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We study existence, uniqueness, and regularity properties of the Dirichlet problem related to fractional Dirichlet energy minimizers in a complete doubling metric measure space $(X,d_X,\mu_X)$ satisfying a $2$-Poincar\'e inequality. Given a bounded domain $\Omega\subset X$ with $\mu_X(X\setminus\Omega)>0$, and a function $f$ in the Besov class $B^\theta_{2,2}(X)\cap L^2(X)$, we study the problem of finding a function $u\in B^\theta_{2,2}(X)$ such that $u=f$ in $X\setminus\Omega$ and $\mathcal{E}_\theta(u,u)\le \mathcal{E}_\theta(h,h)$ whenever $h\in B^\theta_{2,2}(X)$ with $h=f$ in $X\setminus\Omega$. We show that such a solution always exists and that this solution is unique. We also show that the solution is locally H\"older continuous on $\Omega$, and satisfies a non-local maximum and strong maximum principle. Part of the results in this paper extend the work of Caffarelli and Silvestre in the Euclidean setting and Franchi and Ferrari in Carnot groups., Comment: 42 pages, comments welcome, submitted. Revision to add crucial references and attributions to the introduction

We consider two notions of functions of bounded variation in complete metric measure spaces, one due to Martio and the other due to Miranda~Jr. We show that these two notions coincide, if the measure is doubling and supports a $1$-Poincar\'e inequality. In doing so, we also prove that if the measure is doubling and supports a $1$-Poincar\'e inequality, then the metric space supports a \emph{Semmes family of curves} structure., Comment: 20 pages

Funding information: The first author was partially supported by the Fonds de recherche du Québec – Nature et technologies (FRQNT). Part of the work for this project was done while the first author was visiting Aalto University; he would like to thank that institution for their kind hospitality. The second author was partially supported by Academy of Finland, project 308063. Publisher Copyright: © 2022. The Finnish Mathematical Society We prove in the setting of Q-Ahlfors regular PI-spaces the following result: if a domain has uniformly large boundary when measured with respect to the s-dimensional Hausdorff content, then its visible boundary has large t-dimensional Hausdorff content for every 0 < t < s ≤ Q − 1. The visible boundary is the set of points that can be reached by a John curve from a fixed point z0∈Ω. This generalizes recent results by Koskela–Nandi–Nicolau (from R2) and Azzam (Rn). In particular, our approach shows that the phenomenon is independent of the linear structure of the space.

In this paper we establish that the gradient of weak solutions to porous medium-type systems admits the self-improving property of higher integrability.

We study two notions of Dirichlet problem associated with BV energy minimizers (also called functions of least gradient) in bounded domains in metric measure spaces whose measure is doubling and supports a (1, 1)-Poincaré inequality. Since one of the two notions is not amenable to the direct method of the calculus of variations, we construct, based on an approach of Juutinen and Mazón-Rossi–De León, solutions by considering the Dirichlet problem for p-harmonic functions, p>1, and letting p→1. Tools developed and used in this paper include the inner perimeter measure of a domain.

We provide a new geometric proof of Reimann's theorem characterizing quasiconformal mappings as the ones preserving functions of bounded mean oscillation. While our proof is new already in the Euclidean spaces, it is applicable in Heisenberg groups as well as in more general stratified nilpotent Carnot groups., Comment: 8 pages

We study a function space $JN_p$ based on a condition introduced by John and Nirenberg as a variant of BMO. It is known that $L^p\subset JN_{p}\subsetneq L^{p,\infty}$, but otherwise the structure of $JN_p$ is largely a mystery. Our first main result is the construction of a function that belongs to $JN_p$ but not $L^p$, showing that the two spaces are not the same. Nevertheless, we prove that for monotone functions, the classes $JN_{p}$ and $L^p$ do coincide. Our second main result describes $JN_p$ as the dual of a new Hardy kind of space $HK_{p'}$., Comment: corrections, 1 figure added, Section 4 (multidimensional counterexample) is new

We deal with the obstacle problem for the porous medium equation in the slow diffusion regime m > 1 . Our main interest is to treat fairly irregular obstacles assuming only boundedness and lower semicontinuity. In particular, the considered obstacles are not regular enough to work with the classical notion of variational solutions, and a different approach is needed. We prove the existence of a solution in the sense of the minimal supersolution lying above the obstacle. As a consequence, we can show that non-negative weak supersolutions to the porous medium equation can be approximated by a sequence of supersolutions which are bounded away from zero.

