The Hardy inequality ${\int }_{\Omega }{\left|u\left(x\right)\right|}^{p}d{\left(x\right)}^{-p}\ddot{x}\le c{\int }_{\Omega }{\left|\nabla u\left(x\right)\right|}^p\ddot{x}$ with $d\left(x\right)=dist(x,\partial \Omega )$ holds for $u\in C_0^{\infty }\left(\Omega \right)$ if $\Omega \subset {ℝ}^n$ is an open set with a sufficiently smooth boundary and if $1<p<\infty$. P. Hajlasz proved the pointwise counterpart to this inequality involving a maximal function of Hardy-Littlewood type on the right hand side and, as a consequence, obtained the integral Hardy inequality. We extend these results for gradients of higher order and also for $p=1$.

This paper gives a stochastic representation in spectral terms for the absorption time T of a finite Markov chain which is irreducible and reversible outside the absorbing point. This yields quantitative informations on the parameters of a similar representation due to O'Cinneide for general chains admitting real eigenvalues. In the discrete time setting, if the underlying Dirichlet eigenvalues (namely the eigenvalues of the Markov transition operator restricted to the functions vanishing on...

The notion of the extremal length and the module of families of curves has been studied extensively and has given rise to a lot of applications to complex analysis and the potential theory. In particular, the coincidence of the p-module and the p-capacity plays an mportant role. We consider this problem on the Carnot group. The Carnot group G is a simply connected nilpotent Lie group equipped vith an appropriate family of dilations. Let omega be a bounded domain on G and Ko, K1 be disjoint non-empty...

A nonlinear generalization of convergence sets of formal power series, in the sense of Abhyankar-Moh [J. Reine Angew. Math. 241 (1970)], is introduced. Given a family $y={\phi }_s(t,x)=sb₁\left(x\right)t+b₂\left(x\right)t²+\cdots$ of analytic curves in ℂ × ℂⁿ passing through the origin, $Con{v}_{\phi }\left(f\right)$ of a formal power series f(y,t,x) ∈ ℂ[[y,t,x]] is defined to be the set of all s ∈ ℂ for which the power series $f({\phi }_s(t,x),t,x)$ converges as a series in (t,x). We prove that for a subset E ⊂ ℂ there exists a divergent formal power series f(y,t,x) ∈ ℂ[[y,t,x]] such that $E=Con{v}_{\phi }\left(f\right)$ if and only if...

We give several definitions of the pluricomplex Green function and show their equivalence.

We study the integral representation of potentials by exit laws in the framework of sub-Markovian semigroups of bounded operators acting on $L^2\left(m\right)$. We mainly investigate subordinated semigroups in the Bochner sense by means of ${𝒞}^1$-subordinators. By considering the one-sided stable subordinators, we deduce an integral representation for the original semigroup.

In this paper we prove the continuity of fractional integrals acting on nonhomogeneous function spaces defined on spaces of homogeneous type with finite measure. A definition of the molecules which are used in the $H^p$ theory is given. Results are proved for $L^p$, $H^p$, BMO, and Lipschitz spaces.

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.