**CBMS Regional Conference Series in Mathematics**

Volume: 70;
1989;
49 pp;
Softcover

MSC: Primary 60;

Print ISBN: 978-0-8218-0721-7

Product Code: CBMS/70

List Price: $24.00

Individual Price: $19.20

**Electronic ISBN: 978-1-4704-2430-5
Product Code: CBMS/70.E**

List Price: $24.00

Individual Price: $19.20

# Some Topics in Probability and Analysis

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*Richard F. Gundy*

A co-publication of the AMS and CBMS

This book is based on lectures presented by the author at DePaul University in July 1986. The lectures cover three main topics. The first is local time theory for Brownian motion and some geometrical inequalities for harmonic functions in the upper half-plane \(R^{n+1}_+\). The author sketches a proof of the inequalities obtained by Barlow and Yor for the maximal local time functional. The second topic concerns a probabilistic treatment of Riesz transforms in \(R^{n+1}_+\), and semimartingale inequalities. The author proves semimartingale inequalities of the type usually obtained for martingales. The final topic centers on a discussion of the Ornstein-Uhlenbeck semigroup and P. A. Meyer's extension of the Riesz inequalities for the infinite-dimensional version of this semigroup. One of the major results of the book is the establishment of inequalities for the density of the area integral.

#### Table of Contents

# Table of Contents

## Some Topics in Probability and Analysis

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents v6 free
- Introduction 18 free
- The Barlow-Yor Inequalities 29 free
- The Density of the Area Integral 613
- Norm Inequalities for D 916
- Local Estimates 1118
- Terminal Expectations and Singular Integrals 1825
- The Ornstein-Uhlenbeck Semigroup 3138
- P. A. Meyer's Riesz Transform Inequalities 3744
- The Ornstein-Uhlenbeck Process for the Rademacher Functions 4451
- References 4754
- Back Cover Back Cover157