Martin compactification for discrete potential theory and the mean value property

A converse of the well-known theorem on themean value property of harmonic functions is given. It is shown that a positive measurable function is harmonic if it possesses arestricted mean value property. Earlier proofs obtained using the probabilistic techniques were given by Veech, Heath and Baxter. Our approach is based on a Martin type compactification built up with the help of some quite elementarya priori inequalities foraveraging kernels.     

Convex functions and subharmonic functions

It is a classical result that a composition of a convex, increasing function and of a subharmonic function is subharmonic. We give related results for a composition of a convex function of several variables and of several subharmonic functions, thus imporving some recent results in this area.     

Some properties of the riesz charge associated with a δ-subharmonic function

The first property is a refinement of earlier results of Ch. de la Vallée Poussin, M. Brelot, and A. F. Grishin. Let w=u–v with u, v superharmonic on a suitable harmonic space (for example an open subset of R ⁿ ), and let [w]=[u]–[v] denote the associated Riesz charge. If w0, and if E denotes the set of those points of at which the lim inf of w in thefine topology is 0, then the restriction of [w] to E is 0. Another property states that, if e denotes a polar subset of such that the fine lim inf of |w| at each point of e is finite, then the restriction of [w] to e is 0.     

Conditions for separately subharmonic functions to be subharmonic

Letu be a function on ^m × ⁿ , wherem2 andn2, such thatu(x, .) is subharmonic on ⁿ for each fixedx in ^m andu(.,y) is subharmonic on ^m for each fixedy in ⁿ . We give a local integrability condition which ensures the subharmonicity ofu on ^m × ⁿ , and we show that this condition is close to being sharp. In particular, the local integrability of (log⁺ u ⁺) ^m+n–2+ is enough to secure the subharmonicity ofu if >0, but not if <0.     

Fonctions finement harmoniques et l'axiome de domination

In a harmonic space with the domination Axiom (axiom D), B. Fuglede [5] has introduced the sheaf property of the cones of the finely hyperharmonic functions (defined in the fine opens). In [7], [8], J. Luke, J. Malý and L. Zajíek have studied a notion analogous to the finely hyperharmonic functions without supposing axiom D, and have proved ([8] theorem 12.16) that if the cones of the positive finely hyperharmonic functions make a sheaf, then axiom D is satisfied. See N. Boboc, Gh. Bucur and A. Cornea [3] for the first result of this type given within the context of theH-cones.In this paper, we prove axiom D is a consequence of the sheaf property even for the smallest class of the functions (the class of the finely harmonic functions; absolute-value bounded in a convenient sense). This result implies that of Lukeet al. cited above. Consequently, the tow fine properties of the sheaf are equivalent, which has not been evident previously.

     

Separately harmonic and subharmonic functions

Let u(x, y) be defined in B ₁×B ₂ where B ₁ ^m and B ₂ ⁿ , and assume that u(x, ·) harmonic for every fixed x and u(·, y) is subharmonic for every fixed y. We show that if u(·, y) is, in addition, C ² for each y then u is subharmonic in B ₁×B ₂ in both variables jointly.     

Removability of polar sets for harmonic functions

A classical result of G. Bouligand states that bounded harmonic functions can be extended across closed polar sets. F.-Y. Maeda replaced the boundedness assumption by the condition of energy finiteness for harmonic spaces with Green function.This paper proves this result for generalP-harmonic spaces and shows that the extension property for a harmonic functionu and the condition of energy finiteness are equivalent to a majorization property foru ² .     

Absorbent,parabolic, elliptic and quasielliptic balayages in potential theory; relationships with the Green function

In the frame of standard H-cones of functions (the cone of all excessive functions with respect to a submarkovian resolvent of kernels with reference measure on a measurable space) on a Green set we show that the cofine closure of the complement of an absorbent set in coabsorbent. We obtain different characterizations concerning the parabolicity, ellipticity and quasiellipticity in terms of the Green function. We also show that these notions are the same in the direct and the dual theory.     

