Related papers: Amenability and harmonic $L^p$-functions on hyperg…

Let $(X,\mathcal W)$ be a balayage space, $1\in \mathcal W$, or - equivalently - let $\mathcal W$ be the set of excessive functions of a Hunt process on a locally compact space $X$ with countable base such that $\mathcal W$ separates…

We show that the Dirichlet problem at infinity is unsolvable for the p-Laplace equation for any nonconstant continuous boundary data, for certain range of p>n, on an n-dimensional Cartan-Hadamard manifold constructed from a complete…

We motivate and study the class $\mathcal{C}$ of countable groups $G$ such that the conjugacy relation between minimal actions of $G$ on $\mathbb{R}$ by orientation-preserving homeomorphisms is smooth -- that is, admits a Borel transversal.…

Let $G$ be a non-amenable countable group. We show that every almost automorphic $G$-action on a compact Hausdorff space, with a maximal equicontinuous factor whose phase space is a Cantor set, admits invariant probability measures (this…

Let $h$ be a harmonic function defined on a spherical disk. It is shown that $\Delta^k |h|^2$ is nonnegative for all $k\in \mathbb{N}$ where $\Delta$ is the Laplace-Beltrami operator. This fact is generalized to harmonic functions defined…

We prove a Liouville property for uniformly almost localized (up to translations) H 1-global solutions of the Camassa-Holm equation with a momentum density that is a non negative finite measure. More precisely, we show that such solution…

For a compact space $K$ we denote by $C_w(K)$ ($C_p(K)$) the space of continuous real-valued functions on $K$ endowed with the weak (pointwise) topology. In this paper we address the following basic question which seems to be open: Suppose…

For two not necessarily commutative topological groups G and T, let H(G,T) denote the space of all continuous homomorphisms from G to T with the compact-open topology. We prove that if G is metrizable and T is compact then H(G,T) is a…

Let $\mu$ and $\nu$ be probability measures on a group \Gamma and let G_\mu and G_\nu denote Green's function with respect to \mu and \nu . The group \Gamma is said to admit instability of Green's function if there are symmetric, finitely…

We review and give elementary proofs of Liouville type properties of harmonic and subharmonic functions in the plane endowed with a complete Riemannian metric, and prove a gap theorem for the possible growth of harmonic functions when this…

Commability is the finest equivalence relation between locally compact groups such that $G$ and $H$ are equivalent whenever there is a continuous proper homomorphism $G \to H$ with cocompact image. Answering a question of Cornulier, we show…

Recently, Charpentier showed that there exist holomorphic functions $f$ in the unit disk such that, for any proper compact subset $K$ of the unit circle, any continuous function $\phi$ on $K$ and any compact subset $L$ of the unit disk,…

We prove a Liouville property for any $f$-harmonic function with polynomial growth on a complete noncompact smooth metric measure space $(M,g,e^{-f}dv)$ when the Bakry-\'Emery Ricci curvature is nonnegative and its diameter of geodesic…

Given an arbitrary planar $\infty$-harmonic function $u$, for each $\alpha>0$ we establish a quantitative local $W^{1,2}$-estimate of $|Du|^\alpha $, which is sharp as $\alpha\to0$. We also show that the distributional determinant of $u$ is…

We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with…

We study some qualitative properties (including removable singularities and superharmonicity) of non-negative solutions to $$ (-\Delta)^\gamma u=fu^p\quad\text{in }\mathbb R^n\setminus\Sigma $$ which are singular at $\Sigma$. Here $\gamma…

For a compact homogeneous space $G/K$, we study the problem of existence of $G$-invariant Riemannian metrics such that each eigenspace of the Laplacian is a real irreducible representation of $G$. We prove that the normal metric of a…

We prove that a compact space $K$ embeds into a $\sigma$-product of compact metrizable spaces ($\sigma$-product of intervals) if and only if $K$ is (strongly countable-dimensional) hereditarily metalindel\"of and every subspace of $K$ has a…

For a locally compact group $G$ and $p \in (1,\infty)$, we define and study the Beurling-Figa-Talamanca-Herz algebras $A_p(G,\omega)$. For $p=2$ and abelian $G$, these are precisely the Beurling algebras on the dual group $\hat{G}$. For $p…

On a complete Riemannian manifold $(M,g)$, we consider $L^{p}_{loc}$ distributional solutions of the the differential inequality $-\Delta u + \lambda u \geq 0$ with $\lambda >0$ a locally bounded function that may decay to $0$ at infinity.…