Group actions and envelopes of holomorphy

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ENVELOPES OF HOLOMORPHY OF HARTOGS AND CIRCULAR DOMAINS

Conditions are given for the envelope of holomorphy of a Hartogs or circular domain in C(n) to be univalent, together with its explicit construction. The noneliminability of the assumptions is shown by counterexamples

Construction of envelopes of holomorphy for some classes of special domains

The envelope of holomorphy E(OMEGA) of any domain OMEGA in C(n) which is either of circular type or a tube, is constructed in terms of the envelope of a lower dimensional complex space. As consequences, conditions for the univalence of E(OMEGA) are proved

Twist points of planar domains

We establish a potential theoretic approach to the study of twist points in the boundary of simply connected planar domains

Converse mean value theorems on trees and symmetric spaces

Harmonic functions satisfy the mean value property with respect to all integrable radial weights if f is harmonic then hf f h for any such weight h But need a function f that satises this relation with a given nonnegative h b e harmonic By a classical result of Furstenb erg the answer is p ositive for every b ounded f on a Riemannian symmetric space but if the boundedness condition is relaxed then the answer turns out to depend on the weight h In this paper various types of weights are investigated on Euclidean and hyp erb olic spaces as well as on homogeneous and semihomogeneous trees IRf h decays faster than exponentially then the mean value property hf f h does not imply harmonicity of f For weights than exponentially at least a weak converse mean value the eigenfunctions of the Laplace operator which satisfy harmonic The critical case is that of exp onential decay exhibit weights that characterize harmonicity and others that do no

Integral Geometry on Trees

In the hyperbolic disc (or, more generally, in real hyperbolic spaces) we consider the horocyclic Radon transform R and the geodesic Radon transform X. Composition with their respective adjoint operators yields two convolution operators on the disc (with respect to the hyperbolic measure). We describe their convolution kernels in comparison with those of the corresponding operators on a homogeneous tree

Sustainable Development Index in Hong Kong: Approach, Method and Findings

Economic growth, Environmental protection, Societal development, Sustainable development,