Related papers: H\"older continuity of harmonic functions for Hunt…

Within the framework of balayage spaces (the analytical equivalent of nice Hunt processes), we prove equicontinuity of bounded families of harmonic functions and apply it to obtain criteria for compactness of potential kernels.

We show that quasi-minimizers of non-homogeneous energy functionals on metric measure spaces are locally H\"older continuous and satisfy the Harnack inequality. We assume that the spaces are doubling and support a Poincar\'e inequality. The…

Given a "Green function" $G$ on a locally compact space $X$ with countable base, a Borel set $A$ in $X$ is called $G$-semipolar, if there is no measure $\nu\ne 0$ supported by $A$ such that $G\nu:=\int G(\cdot,y)\,d\nu(y)$ is a continuous…

In this paper we consider the Dirichlet form on the half-space $\mathbb{R}^d_+$ defined by the jump kernel $J(x,y)=|x-y|^{-d-\alpha}\mathcal{B}(x,y)$, where $\mathcal{B}(x,y)$ can be degenerate at the boundary. Unlike our previous works…

We consider a family of pseudo differential operators $\{\Delta+ a^\alpha \Delta^{\alpha/2}; a\in [0, 1]\}$ on $\R^d$ that evolves continuously from $\Delta$ to $\Delta + \Delta^{\alpha/2}$, where $d\geq 1$ and $\alpha \in (0, 2)$. It gives…

In this paper, we consider a large class of subordinate Brownian motions $X$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero and at infinity. We first discuss…

Spaces of harmonic functions in upper half-space with controlled growth near the boundary are described in terms of multiresolution approximations. The results are applied to prove the law of the iterated logarithm for the oscillation of…

Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v $ that fulfill a doubling…

We study properties of $\mathcal{A}$-harmonic and $\mathcal{A}$-superharmonic functions involving an operator having generalized Orlicz-growth embracing besides Orlicz case also natural ranges of variable exponent and double-phase cases. In…

Using three hypergeometric identities, we evaluate the harmonic measure of a finite interval and of its complementary for a strictly stable real L{\'e}vy process. This gives a simple and unified proof of several results in the literature,…

In a complete metric space equipped with a doubling measure supporting a $p$-Poincar\'e inequality, we prove sharp growth and integrability results for $p$-harmonic Green functions and their minimal $p$-weak upper gradients. We show that…

Homogeneous and inhomogeneous biharmonic equation are considered on the $n$-dimensional unit sphere. The Green function is given as a series of Gegenbauer polynomials. In the paper, explicit representations of the Green function are found…

In this paper, we study discrete harmonic functions on infinite penny graphs. For an infinite penny graph with bounded facial degree, we prove that the volume doubling property and the Poincar\'e inequality hold, which yields the Harnack…

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function $f$ can be approximated in norm by its dilates $f_r(z):=f(rz)~(r<1)$, in other words, $\lim_{r\to1^-}\|f_r-f\|=0$. We construct a…

We study a notion of generalized H\"older continuity for functions on $\mathbb{R}^d$. We show that for any bounded function $f$ of bounded support and any $r>0$, the $r$-oscillation of $f$ defined as $osc_r f (x):= \sup_{B_r(x)} f -…

In this paper, a necessary and sufficient condition is obtained for the scale invariant boundary Harnack inequality (BHP in abbreviation) for a large class of Hunt processes on metric measure spaces that are in weak duality with another…

This paper focuses on a relation between the growth of harmonic functions and the Hausdorff measure of their zero sets. Let $u$ be a real-valued harmonic function in $\mathbb{R}^n$ with $u(0)=0$ and $n\geq 3$. We prove…

We give a proof of Lipschitz continuity of p-harmonious functions, that are tug-of-war game analogies of ordinary p-harmonic functions. This result is used to obtain a new proof of Harnack's inequality for p-harmonic functions in the case…

For $0<\alpha<1$ let $V(\alpha)$ denote the supremum of the numbers $v$ such that every $\alpha$-H\"older continuous function is of bounded variation on a set of Hausdorff dimension $v$. Kahane and Katznelson (2009) proved the estimate $1/2…

The aim of this paper is twofold. First, we prove $L^p$ estimates for a regularized Green's function in three dimensions. We then establish new estimates for the discrete Green's function and obtain some positivity results. In particular,…