Some properties of the functions of the form $v\left(x\right)={\sum }_{i=0}^m{\left|x\right|}^i{h}_i\left(x\right)$ in ℝⁿ, n ≥ 2, where each $h_i$ is a harmonic function defined outside a compact set, are obtained using the harmonic measures.

Let $u$ be a $\delta$-subharmonic function with associated measure $\mu$, and let $v$ be a superharmonic function with associated measure $\nu$, on an open set $E$. For any closed ball $B(x,r)$, of centre $x$ and radius $r$, contained in $E$, let $ℳ(u,x,r)$ denote the mean value of $u$ over the surface of the ball. We prove that the upper and lower limits as $s,t\to 0$ with $0<s<t$ of the quotient $\left(ℳ\right(u,x,s)-ℳ(u,x,t\left)\right)/\left(ℳ\right(v,x,s)-ℳ(v,x,t\left)\right)$, lie between the upper and lower limits as $r\to 0+$ of the quotient $\mu \left(B\right(x,r\left)\right)/\nu \left(B\right(x,r\left)\right)$. This enables us to use some well-known measure-theoretic results to prove new variants...

A 2p-times continuously differentiable complex-valued function f = u + iv in a domain D ⊆ ℂ is p-harmonic if f satisfies the p-harmonic equation ${\Delta }^{p}f=0$, where p (≥ 1) is a positive integer and Δ represents the complex Laplacian operator. If Ω ⊂ ℂⁿ is a domain, then a function $f:\Omega \to {ℂ}^m$ is said to be p-harmonic in Ω if each component function $f_i$ (i∈ 1,...,m) of $f=(f₁,...,f_m)$ is p-harmonic with respect to each variable separately. In this paper, we prove Landau and Bloch’s theorem for a class of p-harmonic mappings...

Let $\Delta u={\Sigma }_i\frac{{\partial }^2}{{\partial }_{x_i}^2}$, $Lu={\Sigma }_{i,j}{a}_{ij}\frac{{\partial }^2}{\partial x_i\partial x_j}u+{\Sigma }_i{b}_i\frac{\partial }{\partial x_i}u+cu$ be elliptic operators with Hölder continuous coefficients on a bounded domain $\Omega \subset {\mathbf{R}}^n$ of class $C^{1,1}$. There is a constant $c\&gt;0$ depending only on the Hölder norms of the coefficients of $L$ and its constant of ellipticity such that $$c^{-1}{G}_{\Delta }^{\Omega }\le G_L^{\Omega }\le c{G}_{\Delta }^{\Omega }\text{on}\Omega \times \Omega ,$$ where ${\gamma }_{\Delta }^{\Omega }$ (resp. $G_L^{\Omega }$) are the Green functions of $\Delta$ (resp. $L$) on $\Omega$.

New concepts of Lebesgue measure on ${ℝ}^{\infty }$ are proposed and some of their realizations in the ZFC theory are given. Also, it is shown that Baker’s both measures [1], [2], Mankiewicz and Preiss-Tišer generators [6] and the measure of [4] are not α-standard Lebesgue measures on ${ℝ}^{\infty }$ for α = (1,1,...).

Let $K\subset {ℝ}^m$ ($m\ge 2$) be a compact set; assume that each ball centered on the boundary $B$ of $K$ meets $K$ in a set of positive Lebesgue measure. Let $C_0^{\left(1\right)}$ be the class of all continuously differentiable real-valued functions with compact support in ${ℝ}^m$ and denote by ${\sigma }_m$ the area of the unit sphere in ${ℝ}^m$. With each $\varphi \in C_0^{\left(1\right)}$ we associate the function $$W_K\varphi \left(z\right)=\frac{1}{{\sigma }_m}\underset{{ℝ}^m\setminus K}{\to }\int \mathrm{g}rad\varphi \left(x\right)·\frac{z-x}{{|z-x|}^m}x$$ of the variable $z\in K$ (which is continuous in $K$ and harmonic in $K\setminus B$). $W_K\varphi$ depends only on the restriction ${\varphi |}_B$ of $\varphi$ to the boundary $B$ of $K$. This gives rise to a linear operator $W_K$...

Let $U$ be a domain of type $H$ in a Brelot potential theory. A compact $K$ in $U$ is a $G_{\delta }$ in $U$ iff $U-K$ has at most countably many components. If $F$ is a relatively closed locally polar subset of $U$, any $G_{\delta }$ in $F$ is a $G_{\delta }$ in $U$. If $V$ is a domain in $U$, all Borel subsets of $\partial V\cap U$ are Baire even if $\partial V\cap U$ is not metrizable. The known results concerning equivalences between weak thinness, thinness, and strong thinness of a set $A$ at a point $x\notin A$ are extended from the case where $\left\{x\right\}$ is a $G_{\delta }$ to the cases in which $A$ meets only...

We prove that if $f:Z^d\to R$ is harmonic and there exists a polynomial $W:Z^d\to R$ such that f + W is nonnegative, then f is a polynomial.

Let $\Omega \subset {ℝ}^n,n\ge 3$, and let $p,1<p<\infty ,p\ne 2$, be given. In this paper we study the dimension of $p$-harmonic measures that arise from non-negative solutions to the $p$-Laplace equation, vanishing on a portion of $\partial \Omega$, in the setting of $\delta$-Reifenberg flat domains. We prove, for $p\ge n$, that there exists $\tilde{\delta }=\tilde{\delta }(p,n)>0$ small such that if $\Omega$ is a $\delta$-Reifenberg flat domain with $\delta <\tilde{\delta }$, then $p$-harmonic measure is concentrated on a set of $\sigma$-finite $H^{n-1}$-measure. We prove, for $p\ge n$, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$-harmonic...