Wolfhard Hansen

A converse to the mean value theorem for harmonic functions

(A Lebesgue measure on Rd). The converse question to what extent this restricted mean value property implies harmonicity has a long history (we axe indebted to I. Netuka for valuable hints). Volterra [26] and Kellogg [20] noted first that a continuous function f on the closure U of U satisfying (*) is harmonic on U. At least if U is regular there is a very elementary proof for this fact (see Burckel [7]): Let g be the difference between f and the solution of the Dirichlet problem with boundary value f . If g#0 , say a=sup g(U)> 0, choose x6{g=a} having minimal distance to the boundary. Then (*) leads to an immediate contradiction. In fact, for continuous functions on U the question is settled for arbitrary harmonic spaces and arbitrary representing measures #x # ~ for harmonic functions. If f is bounded on U and Borel measurable the answer may be negative unless restrictions on the radius r(x) of the balls B ~ are imposed (Veech [23]): Let U = ] I , 1[, / (0 )=0 , f = i on ] -1 ,0[ , f= l on ]0, 1[, 0~B x for x?t0 (similarly in R d, d/>2)! There are various positive results, sometimes under restrictions on U, but always under restrictions on the function x~-+r(x) (Feller [9], Akcoglu and Sharpe [1], Baxter [2] and [3], Heath [17], Veech [23] and [24]). For example Heath [17] showed for arbitrary U that a bounded Lebesgue measurable function on U having the restricted mean value property (.) is harmonic provided that , for some e>0, cd(x, CU)

Perturbation of harmonic spaces and construction of semigroups

In I-8] and [5] perturbation of harmonic spaces is studied for determining the cohomology groups of sheaves of harmonic functions. We now want to use this perturbation for the construction of associated resolvents and semigroups on a harmonic space X which may fail to be strong-harmonic. This construction applies on every F, n (n> 1) and yields Brownian motion, if we take the natural perturbation of the classical potential theory corresponding to the replacement of the differential equation A u = 0 by A u = 2 ct u (ct e F,+). We note that the underlying space may be compact such as R/Z. The main result is the following: There exists a Hunt process Y on X such that for every open subset U of X and ct > 0 the or-excessive functions of the restriction yv of Y on U are precisely the positive ct-hyperharmonic functions on U. If U is a relatively compact open subset of some strongharmonic subspace of X then yv has a Feller semigroup of transitions if and only if U is regular. The construction of the corresponding resolvents requires some additional facts on the existence of strict sections of potentials and on perturbation of harmonic spaces. Two preliminary sections which may be of interest by themselves are devoted to these facts. The notation is essentially based on the representation given in [5]. For the readers convenience we repeat the introduction of the main notions.

An invariance result for capacities on Wiener space

We prove that tight capacities are invariant if one weakens the underlying topology. As a consequence we obtain a comparison theorem about (r, p)-capacities (and the corresponding notion of (r, p)-quasi-continuity used in the Malliavin calculus) on different abstract Wiener spaces (Ej, H, μj) with common Hilbert space H. Furthermore, we prove tightness of (r, p)-capacities of Ornstein-Uhlenbeck semigroups with general linear drift.

Perturbation by differences of unbounded potentials

Perturbation by differences of unbounded potentials and corresponding eigenvalues to the time-independent Schrodinger equation yields the following: Let μ be a signed measure on R n such that μ does not charge polar sets and μ − generates a finite potential. Let U be a bounded domain in R n such that U is (μ+βλ n )-bounded for some β≥0 (λ n Lebesgue measure). Then the set of all eigenvalues of the equation (1/2Δ-μ)u=0 on U is a non-empty upper bounded subset of R which has no accumulation points

Isoperimetric Inequalities for Capacities

Given a bounded Borel subset A of ℝ n , n ≥ 2, such that the volume λ n (A) is strictly positive, let r e (A) denote the circumradius of A and let r 0(A) be the radius of the open ball A 0 with center 0 such that λ n (A 0) = λ n (A).

A Liouville property for spherical averages in the plane

Abstract. It is shown that any continuous bounded function f on

A Strong Version of Liouville's Theorem

{\mathbb R}^2

Volume densities with the mean value property for harmonic functions

such that \[f(x)=\frac{1}{(2\pi)} \int_0^{2\pi} f(x+r(x)e^{it}) dt\; , \]

Inverse mean value property of harmonic functions

x\in{\mathbb R}^2

Liouville Property, Wiener’s Test and Unavoidable Sets for Hunt Processes

, is constant provided r is a strictly positive real function on