Introduction

In thes e lecture s I concentrate d o n thre e topics : (1) Loca l tim e theor y

for Brownia n motio n an d som e geometrical inequalitie s fo r harmoni c func -

tions i n th e uppe r half-plan e R+

+1

. (2 ) A probabilistic treatmen t o f Ries z

transforms i n R+

+1

an d semimartingal e inequalities . (3 ) A discussion of the

Ornstein-Uhlenbeck semigrou p and P. A. Meyer's extension of the Riesz in-

equalities for th e infinite-dimensional versio n of this semigroup, introduce d

by Malliavin (se e [26, 27]).

Regarding topic (1), we sketch a proof of the inequalities obtained by Bar-

low and Yor in [1] for the maximal local time functional. Thes e inequalitie s

led the autho r t o som e ne w inequalities fo r a geometric functional , define d

on harmonic functions i n R+

+1,

called the density of the area integral

Topic (2) is a probabilistic approach to the Riesz transform inequalitie s in

R". Thi s method o f proo f wa s first introduced b y the author an d Varopou -

los [24] . Th e metho d wa s elaborated furthe r b y the autho r wit h Silverstei n

[23]. W e sho w that th e sam e idea s ar e effectiv e i n provin g semimartingal e

inequalities of the type usually obtained fo r martingales .

The final topic in the series is a discussion of the Ornstein-Uhlenbeck semi-

group. W e give a proof o f Nelson's hypercontractivity inequality , followin g

the ideas of Neveu [29] . Then, we present a proof o f P. A. Meyer's inequal-

ities, the analogues of the classical Riesz transform inequalitie s in which the

role o f th e Laplacia n A is replace d b y th e Ornstein-Uhlenbec k generator ,

A-X-V. Thi s approach, found i n [21], is a direct extension of the methods

discussed in the previous section (topi c (2)).

The author woul d lik e to express his warmest gratitud e t o the organizer s

of th e Conference a t DePau l University , the participants, an d especially , t o

Roger Jones for hi s enthusiastic hospitality .

I