Recently Loeb and Walsh [3] established several results concerning the Royden boundary in the axiomatic setting. This in effect generalizes theorems about H D-f unctions, Dirichlet-finite harmonic functions on surfaces to Riemannian manifolds, which are the most general carriers of these functions. Their results are however restricted to bounded H D-f unctions. Whether Nakai's [4] characterizat...

satisfying zZ\ \^A < °°, if and only if f is) is bounded away from zero in the half-plane 0-3:0. This discovery amounts to a determination of the spectrum of the Banach-algebra element associated with fis), and it thus makes available for the theory of ordinary Dirichlet series the well-known theorem of Gelfand on analytic functions of Banach-algebra elements (see §24D of [7]) and its generaliz...

The paper is devoted to isometric Banach-space-theoretical structure of transportation cost (TC) spaces on finite metric spaces. TC are also known as Arens-Eells, Lipschitz-free, or Wasserstein A new notion a roadmap pertinent problem space has been introduced and used simplify proofs for the results representation quotients $$\ell _1$$ edge set over cycle space. Tolstoi-type theorem roadmaps p...

We simplify proof of the theorem that close to any pseudoholomorphic disk there passes a pseudoholomorphic disk of arbitrary close size with any pre-described sufficiently close direction. We apply these results to the Kobayashi and Hanh pseudodistances. It is shown they coincide in dimensions higher than four. The result is new even in the complex case. We aim here to prove the following state...

If X is a topological space, then we let H(X) denote the group of autohomeomorphisms of X equipped with the compact-open topology. For subsets A and B of X we define [A, B] = {h ∈ H(X) : h(A) ⊂ B}, and we recall that the topology on H(X) is generated by the subbasis SX = {[K , O] : K compact, O open in X}. If X is a compact Hausdorff space, then H(X) is a topological group, that is, composition...

The proofs use the notion of analytic structure in a maximal ideal space. J. Wermer first obtained results along these lines and further contributions were made by E. Bishop and H. Royden and then by G. Stolzenberg [5] who proved STOLZENBERG'S THEOREM. Let XQC be a polynomially convex set. Let KQC be a finite union of Q-curves. Then (XKJK)*—X\JK is a {possibly empty) pure 1-dimensional analytic...

Let e T be a cross product of n Teichmüller spaces of Fuchsian groups, n > 1. From the properties of Kobayashi metric and from the Royden-Gardiner theorem, e T is a complete hyperbolic manifold. Each two distinct points of e T can be joined by a hyperbolic geodesic segment, which is not in general unique. But when e T is finite dimensional or infinite dimensional of a certain kind, then among a...