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

By classical Fatou type theorems in various setups, it is well-known that positive harmonic functions have non-tangential limit at almost every point on the boundary. In this paper, in the setting of non-positively curved Harmonic manifolds…

It is shown that an arbitrary function from $D\subset \R^n$ to $\R^m$ will become $C^{0,\alpha}$-continuous in almost every $x\in D$ after restriction to a certain subset with limit point $x$. For $n\geq m$ differentiability can be…

We prove an uniform boundary Harnack inequality for nonnegative functions harmonic with respect to $\alpha$-stable process on the Sierpi{\'n}ski triangle, where $\alpha \in (0, 1)$. Our result requires no regularity assumptions on the…

We prove the stronger version of Harnack's inequality for positive harmonic functions defined on the unit disc.

One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{SD}$ of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space…

A class of functions involving the divided differences of the psi function and the polygamma functions and originating from Kershaw's double inequality are proved to be completely monotonic. As applications of these results, the…

On a complete $p$-nonparabolic $3$-dimensional manifold with non-negative scalar curvature and vanishing second homology, we establish a sharp monotonicity formula for the proper $p$-Green function along its level sets for $1<p<3$. This can…

We study Widom factors for (a) monic orthogonal polynomials in $L^2$ with respect to the equilibrium measure of a compact set $K\subset\mathbb{R}$ and (b) residual polynomials normalized at an exterior point. Using weakly equilibrium Cantor…

This paper investigates the existence, nonexistence, and qualitative properties of p-harmonic functions in the upper half-space $\mathbb{R}^N_+ \, (N \geq 3)$ satisfying nonlinear boundary conditions for $1<p<N$. Moreover, the symmetry of…

This paper is the continuation of a program, initiated in Grenier-Nguyen [8,9], to derive pointwise estimates on the Green function of Orr Sommerfeld equations. In this paper we focus on long wavelength perturbations, more precisely…

A complex function $f(z)$ is called a Herglotz-Nevanlinna function if it is holomorphic in the upper half-plane ${\mathbb C}_+$ and maps ${\mathbb C}_+$ into itself. By a maximum principle a Herglotz-Nevanlinna function which takes a real…

For $\Omega$ a domain in $\mathbb C^n$, the pluricomplex Green function with poles $a_1, ...,a_N \in \Omega$ is defined as $G(z):=\sup \{u(z): u\in PSH_-(\Omega), u(x)\le \log \|x-a_j\|+C_j \text{when} x \to a_j, j=1,...,N \}$. When there…

GW calculations with fully self-consistent G and W -- based on the iterative solution of the Dyson equation -- provide an approach for consistently describing ground and excited states on the same quantum mechanical level. We show that for…

The Hardy-Littlewood maximal function $\mathcal{M}$ and the trigonometric function $\sin{x}$ are two central objects in harmonic analysis. We prove that $\mathcal{M}$ characterizes $\sin{x}$ in the following way: let $f \in…

We consider a general one-particle Hamiltonian H = - \Delta_r + u(r) defined in a d-dimensional domain. The object of interest is the time-independent Green function G_z(r,r') = < r | (z-H)^{-1} | r' >. Recently, in one dimension (1D), the…

Suppose $(M,g)$ is a Riemannian manifold having dimension $n$, nonnegative Ricci curvature, maximal volume growth and unique tangent cone at infinity. In this case, the tangent cone at infinity $C(X)$ is an Euclidean cone over the…

Given a bounded open set $\Omega\subset \mathbb{R}^n$, we study sequences of quadratic functionals on the Sobolev space $H^1_0(\Omega)$, perturbed by sequences of bounded linear functionals. We prove that their $\Gamma$-limits, in the weak…

For any $p\in[1,\infty)$, we prove that the set of simple functions taking at most $k$ different values is proximinal in B\"ochner spaces $L^p(X)$ whenever $X$ is a dual Banach space with $w^*$-sequentially compact unit ball. With…

The aim of the paper is to highlight some open problems concerning approximation properties of Hardy spaces. We also present some results on the bounded compact and the dual compact approximation properties (shortly, BCAP and DCAP) of such…

Let $\varphi$ be a plurisubharmonic function on a pseudoconvex domain $D \subset \mathbb C^n$. We show that there exists a nonzero holomorphic function $f$ on $D$ such that some local mean value of $\varphi$ with logarithmic additional…