Mobius Invariant QK Spaces

Description

This monograph summarizes the recent major achievements in Möbius invariant QK spaces. First introduced by Hasi Wulan and his collaborators, the theory of QK spaces has developed immensely in the last two decades, and the topics covered in this book will be helpful to graduate students and new researchers interested in the field. Featuring a wide range of subjects, including an overview of QK spaces, QK-Teichmüller spaces, K-Carleson measures and analysis of weight functions, this book serves as an important resource for analysts interested in this area of complex analysis. Notes, numerous exercises, and a comprehensive up-to-date bibliography provide an accessible entry to anyone with a standard graduate background in real and complex analysis. Preface Contents 1 Möbius Invariant Function Spaces 1.1 The Möbius Group 1.2 Möbius Invariant Function Spaces 1.3 General Properties 1.4 Uniqueness of the Dirichlet Space 1.5 The Invariant Dual 1.6 Notes 1.7 Exercises 2 QK Spaces 2.1 Preliminaries 2.2 The Definition of QK 2.3 The Subspace QK,0 2.4 Comparing Two QK Spaces 2.5 Examples of Functions in QK 2.6 Notes 2.7 Exercises 3 Analysis of Weight Functions 3.1 Standing Assumptions 3.2 Modifications of the Weight Function 3.3 Integral Estimates 3.4 Notes 3.5 Exercises 4 K-Carleson Measures 4.1 K-Carleson Measures 4.2 QK Spaces via Carleson Measures 4.3 Inner Functions in QK 4.4 Outer Functions in QK 4.5 Boundary QK Spaces 4.6 Notes 4.7 Exercises 5 QK Spaces via Other Derivatives 5.1 Raising the Order of Derivative 5.2 Lowering the Order of Derivative 5.3 Another Approach to Higher Order Derivatives 5.4 QK Spaces via Fractional Order Derivatives 5.5 Derivative-Free Characterizations 5.6 Notes 5.7 Exercises 6 Taylor Series and QK Spaces 6.1 Lacunary Series in QK 6.2 Mean Lipschitz Spaces and QK 6.3 Hadamard Products 6.4 Notes 6.5 Exercises 7 Decompositions and Interpolation 7.1 Preliminaries 7.2 Atomic Decomposition for QK 7.3 Atomic Decomposition for QK,0 7.4 A Fefferman-Stein Decomposition 7.5 Interpolation in QK Spaces 7.6 Notes 7.7 Exercises 8 QK-Teichmüller Spaces 8.1 Carleson Measures on CC DD 8.2 QK-Teichmüller Spaces 8.3 The Schwarzian Derivative and QK Spaces 8.4 Geometric Descriptions of QK,0 Spaces 8.5 TK and QK 8.6 Notes 8.7 Exercises 9 Additional Topics 9.1 Preduals of QK Spaces 9.2 Morrey K-Spaces 9.3 The Corona Problem for QK 9.4 Distances from Bloch Functions to QK Spaces 9.5 Korenblum's Inequality for QK 9.6 Notes 9.7 Exercises References Index