We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of R n and the relation between these spaces and traces of classical Sobolev spaces. This extends in a certain way the results of Shvartsman (20) to the case of lower dimensional subsets of the Euclidean space.

On metric spaces supporting a geometric version of a Semmes family of curves, we provide a Reshetnyak-type characterization of functions of bounded variation in terms of the total variation on such a family of curves. We then use this characterization to obtain a Federer-type characterization of sets of nite perimeter, that is, we show that a measurable set is of nite perimeter if and only if the Hausdor measure of its measure theoretic boundary is nite. We present a construction of a geometric Semmes family of curves in the rst Heisenberg group.

This paper studies functions of bounded mean oscillation (BMO) on metric spaces equipped with a doubling measure. The main result gives characterizations for mappings that preserve BMO. This extends the corresponding Euclidean results by Gotoh to metric measure spaces. The argument is based on a generalizations Uchiyama's construction of certain extremal BMO-functions and John-Nirenberg's lemma.

We study equivalence between the Poincare inequality and several different relative isoperimetric inequalities on metric measure spaces. We then use these inequalities to establish sufficient conditions for the finite perimeter of sets.

We study pointwise properties of functions of bounded variation on a metric space equipped with a doubling measure and a Poincare inequality. In particular, we obtain a Lebesgue type result for \(BV\) functions. We also study approximations by Lipschitz continuous functions and a version of the Leibniz rule. We give examples which show that our main result is optimal for \(BV\) functions in this generality.

We extend the theory of the thermal capacity for the heat equation to nonlinear parabolic equations of the $$p$$ -Laplacian type. We study definitions and properties of the nonlinear parabolic capacity and show that the capacity of a compact set can be represented via a capacitary potential. As an application, we show that polar sets of superparabolic functions are of zero capacity. The main technical tools used include estimates for equations with measure data and obstacle problems.

We give an intrinsic characterization of the property that the zero extension of a Newtonian function, defined on an open set in a doubling metric measure space supporting a strong relative isoperimetric inequality, belongs to the Newtonian space on the entire metric space. The theory of functions of bounded variation is used extensively in the argument and we also provide a structure theorem for sets of finite perimeter under the assumption of a strong relative isoperimetric inequality. The characterization is used to prove a strong version of quasicontinuity of Newtonian functions.

We study the existence of a set with minimal perimeter that separates two disjoint sets in a metric measure space equipped with a doubling measure and supporting a Poincare inequality. A measure constructed by De Giorgi is used to state a relaxed problem, whose solution coincides with the solution to the original problem for measure theoretically thick sets. Moreover, we study properties of the De Giorgi measure on metric measure spaces and show that it is comparable to the Hausdorff measure of codimension one. We also explore the relationship between the De Giorgi measure and the variational capacity of order one. The theory of functions of bounded variation on metric spaces is used extensively in the arguments.

We present a new proof for a pointwise upper bound in terms of Wolff potential for A-superharmonic functions, which are the pointwise defined solutions to elliptic equations involving nonnegative measure data.

We show for a general class of parabolic equations that every bounded superparabolic function is a weak supersolution and, in particular, has derivatives in a Sobolev sense. To this end, we establish various comparison principles between super- and subparabolic functions, and show that a pointwise limit of uniformly bounded weak supersolutions is a weak supersolution.

We present isocapacitary characterizations of Sobolev inequalities in very general metric measure spaces., Comment: 17 pages, corrected typos

The purpose of this work is to study regularity of Sobolev functions on metric measure spaces equipped with a doubling measure and supporting a weak Poincare inequality. We show that every Sobolev function whose gradient is integrable to power one has Lebesgue points outside a set of 1-capacity zero. We also show that 1-capacity is equivalent to the Hausdorff content of codimension one and study characterizations of 1-capacity in terms of Frostman's lemma and functions of bounded variation. As the main technical tool, we prove a metric space version of Gustin's boxing inequality. Our proofs are based on covering arguments and functions of bounded variation. Perimeter measures, isoperimetric inequalities and coarea formula play an essential role in our approach.

