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+
. (2 ) A probabilistic treatmen t o f Ries z
transforms i n R+
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  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+
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  . Th e metho d wa s elaborated furthe r b y the autho r wit h Silverstei n
. 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  . 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 , 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 .