Stephen J. Gardiner

Sets of determination for harmonic functions

Let h denote a positive harmonic function on the open unit ball B of Euclidean space R n (n ≥ 2). This paper characterizes those subsets E of B for which sup E H/h = sup B H/h or inf E H/h = inf B H/h for all harmonic functions H belonging to a specified class. In this regard we consider the classes of positive harmonic functions, differences of positive harmonic functions, and harmonic functions with a one-sided quasi-boundedness condition. We also consider the closely related question of representing functions on the sphere ∂B as sums of Poisson kernels corresponding to points in E

Conditions for separately subharmonic functions to be subharmonic

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

Convexity and the Exterior Inverse Problem of Potential Theory

Let Ω 1 and Ω 2 be bounded solid domains such that their associated volume potentials agree outside Ω 1 U Ω 2 . Under the assumption that one of the domains is convex, it is deduced that Ω 1 =Ω 2 .

The Lusin-Privalov theorem for subharmonic functions

This paper establishes a generalization of the Lusin-Privalov radial uniqueness theorem which applies to subharmonic functions in all dimensions. In particular, it answers a question of Rippon by showing that no subharmonic function on the upper half-space can have normal limit -oo at every boundary point.

Universal Taylor series, conformal mappings and boundary behaviour@@@Séries de Taylor universelles, transformations conformes et comportement à la frontière

A holomorphic function f on a simply connected domain Ω is said to possess a universal Taylor series about a point in Ω if the partial sums of that series approximate arbitrary polynomials on arbitrary compacta K outside Ω (provided only that K has connected complement). This paper shows that this property is not conformally invariant, and, in the case where Ω is the unit disc, that such functions have extreme angular boundary behaviour.

SMOOTH POTENTIALS WITH PRESCRIBED BOUNDARY BEHAVIOUR

This paper examines when it is possible to find a smooth potential on a C1 domain D with prescribed normal derivatives at the boundary. It is shown that this is always possible when D is a Liapunov-Dini domain, and this restriction on D is essential. An application concerning C1 superharmonic extension is given.

Global approximation in harmonic spaces

This paper characterizes, in terms of thinness, compact sets K in a suitable harmonic space Q which have the following property: functions which are harmonic (resp. continuous and superharmonic) on a neighbourhood of K can be uniformly approximated on K by functions which are harmonic (resp. continuous and superharmonic) on Q. The corresponding problems of approximating functions which are continuous on K and harmonic (resp. su0 perharmonic) on the interior K are also solved.

Boundary behaviour of Dirichlet series with applications to universal series

This paper establishes connections between the boundary behaviour of functions representable as absolutely convergent Dirichlet series in a half-plane and the convergence properties of partial sums of the Dirichlet series on the boundary. This yields insights into the boundary behaviour of Dirichlet series and Taylor series which have universal approximation properties.

Growth properties of superharmonic functions along rays

This paper gives a precise topological description of the set of rays along which a superharmonic function on RI may grow quickly. The corollary that arbitrary growth cannot occur along all rays answers a question posed by Armitage.

Uniform harmonic approximation with continuous extension to the boundary

Let Ω be an open set in ℝn andE be a relatively closed subset of Ω. Further, letCe(E) be the collection of real-valued continuous functions onE which extend continuously to the closure ofE in ℝn. We characterize those pairs (Ω,E) which have the following property: every function inCe(E) which is harmonic onE0 can be uniformly approximated onE by functions which are harmonic on Ω and whose restrictions toE belong toCe(E).