Bounded Subsets of Smirnov and Privalov Classes on the Upper Half Plane

Some characterizations of boundedness in N⁎(D) and Np(D) (1<p<∞) will be described, where N⁎(D) denote the Smirnov class and Np(D) the Privalov class on the upper half plane D={z∈C∣Im⁡ z>0}, respectively.

Multiplicative Isometries on ClassesMp(X)of Holomorphic Functions

In (Iida and Kasuga 2013), the authors described multiplicative (but not necessarily linear) isometries ofMp(X)ontoMp(X)in the case of positive integerp∈N, whereMp(X) (p≥1)is included in the Smirnov classN∗(X). In this paper, we will generalize the result to arbitrary (not necessarily positive integer) value of the exponents0<p<∞.

Multiplicative Isometries on SomeF-Algebras of Holomorphic Functions

Multiplicative(but not necessarily linear) isometries ofMp(X)ontoMp(X)will be described, whereMp(X)(p≥1) areF-algebras included in the Smirnov classN∗(X).

Multiplicative Isometries onF-Algebras of Holomorphic Functions

We study multiplicative isometries on the followingF-algebras of holomorphic functions: Smirnov classN*(X), Privalov classNp(X), Bergman-Privalov classANαp(X),and ZygmundF-algebraNlogβN(X),whereXis the open unit ball&#x1D539;nor the open unit polydisk&#x1D53B;ninℂn.

The Dirichlet Problem for Harmonic Functions in the Smirnov Class with Variable Exponent

A solution of the Dirichlet problem for harmonic functions from the Smirnov class is obtained in the framework of functional spaces with a nonstandard growth condition. It is found that the domain boundary geometry influences the character of a problem solution. In the case of solvability, solutions are constructed in explicit form.

Rouché type theorems and a theorem of Adamyan, Arov and Krein

We show Rouché type theorems using a theorem of Adamyan, Arov and Krein. As applications, we obtain a certain characterization of self-maps of the unit disc in terms of the location of the Denjoy-Wolf point and we study a function in the Smirnov class whose real part is positive.