Some characterizations of 𝐶(ℳ)

Pacific Journal of Mathematics, 1987

It is known that for 0 < p < oo the Hardy space H p contains a residual set of functions, each of which has range equal to the whole plane at every boundary point of the unit disk. With quite new general techniques, we are able to show that this result holds for numerous other spaces. The space BMOA of analytic functions of bounded mean oscillation, the Bloch spaces, the Nevanlinna space and the Dirichlet spaces D a f or 0 < a < 1/2 are examples. Our methods involve hyperbolic geometry, cluster set analysis and the "depth" function which we have used previously for determining geometric properties of the image surfaces of functions. Denote by D(a, r) the open disc in C centered at a and of radius r. Denote by D the unit disc D(0,l) and let Δ(a,r) = D Π D{a,r) for a e 3D. Brown and Hansen [4] proved that each Hardy space H p 9

Proceedings of the American Mathematical Society, 1983

A method is presented for characterizing Carleson-type measures relative to Bergman spaces. This method applies to the standard weighted and unweighted Bergman spaces on the unit ball in C" to yield simple proofs of the known results. It also extends these results to domains more general than balls Let D be a domain in C and A a space of analytic functions with a norm || /1| defined by some integral or integrals of \f\f being bounded. For example if D is the unit disk in C', A could be an Hp space or a Bergman space. If p is a positive measure on D, a common terminology is to call p an A-Carleson measure if there is a constant C > 0 such that (/j/l'^j '<CH/II. This note will deal only with Bergman spaces Ap(w, D) defined for p > 0 and nonnegative w by Ap(w, D) = {/: / is analytic in D and jD \f\pwdm < +00}. The integral is with respect to Lebesgue 2 «-dimensional volume measure (or area if n = 1). Associate with each z E D an open set E(z) containing 2 with the following properties: (1) E(z) E D and XeoÍÜ) 's measurable in D X D. (2) There is a constant C, > 0 such that I U {E(z): E(z) n E(a) ¥> 0)\<Cx\E(a)\. (3) There is a constant C2 > 0 such that for all a E D, w(z,) < C2w(z2) when z,, z2 £ E(a). (| • I denotes the Lebesgue measure of a set.) Example. Let D be the unit disk in C and let r > 0. Let E(z) be the hyperbolic disk about z with hyperbolic radius r. If w(z) = (1-| z |)a for a >-1 then (l)-(3) are satisfied. Let E2(y) = U [E(z): E(y) D E(z) # 0 }. Lemma 1. Suppose there is a constant C > 0 such that \K2W<7rhr\S i^w< fEAp(w,D), IM*) I JE(z) and suppose ft(E2(z)) < CjEi,z)wdm, z ED. Then fi is an Ap(w)-Carleson measure.

Pacific Journal of Mathematics, 1990

This paper studies the extensions of harmonic and analytic functions defined on the unit disk to continuous functions defined on a certain compactification of the disk.

Journal für die reine und angewandte Mathematik (Crelles Journal), 1969

Indiana University Mathematics Journal, 1998

We present an example of a complex manifold X-in fact, a pseudoconvex open set in C 2-such that X is not Kobayashi-hyperbolic, but any holomorphic map from the punctured unit disk to X extends to a map from the whole unit disk to X. 0. Introduction. Let X be a complex manifold. We say that X has the D *-extension property (D *-EP) iff for any holomorphic map f from D * = {z ∈ C : 0 < |z| < 1} to X, there exists a mapf ∈ Hol(D,X) (where D = {z ∈ C : |z| < 1}) such thatf | D * = f. Kwack [4] and Thai [8] proved that in the case where X is compact, then the D *-EP is equivalent to hyperbolicity in the sense of Kobayashi, as well as in the sense of Brody (see precise definitions below). If X is not assumed to be compact, X = D * , f (z) = z provides an immediate example of a hyperbolic (indeed, complete hyperbolic) manifold which does not have the D *-EP. But if X has the D *-EP, then it must be Brody-hyperbolic [8]. The purpose of this note is to give an example of a complex manifold X (in fact, an open set in C 2) which has the D *-EP but is not Kobayashihyperbolic. Note that our example is pseudo-convex, and in fact any Riemann domain spread over C n having the D *-EP must be pseudo-convex. Whether an arbitrary complex manifold X having the D *-EP is also pseudoconvex, or verifies the Kontinuitätssatz, is an interesting open problem.

2021

The classical Poincaré’s hyperbolic metric was first introduced in the early nineties. Since then, a family of conformal metrics that are closely related to the hyperbolic metric was introduced by several mathematicians. Just to name a few, they are the Hurwitz metric [13], the Gardiner-Lakic metric [3], the Hahn metric [6], the quasihyperbolic metric [4], and many more. As the hyperbolic metric is valid for only hyperbolic domains of the plane due to its very difficult nature to compute explicitly, these metrics play a vital role to enhance the study of the hyperbolic metric in several purposes. One such important problem is biLipschitz equivalence of the hyperbolic metric with the other conformal metrics. For instance, the Gardiner-Lakic metric in a hyperbolic domain is defined by taking the supremum of the hyperbolic metric in twice punctured plane, punctured at the distinct pair of points in the complement of the domain and the supremum is taken over these pair of points. Note t...

Annales Polonici Mathematici, 2019

We study the notions of extendability and domain of holomorphy in the infinite-dimensional case. In this setting it is also true that the notions of domain of holomorphy and weak domain of holomorphy are equivalent. We also prove that the set of non-extendable functions belonging to some classes X(B) ⊂ H(B), B being the open unit ball in a separable complex Banach space, is a lineable and dense G δ. Moreover, when Ω is H b-holomorphically convex (defined in the text), it is shown that the set of non-extendable holomorphic functions on Ω is a lineable and dense G δ set.

Reports of the National Academy of Sciences of Ukraine, 2019

We study the Hilbert boundary-value problem for analytic functions in the Jordan domains satisfying the quasihyperbolic boundary condition by Gehring-Martio. Assuming that the coefficients of the problem are functions of the countably bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of solutions of the problem in terms of angular limits. As consequences, we derive the corresponding results concerning the Dirichlet, Neumann, and Poincaré boundary-value problems for harmonic functions.

2020

In 2012, R.M. Aron, D. Carando, T.W. Gamelin, S. Lassalle, and M. Maestre presented that the Cluster Value Theorem in the infinite dimensional Banach space setting holds for the Banach algebra $\mathcal{H}^\infty (B_{c_0})$. On the other hand, B.J. Cole and T.W. Gamelin showed in 1986 that $\mathcal{H}^\infty (\ell_2 \cap B_{c_0})$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$ in the sense of an algebra. Motivated by this work, we are interested in a class of open subsets $U$ of a Banach space $X$ for which $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$. We prove that there exist polydisk type domains $U$ of any infinite dimensional Banach space $X$ with a Schauder basis such that $\mathcal{H}^\infty (U)$ is isometrically isomorphic to $\mathcal{H}^\infty (B_{c_0})$, which also generalizes the result by Cole and Gamelin. We also show that the Cluster Value Theorem is true for $\mathcal{H}^\infty (U)$. As the dual space $X^*$ is...

2005

One of the main covering results asserts that if a holomorphic function f in the unit disk satisfies |f (0)| ≥ A|f (0)| with A > 4, then f covers an annulus of the form r < |w| < Kr for some r > 0, where K is a certain function of A . Extremals are furnished by universal covering maps onto complements of certin discrete sets. The covering theorems are proved by solving minimum problems for hyperbolic metrics.