The purpose of this work is to prove the following result: If a doubling metric measure space supports a weak (1, p)–Poincare inequality with p sufficiently small, then annuli are almost quasiconvex. We also obtain estimates for the Hausdorff s-content and the diameter of the spheres.

We show that, unlike minima of superharmonic functions which are again superharmonic, the same property fails for Q -quasisuperminimizers. More precisely, if u i is a Q i -quasisuperminimizer, i = 1 , 2 , where 1 Q 1 ≤ Q 2 , then u = min { u 1 , u 2 } is a Q -quasisuperminimizer, but there is an increase in the optimal quasisuperminimizing constant Q . We provide the first examples of this phenomenon, i.e. that Q > Q 2 . In addition to lower bounds for the optimal quasisuperminimizing constant of u we also improve upon the earlier upper bounds due to Kinnunen and Martio. Moreover, our lower and upper bounds turn out to be quite close. We also study a similar phenomenon in pasting lemmas for quasisuperminimizers, where Q = Q 1 Q 2 turns out to be optimal, and provide results on exact quasiminimizing constants of piecewise linear functions on the real line, which can serve as approximations of more general quasiminimizers.

We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting. As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, BV, and maximal functions.

In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem., Comment: 29 pages

We characterize quasiconformal mappings as those homeomorphisms between two metric measure spaces of locally bounded geometry that preserve a class of quasiminimizers. We also consider quasiconformal mappings and densities in metric spaces and give a characterization of quasiconformal mappings in terms of the uniform density property introduced by Gehring and Kelly.

We consider continuously differentiable curves embedded in graded groups, along with the points of these curves that have the maximal transversality with respect to the grading of the group. We prove a blow-up theorem at these points and we show that the remaining points are negligible with respect to the Hausdorff measure whose dimension equals the Hausdorff dimension of the curve. This leads us to an area-type formula for the intrinsic spherical Hausdorff measure of any of these curves embedded in an arbitrary graded group.

This paper studies regularity of perimiter quasiminimizing sets in metric measure spaces with a doubling measure and a Poincare inequality. The main result shows that the measure theoretic boundary of a quasiminimizing set coincides with the topological boundary. We also show that such a set has finite Minkowski content and apply the regularity theory to study rectifiability issues related to quasiminimal sets in strong A_{\infty}-weighted Euclidean case., Comment: 31 pages

We prove that every strong A(infinity)-weight is a Muckenhoupt weight in Ahlfors-regular metric measure spaces that support a Poincare inequality. We also explore the relations between various definitions for A(infinity)-weights in this setting, since some of these characterizations are needed in the proof of the main result.

Necessary and sufficient conditions for the validity of a multidimensional version of the Hardy inequality are discussed. A characterization through a boundary Poincare inequality is considered.

We prove an equivalence result between the validity of a pointwise Hardy inequality in a domain and uniform capacity density of the complement. This result is new even in Euclidean spaces, but our methods apply in general metric spaces as well. We also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the Hardy inequality in the setting of metric spaces. In addition, we consider the relations between the above concepts and certain Hausdorff content conditions., 18 pages

We present an easy proof that $p$--Hardy's inequality implies uniform $p$--fatness of the boundary when $p=n$. The proof works also in metric space setting and demonstrates the self--improving phenomenon of the $p$--fatness. We also explore the relationship between $p$-fatness, $p$-Hardy inequality, and the uniform perfectness for all $p\ge 1$, and demonstrate that in the Ahlfors $Q$-regular metric measure space setting with $p=Q$, these three properties are equivalent., 12 pages

We study Sobolev type spaces defined in terms of sharp maximal functions on Ahlfors regular subsets of the Euclidean space and the relation between these spaces and traces of classical Sobolev spaces., Comment: 18 pages

We show the existence of a continuous solution to a nonlinear parabolic obstacle problem with a continuous time-dependent obstacle. The solution is constructed by an adaptation of the Schwarz alternating method. Moreover, if the obstacle is Holder continuous, we prove that the solution inherits the same property.