RETRACTED ARTICLE: A new application of boundary integral behaviors of harmonic functions to the least harmonic majorant

Our main aim in this paper is to obtain a new type of boundary integral behaviors of harmonic functions in a smooth cone. As an application, the least harmonic majorant of a nonnegative subharmonic function is also given.

Theorems of the Phragmén–Lindelöf Type for Subfunctions in a Cone

Our first aim in this paper is to deal with the maximum principle for subfunctions in an arbitrary unbounded domain. As an application, we next give a result concerning the classical Phragmén–Lindelöf theorem for subfunctions in a cone. For a subfunction defined in a cone that is dominated on the boundary by a certain function, we finally generalize the Phragmén–Lindelöf type theorem by making a generalized harmonic majorant of it.

A THEOREM OF PHRAGMÉN-LINDELÖF TYPE FOR SUBFUNCTIONS IN A CONE

For a subfunction u, associated with the stationary Schrödinger operator, which is dominated on the boundary by a certain function on a cone, we generalise the classical Phragmén-Lindelöf theorem by making an a-harmonic majorant of u.

Some types of regularity for the Dirichlet problem

The question of whether the existence of a harmonic majorant in a relative neighbourhood of each point of a boundary of a domain D implies the existence of a harmonic majorant in the whole of D has received great attention in recent years and has been dealt with by several authors in different settings. The most general results to date have been achieved in [10] with the Martin boundary. In [9], the author arrives, by independent means, at the conclusions of [10] in the particular case where D is a Lipschitz domain.In this paper, we answer the question in domains with suitably regular topological frontiers. Our methods rely heavily on the possibility of obtaining an extented-representation for nonnegative superharmonic functions defined near a frontier point. This naturally led to the introduction and the study of new types of regularity for the generalised Dirichlet problem. As well as their suitability in dealing with the question of harmonic majorisation, they present an intrinsic importance as natural extensions of the (classical) regularity. For simplicity reasons, we will treat the finite boundary points and the point at infinity separately.

A note on n-harmonic majorants

Suppose D (υ) is the Dirichlet integral of a function υ defined on the unit disc U in the complex plane. It is well known that if υ is a harmonic function in U with D (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has a harmonic majorant in U.We define the “iterated” Dirichlet integral Dn (υ) for a function υ on the polydisc Un of Cn and prove the polydisc version of the well known fact above:If υ is an n-harmonic function in Un with Dn (υ) < ∞, then for each p, 0 < p < ∞, |υ|p has an n-harmonic majorant in Un.

On a Class of Analytic Functions of Smirnov

The class S of functions under study in this paper was introduced by V. I. Smirnov in 1932. This class was subsequently investigated by various authors, a pertinent paper to the present wrork being that of Tumarkin and Havinson [2], who showed that a plane compact set of logarithmic capacity zero is 5-removable. Another important development, due to Yamashita [3], wras that the class 5 could be characterized as those analytic functions ƒ for which log+ |ƒ| has a quasi-bounded harmonic majorant.In what follows, we discuss the Smirnov class in the context of planar surfaces, exploiting some ideas in the work of Hejhal [1] to establish that a closed, bounded, totally disconnected set is S-removable if and only if its complement belongs to the null class Os.