques of Lerner to define an explicit weight class $\widetilde{C}_p$ such that $\bigcup_{q > p} C_q \subsetneq \widetilde{C}_p \subsetneq C_p$ and every $w \in \widetilde{C}_p$ satisfies Muckenhoupt's conjecture. In particular, we give a different, self-contained proof for the fact that the $C_{p+\varepsilon}$ condition is not necessary for the Coifman--Fefferman inequality and our ideas allow us to consider also dimensions higher than $1$.

math.CA↗
arXiv2019

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

Suppose that $E \subset \mathbb{R}^{n+1}$ is a uniformly rectifiable set of codimension $1$. We show that every harmonic function is $\varepsilon$-approximable in $L^p(Ω)$ for every $p \in (1,\infty)$, where $Ω:= \mathbb{R}^{n+1} \setminus E$. Together with results of many authors this shows that pointwise, $L^\infty$ and $L^p$ type $\varepsilon$-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.

math.CA↗
arXiv2018

$\varepsilon$-Approximability of Harmonic Functions in $L^p$ Implies Uniform Rectifiability

Suppose that $Ω\subset \mathbb{R}^{n+1}$, $n \ge 2$, is an open set satisfying the corkscrew condition with an $n$-dimensional ADR boundary, $\partial Ω$. In this note, we show that if harmonic functions are $\varepsilon$-approximable in $L^p$ for any $p > n/(n-1)$, then $\partial Ω$ is uniformly rectifiable. Combining our results with those in [HT] (Hofmann-Tapiola) gives us a new characterization of uniform rectifiability which complements the recent results in [HMM] (Hofmann-Martell-Mayboroda), [GMT] (Garnett-Mourgoglou-Tolsa) and [AGMT] (Azzam-Garnett-Mourgoglou-Tolsa).

math.AP↗
arXiv2015

Weak A_\infty weights and weak Reverse Hölder property in a space of homogeneous type

In the Euclidean setting, the Fujii-Wilson-type $A_\infty$ weights satisfy a Reverse Hölder Inequality (RHI) but in spaces of homogeneous type the best known result has been that $A_\infty$ weights satisfy only a weak Reverse Hölder Inequality. In this paper, we compliment the results of Hytönen, Pérez and Rela and show that there exist both $A_\infty$ weights that do not satisfy an RHI and a genuinely weaker weight class that still satisfies a weak RHI. We also show that all the weights that satisfy a weak RHI have a self-improving property but the self-improving property of the strong Reverse Hölder weights fails in a general space of homogeneous type. We prove most of these purely non-dyadic results using convenient dyadic systems and techniques.

math.CA↗
arXiv2015

Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space $L^2(w)$, we obtain a bound that is quadratic in the $A_2$ constant $[w]_{A_2}$. We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.

math.CA↗
arXiv2013

Almost Lipschitz-continuous wavelets in metric spaces via a new randomization of dyadic cubes

In any quasi-metric space of homogeneous type, Auscher and Hytönen recently gave a construction of orthonormal wavelets with Hölder-continuity exponent $η>0$. However, even in a metric space, their exponent is in general quite small. In this paper, we show that the Hölder-exponent can be taken arbitrarily close to 1 in a metric space. We do so by revisiting and improving the underlying construction of random dyadic cubes, which also has other applications.

math.CA↗