We study uniqueness of p-harmonic Green functions in domains ohm in a complete metric space equipped with a doubling measure supporting a p-Poincar & eacute; inequality, with 1 < p < infinity. For bounded domains in unweighted R-n, the uniqueness was shown for the p-Laplace operator Delta p and all p by Kichenassamy and V & eacute;ron (1986) [25], while for p = 2 it is an easy consequence of the linearity of the Laplace operator Delta. Beyond that, uniqueness is only known in some particular cases, such as in Ahlfors p-regular spaces, as shown by Bonk et al. (2022) [10]. When the singularity x0 has positive p-capacity, the Green function is a particular multiple of the capacitary potential for cap(p)({x(0)}, ohm) and is therefore unique. Here we give a sufficient condition for uniqueness in metric spaces, and provide an example showing that the range of p for which it holds (while x0 has zero p-capacity) can be a nondegenerate interval. In the opposite direction, we give the first example showing that uniqueness can fail in metric spaces, even for p = 2. (c) 2025 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

This note is a follow up on our recent paper with L. Malý (to appear in Rev. Mat. Complut.). We provide a simple example of a compact metric space 𝒫 for which L^∞(𝒫) has the Vitali-Carathéodory property, the Sobolev C_∞-capacity is an outer capacity, but the Newtonian space N^1,∞(𝒫) contains functions which are not weakly quasicontinuous. The novelty here is that the Vitali-Carathéodory property is satified. We also obtain some related results about quasicontinuous functions in N^1,∞(𝒫) and a characterization of when L^∞(𝒫) has the Vitali-Carathéodory property.

For connected proper metric spaces X, equipped with a locally doubling measure supporting a local p-Poincaré inequality, we completely characterize which compact sets K with positive capacity are removable for bounded p-harmonic functions, p>1. Similar results are proved also for bounded quasiharmonic functions. The characterization is both in geometric and analytic terms. In particular, removability is shown to be equivalent to the validity of a Liouville type theorem in X∖ K. Properties such as local connectedness, sequential annular quasiconvexity, concentration of capacity and p-parabolicity are identified as crucial for removability. Along the way, we give a rather elementary proof of the Liouville theorem for quasisuperharmonic functions in p-parabolic spaces. Our results apply in particular to manifolds and 𝐑^n equipped with (locally) p-admissible weights.

We introduce a new sphericalization mapping for metric spaces that is applicable in very general situations, including totally disconnected fractal type sets. For an unbounded complete metric space which is uniformly perfect at a base point for large radii and equipped with a doubling measure, we make a more specific construction based on the measure and equip it with a weighted measure. This mapping is then shown to preserve the doubling property of the measure and the Besov (fractional Sobolev) energy. The corresponding results for flattening of bounded complete metric spaces are also obtained. Finally, it is shown that for the composition of a sphericalization with a flattening, or vice versa, the obtained space is biLipschitz equivalent with the original space and the resulting measure is comparable to the original measure.

We construct various examples of Sobolev-type functions, defined via upper gradients in metric spaces, that fail to be quasicontinuous or weakly quasicontinuous. This is done with quasi-Banach function lattices X as the function spaces defining the smoothness of the Sobolev-type functions. These results are in contrast to the case X=L^p with 1≤ p<∞ , where all Sobolev-type functions in N^1,p are known to be quasicontinuous, provided that the underlying metric space 𝒫 is locally complete. In most of our examples, 𝒫 is a compact subset of R^2 and X=L^∞ . Four particular examples are the damped topologist’s sine curve, the von Koch snowflake curve, the Cantor ternary set and the Sierpiński carpet. We also discuss several related properties, such as whether the Sobolev capacity is an outer capacity, and how these properties are related. A fundamental role in these considerations is played by the lack of the Vitali–Carathéodory property.

In this paper we study when the origin (0,0) is a regular (or irregular) boundary point for the so-called soda can domains of the type Θ_l,θ:= {(x,t) ∈𝐑^n+1: 0<-t < θ|x|^l <θ}, with l,θ>0, for the p-parabolic equation ∂_t u- Δ_p u=0, where 1

In this paper we study nonlocal nonlinear equations of s-fractional p-Laplacian type in open subsets of 𝐑^n. We investigate how the boundary regularity of solutions depends on the parameters s and p, including the local case s=1. Specifically, we show exactly when regularity for (s_1, p_1) implies regularity for (s_2, p_2). The proof relies on the equivalence between Wiener criteria formulated with condenser and Sobolev capacities. To establish this equivalence, we derive precise comparison estimates between the two capacities. The Wiener integral defines thinness and the fine topology. We show that every superharmonic function associated with a nonlocal nonlinear operator is finely continuous. Moreover, we prove that polar sets in this fractional setting coincide with sets of zero capacity.

