The book The E. M. Stein Lectures on Hardy Spaces is based on a graduate course on real variable Hardy spaces which was given by E.M. Stein at Princeton University in the academic year 1973-1974. Stein, along with C. Fefferman and G. Weiss, pioneered this subject area, removing the theory of Hardy spaces from its traditional dependence on complex variables, and to reveal its real-variable underpinnings.
This book is based on Steven G. Krantz’s notes from the course given by Stein. The text builds on Fefferman's theorem that BMO is the dual of the Hardy space. Using maximal functions, singular integrals, and related ideas, Stein offers many new characterizations of the Hardy spaces. The result is a rich tapestry of ideas that develops the theory of singular integrals to a new level. The final chapter describes the major developments since 1974.
This monograph is of broad interest to graduate students and researchers in mathematical analysis. Prerequisites for the book include a solid understanding of real variable theory and complex variable theory. A basic knowledge of functional analysis would also be useful.
Author(s): Steven G. Krantz
Series: Lecture Notes in Mathematics, 2326
Publisher: Springer
Year: 2023
Language: English
Pages: 256
City: Cham
Preface
Contents
1 Introductory Material
1.1 Various Maximal Functions
1.2 Nontangential Convergence
1.3 Unrestricted Convergence
1.4 The Area Integral
1.5 Generalizations of R+ and Hp(R+)
1.6 Relationships Among Domains
2 More on Hardy Spaces
2.1 Hardy Spaces and Maximal Functions
2.2 More Maximal Functions
2.3 Real Variable Hp
2.4 Some Thoughts on Summability
3 Background on Hp Spaces
3.1 Where Did Hp Spaces Get Started?
3.2 Hardy Spaces in C1
3.3 The Hardy–Littlewood Maximal Function
3.4 The Poisson Kernel and Fourier Inversion
4 Hardy Spaces on D
4.1 The Role of the Hilbert Transform
4.2 Blaschke Products
4.3 Passage from D to R2+
5 Hardy Spaces on Rn
5.1 The Poisson Kernel on the Ball
5.2 The Poisson Kernel on the Upper Halfspace
5.3 Cauchy–Riemann Systems
5.4 A Characterization of Hp, p > 1
5.5 The Area Integral
5.6 Applications of the Maximal Function Characterization
5.7 H1(Rn) and Duality with Respect to BMO
6 Developments Since 1974
6.1 The Atomic Theory
6.2 The Local Theory of Hardy Spaces
6.3 The Work of Chang/Krantz/Stein on Hardy Spaces for Elliptic Boundary Value Problems
6.4 Multi-Parameter Harmonic Analysis
6.5 The T1 Theorem of David/Journé
6.6 Contributions of Tom Wolff
6.7 Wavelets
6.7.1 Localization in the Time and Space Variables
6.7.2 Building a Custom Fourier Analysis
6.7.3 The Haar Basis
Axioms for a Multi-Resolution Analysis (MRA)
7 Concluding Remarks
References
Index
