The Barlow-Yor Inequalities
For //^-theory o f harmonic function s i n
R++1,
two functionals hav e been
studied i n detail : th e nontangentia l maxima l functio n an d th e Lusi n are a
function. Thes e two geometric objects are analogs of two probabilistic func -
tionals associate d wit h continuou s martingales . I f X = {X tJ 0} is a con-
tinuous martingale, then
X* - su p \Xt\ an d S(X) = (X)
in
t
(where (X) is the quadratic variation of X), Th e basic relation between S(X)
and X* is well known now:
\\S{X)\\P*\\X*\\P
for al l 0 p oo. Unti l th e appearance o f Barlow and Yor's paper [1], no
other functional "o f significance" had been found. The y discovered anothe r
functional tha t is , in some sense , in between the maximal functio n an d the
quadratic variation : th e maxima l loca l time . Suppos e fo r simplicit y tha t
X i s Brownian motio n ru n up to a stoppin g tim e r . Th e local tim e o f the
Brownian motio n may be defined a s follows. Fo r each trajectory co consider
the mappin g X(co) : [0,T ]
R1,
an d the "push-forward " ma p X*. Tha t
is, X*(dt) i s the image measure o f Lebesgue measure on [0, T] on
R1,
under
the mappin g X. Th e measure X*(dt) o n R
1
i s absolutely continuous , b y a
result du e to P . Levy , an d it s densit y L(r) = L{r,o),x) i s called th e local
time at r (up to time t fo r the trajectory co). Consider L(r) a s a process in
the real parameter r . O f course, the filtration, based on the space parameter
r, i s completely differen t fro m th e time filtration. Now , take th e maxima l
function L * = sup
r
L(r).
We seek to prove a good-A inequality of the form
(1) P{L* 0X,S(X) SX) s{pfS)P{L* X)
with fi 1, S 1, valid for all k 0. Fro m this we can obtain an inequality
for norm s of the form
W\\pcp\\s(X)\\p,
p 1, by now standard argument s [7] . The converse inequalit y i s obtained
by another good-A inequality with the roles of L* and S(X) interchanged .
2
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