Given a bounded finely open set $V$ and a function $f$ on the fine boundary of $V$, we introduce four types of upper Perron solutions to the nonlinear Dirichlet problem for $p$-energy minimizers, $1

The identification between the complex plane and the Riemann sphere preserves holomorphic and harmonic functions and is a classical tool. In this paper we consider a similar mapping from an unbounded metric space X to a bounded space and show how it preserves p-harmonic functions and Poincaré inequalities. When X is Ahlfors regular, this was shown in our earlier paper (J. Math. Anal. Appl. 474 (2019), 852–875). Here we only require the much weaker (and more natural) doubling property of the measure. Furthermore, we consider a broader class of transformed measures. The sphericalization is then applied to obtain new results for the Dirichlet boundary value problem in unbounded sets and for boundary regularity at infinity for p-harmonic functions. Some of these results are new also for unweighted 𝐑^n, n ≥ 2 and p2.

We study boundary regularity at the infinity point ∞ for nonlinear elliptic equations of p-Laplace type in unbounded open sets Ω⊂𝐑^n. We consider the case p ≥ n ≥ 2 and characterize the regularity at ∞ by means of Wiener-type integrals. Our approach uses circular inversion, which maps ∞ to the origin and the original nonlinear equation to a similar weighted equation. The Wiener criterion at the origin for such equations is then transformed back to provide Wiener-type criteria at ∞. When p>n, the criteria simplify so that ∞ is regular if and only if the boundary is unbounded. For p=n this is not true, as shown by an example. This simplified criterion is also proved for p-harmonic functions in unbounded open subsets of Ahlfors Q-regular metric measure spaces with Q<p, supporting a Poincaré inequality.

In this paper we study nonlocal nonlinear equations of fractional (s,p)-Laplacian type on 𝐑^n. We show that the irregular boundary points for the Dirichlet problem can be divided into two disjoint classes: semiregular and strongly irregular boundary points, with very different behaviour. Two fundamental tools needed to show this are the Kellogg property (from our previous paper) and a new removability result for solutions in the V^s,p Sobolev type space, which we deduce more generally also for supersolutions of equations with a right-hand side. Semiregular and strongly irregular points are also characterized in various ways. Finally, it is explained how semiregularity depends on s and p.

We study the condenser capacity cap(p)(E, Omega)on unbounded open sets Omega in a proper connected metric space X equipped with a locally doubling measure supporting a local p-Poincare inequality, where 1 < p < infinity.Using a new definition of capacitary potentials, we show that cap(p) is countably sub additive and that it is a Choquet capacity. We next obtain formulas for the capacity of superlevel sets for the capacitary potential. These are then used to show that p-harmonic Green functions exist in an unbounded domain Omega if and only if either X is p-hyperbolic or the Sobolev capacity C-p(X\Omega) >0. As an application, we deduce new results for Perron solutions and boundary regularity for the Dirichletboundary value problem for p-harmonic functions in unbounded open sets.

A metric space X is called a bow-tie if it can be written as X = X+ boolean OR X-, where X+ boolean AND X- = {x(0)} and X-+/- not equal {x(0)} are closed subsets of X. We show that a doubling measure mu on X supports a (q, p)-Poincare inequality on X if and only if X satisfies a quasiconvexity-type condition, mu supports a (q, p)-Poincare inequality on both X+ and X-, and a variational p-capacity condition holds. This capacity condition is in turn characterized by a sharp measure decay condition at the point x(0). In particular, we study the bow-tie X-Rn consisting of the positive and negative hyperquadrants in R-n equipped with a radial doubling weight and characterize the validity of the p-Poincare inequality on X-Rn in several ways. For such weights, we also give a general formula for the capacity of annuli around the origin. (c) 2024 The Author(s). Published by Elsevier Inc.

For nonlinear operators of fractional -Laplace type, we consider two types of solutions to the nonlocal Dirichlet problem: Sobolev solutions based on fractional Sobolev spaces and Perron solutions based on superharmonic functions. These solutions give rise to two different concepts of regularity for boundary points, namely Sobolev and Perron regularity. We show that these two notions are equivalent and we also provide several characterizations of regular boundary points. Along the way, we give a new definition of Perron solutions, which is applicable to arbitrary exterior Dirichlet data g: Ω^c → [-∞,∞]. We obtain resolutivity results for these Perron solutions, and show that the Sobolev and Perron solutions coincide for a large class of exterior Dirichlet data. This also implies invariance of the Perron solutions under perturbations on sets of zero fractional capacity. A uniqueness result for the Dirichlet problem is also obtained for the class of bounded solutions taking prescribed continuous exterior data quasieverywhere on the boundary.

