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Representing measures for the disc algebra and for the ball algebra

Raymond Brummelhuis, Jan Wiegerinck (1991)

Annales Polonici Mathematici

We consider the set of representing measures at 0 for the disc and the ball algebra. The structure of the extreme elements of these sets is investigated. We give particular attention to representing measures for the 2-ball algebra which arise by lifting representing measures for the disc algebra.

Some Hölder-logarithmic estimates on Hardy-Sobolev spaces

Imed Feki, Ameni Massoudi (2024)

Czechoslovak Mathematical Journal

We prove some optimal estimates of Hölder-logarithmic type in the Hardy-Sobolev spaces H k , p ( G ) , where k * , 1 p and G is either the open unit disk 𝔻 or the annular domain G s , 0 < s < 1 of the complex space . More precisely, we study the behavior on the interior of G of any function f belonging to the unit ball of the Hardy-Sobolev spaces H k , p ( G ) from its behavior on any open connected subset I of the boundary G of G with respect to the L 1 -norm. Our results can be viewed as an improvement and generalization of those established...

The angular distribution of mass by Bergman functions.

Donald E. Marshall, Wayne Smith (1999)

Revista Matemática Iberoamericana

Let D = {z: |z| &lt; 1} be the unit disk in the complex plane and denote by dA two-dimensional Lebesgue measure on D. For ε &gt; 0 we define Σε = {z: |arg z| &lt; ε}. We prove that for every ε &gt; 0 there exists a δ &gt; 0 such that if f is analytic, univalent and area-integrable on D, and f(0) = 0 thenThis problem arose in connection with a characterization by Hamilton, Reich and Strebel of extremal dilatation for quasiconformal homeomorphisms of D.

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