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Let $S$ be a Riemann surface. Let $ℍ^{3}$ be the $3$ -dimensional hyperbolic space and let $\partial_{\infty} ℍ^{3}$ be its ideal boundary. In our context, a Plateau problem is a locally holomorphic mapping $ϕ : S \to \partial_{\infty} ℍ^{3} = \hat{ℂ}$ . If $i : S \to ℍ^{3}$ is a convex immersion, and if $N$ is its exterior normal vector field, we define the Gauss lifting, $\hat{ı}$ , of $i$ by $\hat{ı} = N$ . Let $\vec{n} : U ℍ^{3} \to \partial_{\infty} ℍ^{3}$ be the Gauss-Minkowski mapping. A solution to the Plateau problem $(S, ϕ)$ is a convex immersion $i$ of constant Gaussian curvature equal to $k \in (0, 1)$ such that the Gauss lifting $(S, \hat{ı})$ is complete and $\vec{n} \circ \hat{ı} = ϕ$ . In this paper, we show...

Pointwise and locally uniform convergence of holomorphic and harmonic functions.

Šťepničková, Libuše (1999)

Commentationes Mathematicae Universitatis Carolinae

Pointwise and locally uniform convergence of holomorphic and harmonic functions

Libuše Štěpničková (1999)

Commentationes Mathematicae Universitatis Carolinae

We shall characterize the sets of locally uniform convergence of pointwise convergent sequences. Results obtained for sequences of holomorphic functions by Hartogs and Rosenthal in 1928 will be generalized for many other sheaves of functions. In particular, our Hartogs-Rosenthal type theorem holds for the sheaf of solutions to the second order elliptic PDE's as well as it has applications to the theory of harmonic spaces.

Pointwise inequalities of logarithmic type in Hardy-Hölder spaces

Slim Chaabane, Imed Feki (2014)

Czechoslovak Mathematical Journal

We prove some optimal logarithmic estimates in the Hardy space $H^{\infty} (G)$ with Hölder regularity, where $G$ is the open unit disk or an annular domain of $ℂ$ . These estimates extend the results established by S. Chaabane and I. Feki in the Hardy-Sobolev space $H^{k, \infty}$ of the unit disk and those of I. Feki in the case of an annular domain. The proofs are based on a variant of Hardy-Landau-Littlewood inequality for Hölder functions. As an application of these estimates, we study the stability of both the Cauchy problem...

Polarization, conformal invariants and Brownian motion.

Betsakos, Dimitrios (1998)

Annales Academiae Scientiarum Fennicae. Mathematica

Polyanalytic Besov spaces and approximation by dilatations

Ali Abkar (2024)

Czechoslovak Mathematical Journal

Using partial derivatives $\partial f / \partial z$ and $\partial f / \partial \bar{z}$ , we introduce Besov spaces of polyanalytic functions in the open unit disk, as well as in the upper half-plane. We then prove that the dilatations of functions in certain weighted polyanalytic Besov spaces converge to the same functions in norm. When restricted to the open unit disk, we prove that each polyanalytic function of degree $q$ can be approximated in norm by polyanalytic polynomials of degree at most $q$ .

Polya's theorem for non-entire functions

Yoshino, Kunio (1984)

Proceedings of the 11th Winter School on Abstract Analysis

Polygone de Newton et théorème de préparation

Michel Lazard (1972/1973)

Séminaire Dubreil. Algèbre et théorie des nombres

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