Superharmonic extensions,mean values and Riesz measures

There are two interrelated themes to this paper. One is the generalization of recent harmonic and superharmonic extension theorems to the case where the removable set is not relatively closed, with the simultaneous weakening of other hypotheses in the harmonic case. The other is the use of results which are well-known in geometric measure theory, to prove theorems on the relative behaviour of the spherical mean values of a -subharmonic and a superharmonic function, and to establish new criteria for harmonic and superharmonic extensions. Some related theorems establish sufficient conditions for a polar set to be positive for the Riesz measure of a -subharmonic function, a useful formula for the restriction of such a measure to the infinity set of a superharmonic function, and a condition for such a restriction to be absolutely continuous with respect to an appropriate Hausdorff measure.     

Integral representations of harmonic functions in half spaces

In this paper, using a modified Poisson kernel in an upper half-space, we prove that a harmonic function u(z) in a upper half space with its positive part u⁺(x)=max{u(x),0} satisfying a slowly growing condition can be represented by its integral in the boundary of the upper half space, the integral representation is unique up to the addition of a harmonic polynomial, vanishing in the boundary of the upper half space and that its negative part u⁻(x)=max{−u(x),0} can be dominated by a similar slowly growing condition, this improves some classical result about harmonic functions in the upper half space.     

An introduction to the k-defect polynomials

The 0-defect polynomial of a graph is just the chromatic polynomial. This polynomial has been widely studied in the literature. Yet little is known about the properties of k-defect polynomials of graphs in general, when 0 < k ≤ |E(G)|. In this survey we give some properties of k-defect polynomials, in particular we highlight the properties of chromatic polynomials which also apply to k-defect polynomials. We discuss further research which can be done on the k-defect polynomials.     

Gallai–Edmonds structure theorem for weighted matching polynomial

In this paper, we prove the Gallai–Edmonds structure theorem for weighted matching polynomials. Our result implies the Parter–Wiener theorem and its recent generalization about the existence of principal submatrices of a Hermitian matrix whose graph is a tree.     

Existence of Tangential Limits for α-Harmonic Functions on Half Spaces

Our aim in this paper is to prove the existence of tangential limits for Poisson integrals of the fractional order of functions in the L ^p Hölder space on half spaces.     

On the riesz charge of the lower envelope of δ-subharmonic functions

By potential theoretic methods involving the Cartan fine topology a recent result by two of the authors is extended as follows: The Riesz charge of the lower envelope of a family of 3 or more -subharmonic functions (no longer supposed continuous) in the plane equals the infimum of the charges of the lower envelopes of all pairs of functions from the family. As a key to this it is shown in two different ways that the (fine) harmonic measures of any 3 pairwise disjoint finely open planar sets have Borel supports with empty intersection. One proof of this uses the Jordan curve theorem and the fact that the set of inaccessible points of the fine boundary of a fine domain is Borel and has zero harmonic measure; the other involves Carleman-Tsuji type estimates together with a fine topology version of a recent result of P. Jones and T. Wolff on harmonic measure and Hausdorff dimension.     

Potential theory of Monge-Ampère on a Banach space. Minimum principle and poisson property

We prove the minimum principle and the Poisson property for the potential theory of the homogeneous Monge-Ampère equation on a reflexive Banach space.     

Effilement minimal en une singularité isolée de l'équation de Schrödinger et application au principe de Picard

In this paper, the generalized Schrödinger equation (–)u=0 on the punctured unit disk of ² is investigated. If is rotation free and satisfies the Picard principle at the origin, it is shown that if a setE is minimal thin relatively to an extremal harmonic functionh with zero boundary values at {|x|=1}, there exists a sequence (r _n ) converging to zero such that B(O,r _n ) C E. Lete be the -unit. It is proved that if a measure satisfies _\E e h d<, for a minimal thin, relatively toh , setE then the Picard principle is valid for the measure + .