Integration on then th power of a hyperbolic space in terms of invariants under diagonal action of isometries (Lorentz transformations)

The integral of a function over then'th power of hyperbolicd-dimensional spaceH is decomposed into integration along each orbit under diagonal action onH ⁿ of the isometry groupG onH, followed by integration over the orbit space, parametrized in terms of a complete set of invariants. The Jacobian entering in this last integral is expressed explicitly in terms of certain determinants. When viewingH as a half-hyperboloid in _d+1 ,G is induced by the homogeneous Lorentz groupO (1,d) acting on _d+1 .     

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.     

Bonnesen's inequality for the isoperimetric deficiency of closed curves in the plane

An inequality of T. Bonnesen for the isoperimetric deficiency of a convex closed curve in the plane is extended to arbitrary simple closed curves. As a primary tool it is shown that, for any such curve, there exist two concentric circles such that the curve is between these and passes at least four times between them. Next, it is shown that Bonnesen's inequality holds for any such pair of concentric circles.This work was supported by The Danish Natural Science Council and carried out during the author's stay 1989–90 as a Visiting Member at the Institute for Advanced Study, Princeton, N.J., U.S.A.     

Martin Boundary of a Fine Domain and a Fatou-Naïm-Doob Theorem for Finely Superharmonic Functions

Let Γ be a graph with the doubling property for the volume of balls and P a reversible random walk on Γ. We introduce H¹ Hardy spaces of functions and 1-forms adapted to P and prove various characterizations of these spaces. We also characterize the dual space of H¹ as a BMO-type space adapted to P. As an application, we establish H¹ and H¹- L¹ boundedness of the Riesz transform.     

Lower estimate of the isoperimetric deficit of convex domains inR n in terms of asymmetry

For convex bodiesD inR ⁿ it is shown that the isoperimetric deficit ofD is minorized by a constant times the square of thebarycentric asymmetry (D) ofD. Here (D) is defined as the volume ofDB _D divided by the volume ofD, whereB _D denotes the ball centred at the barycentre ofD and having the same volume asD.Dedicated to the memory of Børge Jessen     

Polyhedra as domains of harmonic maps

Certain more or less known properties of polyhedra in combinatorial topology are established. In particular, the concept of normal pseudomanifolds is extended to so-called admissible polyhedra, whereby branching may occur. Admissible Riemannian polyhedra serve as domains of (generalized) harmonic functions or maps.