We prove that every entire solution of the minimal graph equation that is bounded from below and has at most linear growth must be constant on a complete Riemannian manifold M with only one end if M has asymptotically non-negative sectional curvature. On the other hand, we prove the existence of bounded non-constant minimal graphic and p-harmonic functions on rotationally symmetric Cartan-Hadamard manifolds under optimal assumptions on the sectional curvatures.

Let $p$ be a real number greater than one and let $\Gamma $ be a graph of bounded degree. We investigate links between the $p$-harmonic boundary of $\Gamma $ and the ${D}_{p} $-massive subsets of $\Gamma $. In particular, if there are $n$ pairwise disjoint ${D}_{p} $-massive subsets of $\Gamma $, then the $p$-harmonic boundary of $\Gamma $ consists of at least $n$ elements. We show that the converse of this statement is also true.

Let (X,d) be a compact metric space and let ℳ(X) denote the space of all finite signed Borel measures on X. Define I:ℳ(X)→ℝ by and set M(X)=sup I(μ), where μ ranges over the collection of signed measures in ℳ(X) of total mass 1. The metric space (X,d) is quasihypermetric if for all n∈ℕ, all α1,…,αn∈ℝ satisfying ∑ i=1nαi=0 and all x1,…,xn∈X, the inequality ∑ i,j=1nαiαjd(xi,xj)≤0 holds. Without the quasihypermetric property M(X) is infinite, while with the property a natural semi-inner product structure becomes available on ℳ0(X), the subspace of ℳ(X) of all measures of total mass 0. This paper explores: operators and functionals which provide natural links between the metric structure of (X,d), the semi-inner product space structure of ℳ0(X) and the Banach space C(X) of continuous real-valued functions on X; conditions equivalent to the quasihypermetric property; the topological properties of ℳ0(X) with the topology induced by the semi-inner product, and especially the relation of this topology to the weak-* topology and the measure-norm topology on ℳ0(X); and the functional-analytic properties of ℳ0(X) as a semi-inner product space, including the question of its completeness. A later paper [P. Nickolas and R. Wolf, Distance geometry in quasihypermetric spaces. II, Math. Nachr., accepted] will apply the work of this paper to a detailed analysis of the constant M(X).

We prove there exist exponentially decaying generalized eigenfunctions on a blow-up of the Sierpinski gasket with boundary. These are used to show a Borel-type theorem, specifically that for a prescribed jet at the boundary point there is a smooth function having that jet.

We answer a question posed in [12] on exponential integrability of functions of restricted n-energy. We use geometric methods to obtain a sharp exponential integrability result for boundary traces of monotone Sobolev functions defined on the unit ball.

We study when characteristic and Hölder continuous functions are traces of Sobolev functions on doubling metric measure spaces. We provide analytic and geometric conditions sufficient for extending characteristic and Hölder continuous functions into globally defined Sobolev functions.

We give necessary and sufficient conditions in order that inequalities of the type

\[ \| T_K f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d\sigma)}, \quad f \in L^p(d\sigma), \]

hold for a class of integral operators $T_K f(x) = \int_{R^n} K(x,y) f(y)\,d \sigma(y)$ with nonnegative kernels, and measures $d \mu$ and $d\sigma$ on $\mathbb{R}^n$, in the case where $p>q>0$ and $p>1$.

An important model is provided by the dyadic integral operator with kernel $K_{\mathcal D}(x, y)= \sum_{Q\in{\mathcal D}} K(Q)\chi_Q(x) \chi_Q(y)$, where $\mathcal D=\{Q\}$ is the family of all dyadic cubes in $\mathbb{R}^n$, and $K(Q)$ are arbitrary nonnegative constants associated with $Q \in{\mathcal D}$.

The corresponding continuous versions are deduced from their dyadic counterparts. In particular, we show that, for the convolution operator $T_k f = k \star f$ with positive radially decreasing kernel $k(|x-y|)$, the trace inequality

\[\| T_k f\|_{L^q(d\mu)}\leq C \|f\|_{L^p(d x)}, \quad f \in L^p(dx),\]

holds if and only if ${\mathcal W}_{k}[\mu] \in L^s (d\mu)$, where $s = q(p-1)/(p-q)$. Here ${\mathcal W}_{k}[\mu]$ is a nonlinear Wolff potential defined by ${\mathcal W}_{k}[\mu](x)=\int_0^{+\infty} k(r) \bar{k}(r)^{1/(p-1)} \mu(B(x,r))^{1/(p-1)} r^{n-1} \, dr$, and $\bar{k}(r)=(1/r^n)\int_0^r k(t) t^{n-1} \, dt$. Analogous inequalities for $1\le q < p$ were characterized earlier by the authors using a different method which is not applicable when $q<1$.

In this note we consider a stability property of p-harmonic functions in general metric measure spaces. Despite the fact that there may not be a corresponding differential equation, it is shown here that a family of uniformly convergent p-harmonic functions converges to a p-harmonic function; that is, the collection of p-harmonic functions forms a closed subspace in a certain Ho¨lder class. As a consequence, it is shown that a family of p-harmonic functions bounded in a Sobolev-type space (called the Newtonian space) has a sequence that converges locally uniformly in the domain of harmonicity to a p-harmonic function. This result is used to construct p-harmonic functions on unbounded domains. We also use this convergence result to prove a characterization of a parabolicity property of metric measure spaces. This characterization has been given for Riemannian manifolds by Holopainen, and the result here is a generalization of Holopainen's result.

We introduce a framework for a nonlinear potential theory without a kernel on a reflexive, strictly convex and smooth Banach space of functions. Nonlinear potentials are defined as images of nonnegative continuous linear functionals on that space under the duality mapping. We study potentials and reduced functions by using a variant of the Gauss-Frostman quadratic functional. The framework allows a development of other main concepts of nonlinear potential theory such as capacities, equilibrium potentials and measures of finite energy.