Related papers: Estimates and structure of $\alpha$-harmonic funct…

We prove a sharp integral gradient estimate for harmonic functions on noncompact K\"ahler manifolds. As application, we obtain a sharp estimate for the bottom of spectrum of the p-Laplacian and prove a splitting theorem for manifolds…

We prove the scale invariant Elliptic Harnack Inequality (EHI) for non-negative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of EHI in ${\mathbb{Z}}^d$ that…

We show that two classically known properties of positive supersolutions of uniformly elliptic PDEs, the boundary point principle (Hopf lemma) and global integrability, can be quantified with respect to each other. We obtain an extension to…

In this paper, we discuss tangential limits for regular harmonic functions with respect to $\phi(\Delta):=-\phi(-\Delta)$ in the $C^{1,1}$ open set $D$ in $\mathbb{R}^d$, where $\phi$ is the complete Bernstein function and $d \ge 2$. When…

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain satisfying the uniform exterior cone condition. We establish existence and uniqueness of continuous solutions of the Dirichlet Problem associated to certain intrinsic nonlinear mean value…

The solutions of a kind of second-order homogeneous partial differential equation are called (real kernel) alpha-harmonic functions. In this paper, the boundary correspondence and boundary behavior of alpha-harmonic functions are studied,…

Let $\Omega \subset {R}^n,$ $n \geq 3,$ be a bounded open set, $x=(x_1,x_2,\ldots,x_n)$ a generic point which belongs to $\Omega,$ $u \colon \Omega \to {R}^N ,$ $N>1,$ and $ Du=(D_\alpha u^i)$, $D_\alpha = \partial/\partial x_\alpha, $…

This paper is concerned with nonlinear elliptic equations in nondivergence form where the operator has a first order drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative…

We give Martin representation of nonnegative functions caloric with respect to the fractional Laplacian in Lipschitz open sets. The caloric functions are defined in terms of the mean value property for the space-time isotropic…

We obtain Schwarz-Pick lemma for $(\alpha, \beta)$-harmonic functions u in the disc, where $\alpha$ and $\beta$ are complex parameters satisfying $\Re \alpha + \Re \beta > -1$. We prove sharp estimate of derivative at the origin for such…

We present two companion results: Phragm\'en-Lindel\"of type tight bounds on the minimal possible growth of subharmonic functions with recurrent zero set, and tight bounds on the maximal possible decay of the harmonic measure of the outer…

Let $\alpha\in(0,1)$, $\Omega$ be a bounded open domain in $R^N$ ($N\ge 2$) with $C^2$ boundary $\partial\Omega$ and $\omega$ be the Hausdorff measure on $\partial\Omega$. We denote by $\frac{\partial^\alpha \omega}{\partial…

We prove a scale-invariant boundary Harnack principle for inner uniform domains over a large family of Dirichlet spaces. A novel feature of our work is that our assumptions are robust to time changes of the corresponding diffusions. In…

We prove that bounded harmonic functions of anisotropic fractional Laplacians are H\"older continuous under mild regularity assumptions on the corresponding L\'evy measure. Under some stronger assumptions the Green function, Poisson kernel…

We prove Fatou's theorem for nonnegative harmonic functions with respect to subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such…

We begin by shortly recalling a generalized mean value inequality for subharmonic functions, and two applications of it: first a weighted boundary behavior result (with some new references and remarks), and then a borderline case result to…

Suppose that $d\geq2$ and $\alpha\in(1,2)$. Let D be a bounded $C^{1,1}$ open set in $\mathbb{R}^d$ and b an $\mathbb{R}^d$-valued function on $\mathbb{R}^d$ whose components are in a certain Kato class of the rotationally symmetric…

We study the regularity of Radon measures $\mu$ which satisfy that there exists a function $h_\mu$ in $H^1(\Omega)$, stationary harmonic such that $\Delta h_\mu =\mu$ in $\Omega$ (here $\Omega$ is an open set of $\mathbb{R}^2$). Such…

Let $D$ be a non-empty open subset of $\R^m,\,m\ge 2$, with boundary $\partial D$, with finite Lebesgue measure $|D|$, and which satisfies a parabolic Harnack principle. Let $K$ be a compact, non-polar subset of $D$. We obtain the leading…

Let $ \mathbb{R}^{n} $ denote Euclidean $ n $ space and given $k$ a positive integer let $ \Lambda_k \subset \mathbb{R}^{n} $, $ 1 \leq k < n - 1, n \geq 3, $ be a $k$-dimensional plane with $ 0 \in \Lambda_k.$ If $n-k < p <\infty$, we…