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Start Over Subject Riesz measure

1. A Nevanlinna theorem for superharmonic functions on Dirichlet regular Greenian sets

Creator:
Watson, Neil A.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Nevanlinna theorem, superharmonic function, δ-subharmonic function, Riesz measure, and mean value
Language:
English
Description:
A generalization of Nevanlinna’s First Fundamental Theorem to superharmonic functions on Green balls is proved. This enables us to generalize many other theorems, on the behaviour of mean values of superharmonic functions over Green spheres, on the Hausdorff measures of certain sets, on the Riesz measures of superharmonic functions, and on differences of positive superharmonic functions.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. Inequalities involving heat potentials and Green functions

Creator:
Watson, Neil A.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
heat potential, supertemperature, Green function, and Riesz measure
Language:
English
Description:
We take some well-known inequalities for Green functions relative to Laplace’s equation, and prove not only analogues of them relative to the heat equation, but generalizations of those analogues to the heat potentials of nonnegative measures on an arbitrary open set E whose supports are compact polar subsets of E. We then use the special case where the measure associated to the potential has point support, in the following situation. Given a nonnegative supertemperature on an open set E, we prove a formula for the associated Riesz measure of any point of E in terms of a limit inferior of the quotient of the supertemperature and the Green function for E with a pole at that point.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. Mean values and associated measures of δ-subharmonic functions

Creator:
Watson, Neil A.
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
superharmonic, δ-subharmonic, Riesz measure, and spherical mean values
Language:
English
Description:
Let u be a δ-subharmonic function with associated measure µ, and let v be a superharmonic function with associated measure ν, 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 → 0 with 0 <s<t of the quotient (M(u, x,s)−M(u, x,t))/(M(v,x,s)−M(v,x,t)), lie between the upper and lower limits as r → 0+ of the quotient µ(B(x,r))/ν(B(x,r)). This enables us to use some well-known measure-theoretic results to prove new variants and generalizations of several theorems about δ-subharmonic functions.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public