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

Yasuo Iida

doi:10.1155/2017/9134768

Abstract

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.

Highlights

Let U and T denote the unit disk and the unit circle in C, respectively
We denote the Smirnov class by N∗(U), which consists of all holomorphic functions f on U such that log(1 + |f(z)|) ≤ Q[φ](z) (z ∈ U) for some φ ∈ L1(T), φ ≥ 0, where the right side denotes the Poisson integral of φ on U
In [13], the class Np(D) was introduced, analogous to Np(U); that is, we denote by Np(D) (p > 1) the set of all holomorphic functions f on D such that sup ∫ (log (1 + 󵄨󵄨󵄨󵄨f (x + iy)󵄨󵄨󵄨󵄨))p dx < +∞

Summary

Introduction

Let U and T denote the unit disk and the unit circle in C, respectively. The Privalov class Np(U), 1 < p < ∞, is defined as the set of all holomorphic functions f on U, satisfying sup ∫ (log (1 + 󵄨󵄨󵄨󵄨f (rζ)󵄨󵄨󵄨󵄨))p dσ (ζ) < +∞, (1)0

Results

Conclusion