Let X×Y be the Cartesian product of two locally finite, connected networks that need not have reversible conductance. If X,Y represent random walks, it is known that if X×Y is recurrent, then X,Y are both recurrent. This fact is proved here by non-probabilistic methods, by using the properties of separately superharmonic functions. For this class of functions on the product network X×Y, the Dirichlet solution, balayage, minimum principle etc. are obtained. A unique integral representation is given...

We characterize all subsets $M$ of ${ℝ}^n\times {ℝ}^{+}$ such that $$\underset{X\in {ℝ}^n\times {ℝ}^{+}}{sup}u\left(X\right)=\underset{X\in M}{sup}u\left(X\right)$$ for every bounded parabolic function $u$ on ${ℝ}^n\times {ℝ}^{+}$. The closely related problem of representing functions as sums of Weierstrass kernels corresponding to points of $M$ is also considered. The results provide a parabolic counterpart to results for classical harmonic functions in a ball, see References. As a by-product the question of representability of probability continuous distributions as sums of multiples of normal distributions is investigated.

Let $\alpha >0$, $\lambda ={\left(2\alpha \right)}^{-1/2}$, $S^{n-1}$ be the $(n-1)$-dimensional unit sphere, $\sigma$ be the surface measure on $S^{n-1}$ and $h\left(x\right)={\int }_{S^{n-1}}{e}^{\lambda \langle x,y\rangle }d\sigma \left(y\right)$. We characterize all subsets $M$ of ${ℝ}^n$ such that $$\underset{x\in {ℝ}^n}{inf}\frac{u\left(x\right)}{h\left(x\right)}=\underset{x\in M}{inf}\frac{u\left(x\right)}{h\left(x\right)}$$ for every positive solution $u$ of the Helmholtz equation on ${ℝ}^n$. A closely related problem of representing functions of $L_1\left(S^{n-1}\right)$ as sums of blocks of the form $e^{\lambda \langle x_k,.\rangle }/h\left(x_k\right)$ corresponding to points of $M$ is also considered. The results provide a counterpart to results for classical harmonic functions in a ball, and for parabolic functions on a slab, see References.

We give a method for constructing functions $\phi$ and $\psi$ for which $H(x,y)=\phi \left(x\right)-\psi \left(y\right)$ has a specified subharmonic minorant $h(x,y)$. By a theorem of B. Cole, this procedure establishes integral mean inequalities for conjugate functions. We apply this method to deduce sharp inequalities for conjugates of functions in the class $L{log}^{\alpha }L$, for $-1\le \alpha \<\infty$. In particular, the case $\alpha =1$ yields an improvement of Pichorides’ form of Zygmund’s classical inequality for the conjugates of functions in $LlogL$. We also apply the method to produce a new proof of the...

Let ${\left(h_k\right)}_{k\ge 0}$ be the Haar system on [0,1]. We show that for any vectors $a_k$ from a separable Hilbert space and any ${\epsilon }_k\in [-1,1]$, k = 0,1,2,..., we have the sharp inequality $\left|\right|{\sum }_{k=0}^n{\epsilon }_k{a}_k{h}_k{\left|\right|}_{W\left(\right[0,1\left]\right)}\le 2\left|\right|{\sum }_{k=0}^n{a}_k{h}_k{\left|\right|}_{L^{\infty }\left([0,1]\right)}$, n = 0,1,2,..., where W([0,1]) is the weak-$L^{\infty }$ space introduced by Bennett, DeVore and Sharpley. The above estimate is generalized to the sharp weak-type bound ${\left|\right|Y\left|\right|}_{W\left(\Omega \right)}{\le 2\left|\right|X\left|\right|}_{L^{\infty }\left(\Omega \right)}$, where X and Y stand for -valued martingales such that Y is differentially subordinate to X. An application to harmonic functions on Euclidean domains is presented.

On relatively compact domains in metric measure spaces we construct singular functions that play the role of Green functions of the p-Laplacian. We give a characterization of metric spaces that support a global version of such singular function, in terms of capacity estimates at infinity of such metric spaces. In addition, when the measure of the space is locally Q-regular, we study quasiconformal invariance property associated with the existence of global singular functions.

We complete the characterization of singular sets of separately analytic functions. In the case of functions of two variables this was earlier done by J. Saint Raymond and J. Siciak.

On étudie les singularités et l’intégrabilité d’une classe de fonctions plurisousharmoniques sur une variété analytique $X$ de dimension $n\ge 1$. Pour étudier ce problème, nous commençons par contrôler les nombres de Lelong de certains types de fonctions plurisousharmoniques $\varphi$. Ensuite, nous étudions les singularités du transformé strict du courant $d{d}^c\varphi$ par un éclatement de $X$ au dessus d’un point. Nous répondons ainsi positivement au problème d’intégrabilité locale de ${\mathrm{e}}^{-\varphi }$, lorsque $\dim X=2$, et lorsque $\varphi$ est une fonction plurisousharmonique...