Weighted Littlewood-Paley Theory and Exponential-Square Integrability

Michael Wilson (auth.)3540745823, 978-3-540-74582-2

Littlewood-Paley theory is an essential tool of Fourier analysis, with applications and connections to PDEs, signal processing, and probability. It extends some of the benefits of orthogonality to situations where orthogonality doesn’t really make sense. It does so by letting us control certain oscillatory infinite series of functions in terms of infinite series of non-negative functions. Beginning in the 1980s, it was discovered that this control could be made much sharper than was previously suspected. The present book tries to give a gentle, well-motivated introduction to those discoveries, the methods behind them, their consequences, and some of their applications.

Table of contents :
Front Matter….Pages I-XII
Some Assumptions….Pages 1-7
An Elementary Introduction….Pages 9-37
Exponential Square….Pages 39-68
Many Dimensions; Smoothing….Pages 69-84
The Calderón Reproducing Formula I….Pages 85-100
The Calderón Reproducing Formula II….Pages 101-127
The Calderón Reproducing Formula III….Pages 129-143
Schrödinger Operators….Pages 145-150
Some Singular Integrals….Pages 151-160
Orlicz Spaces….Pages 161-188
Goodbye to Good-λ….Pages 189-195
A Fourier Multiplier Theorem….Pages 197-202
Vector-Valued Inequalities….Pages 203-212
Random Pointwise Errors….Pages 213-218
Back Matter….Pages 219-228