In this paper, several convergence results for fine p-(super)minimizers on quasiopen sets in metric spaces are obtained. For this purpose, we deduce a Caccioppoli-type inequality and local-to-global principles for fine p-(super)minimizers on quasiopen sets. A substantial part of these considerations is to show that the functions belong to a suitable local fine Sobolev space. We prove our results for a complete metric space equipped with a doubling measure supporting a p-Poincare inequality with 1 < p < & INFIN;. However, most of & COPY; 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons .org /licenses /by /4 .0/).

In a complete metric space equipped with a doubling measure supporting a p -Poincaré inequality, we prove sharp growth and integrability results for p -harmonic Green functions and their minimal p -weak upper gradients. We show that these properties are determined by the growth of the underlying measure near the singularity. Corresponding results are obtained also for more general p -harmonic functions with poles, as well as for singular solutions of elliptic differential equations in divergence form on weighted R n and on manifolds. The proofs are based on a new general capacity estimate for annuli, which implies precise pointwise estimates for p -harmonic Green functions. The capacity estimate is valid under considerably milder assumptions than above. We also use it, under these milder assumptions, to characterize singletons of zero capacity and the p -parabolicity of the space. This generalizes and improves earlier results that have been important especially in the context of Riemannian manifolds.

We obtain precise estimates, in terms of the measure of balls, for the Besov capacity of annuli and singletons in complete metric spaces. The spaces are only assumed to be uniformly perfect with respect to the centre of the annuli and equipped with a doubling measure.

We study removable sets for Newtonian Sobolev functions in metric measure spaces satisfying the usual (local) assumptions of a doubling measure and a Poincar\'e inequality. In particular, when restricted to Euclidean spaces, a closed set $E\subset \mathbf{R}^n$ with zero Lebesgue measure is shown to be removable for $W^{1,p}(\mathbf{R}^n \setminus E)$ if and only if $\mathbf{R}^n \setminus E$ supports a $p$-Poincar\'e inequality as a metric space. When $p>1$, this recovers Koskela's result (Ark. Mat. 37 (1999), 291--304), but for $p=1$, as well as for metric spaces, it seems to be new. We also obtain the corresponding characterization for the Dirichlet spaces $L^{1,p}$. To be able to include $p=1$, we first study extensions of Newtonian Sobolev functions in the case $p=1$ from a noncomplete space $X$ to its completion $\widehat{X}$. In these results, $p$-path almost open sets play an important role, and we provide a characterization of them by means of $p$-path open, $p$-quasiopen and $p$-finely open sets. We also show that there are nonmeasurable $p$-path almost open subsets of $\mathbf{R}^n$, $n \geq 2$, provided that the continuum hypothesis is assumed to be true. Furthermore, we extend earlier results about measurability of functions with $L^p$-integrable upper gradients, about $p$-quasiopen, $p$-path and $p$-finely open sets, and about Lebesgue points for $N^{1,1}$-functions, to spaces that only satisfy local assumptions.

By seeing whether a Liouville type theorem holds for positive, bounded, and/or finite \(p\)-energy \(p\)-harmonic and \(p\)-quasiharmonic functions, we classify proper metric spaces equipped with a locally doubling measure supporting a local \(p\)-Poincaré inequality. Similar classifications have earlier been obtained for Riemann surfaces and Riemannian manifolds. We study the inclusions between these classes of metric measure spaces, and their relationship to the \(p\)-hyperbolicity of the metric space and its ends. In particular, we characterize spaces that carry nonconstant \(p\)-harmonic functions with finite \(p\)-energy as spaces having at least two well-separated \(p\)-hyperbolic sequences of sets towards infinity. We also show that every such space \(X\) has a function \(f \notin L^p(X) + \mathbf{R}\) with finite \(p\)-energy.

We initiate the study of fine p -(super)minimizers, associated with p -harmonic functions, on finely open sets in metric spaces, where 1 < p < ∞ . After having developed their basic theory, we obtain the p -fine continuity of the solution of the Dirichlet problem on a finely open set with continuous Sobolev boundary values, as a by-product of similar pointwise results. These results are new also on unweighted R n . We build this theory in a complete metric space equipped with a doubling measure supporting a p -Poincaré inequality.