Seminar on Stochastic Processes, 1988
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Series Editors Loren Pitt Thomas Liggett Charles Newman
Seminar on Stochastic Processes, 1988
E. 0, the function Ihl q is sub-harmonic in the disk, which impies that the process Zt = Ihlq(BtAT) is a submartingale. Hence, choosing any q in (0,1), we may use DOOB'S inequality to assert that
But IZtll/q majorizes IMtl, where M t is the martingale associated to f, while IZTII/q = [f(BT)2 + g(BT)2P/2. It remains to remark that BT is uniformely distributed on the circle to see that E[IZTll/q] is dominated by IlfllHl.
5
On the real line, one can rephrase all the results stated on the circle. There, the HILBERT transform is given by the singular kernel l/7r(x - y), which means that one has
'Hf(x) = lim [ fey) dy; ~-o Jlx-YI>~ 7r( x - y) By FOURRIER transform, one gets 'Hf(x) = -i(sgx)i(x) where f denotes the FOURRIER transform of a function f. Once again, the HILBERT transform is related to harmonic prolongations, this time to the upper half-plane: 'Hf is the boundary value of the harmonic conjuguate of the harmonic prolongation of f. It is still equal to
!
o( ~:: )-1/2 ; it shares the same properties that the HILBERT trans-
form on the circle, the main difference being that, since the measure of the line is infinite, one does not need to restrict himself to the functions with mean O. For the analoguous of the results in L1, we have to replace the Brownian motion starting from 0 in the disk by a Brownian motion starting from a point (t, Xo) in the upper half-plane, with t> 0 and Xo uniformely distributed, and we stop it when it hits the real axis. Then we let t tends to infinity: we get the "white noise of the universe" of D.GUNDY.
b)- The
RIESZ
transforms on
nn
The classical RrESZ transforms on nn are the analoguous of the HILBERT transform on the real line or on the circle ; there are n of them, defined by
where, as before, the square root of .6. has to be understood in the sense of L2(nn). In terms of FOURRIER transforms, one has
They also are defined by mean of singular integral kernels, namely Ri .
.
is the convolution with the function n~(x )/Ixl n , where n~(x) = with a normalizing constant
Cn
xi
Cn
\xl'
whose value is unimportant for us.
6
This is the basic example of CALDERON-ZYGMUND operator, and in fact the RIESZ transforms are the fundamental bricks with which one builts all the singular integral operators in nn. (cf [8], for example.) The interpretation of the RIESZ transforms in terms of holomorphic functions is still valid, once we have defined what will play that role. In an open set of n, this notion is replaced by a system (ft,ยทยทยท, fn) of Coo functions satisfying the CAUCHy-RIEMANN equations :
n
Ofi = of~ ox) ox' { ~ of~ = 0 ~ ox' i
(1)
Now, let us consider the POISSON kernel on nn, which associates to each bounded function f on nn its harmonic extension j on nn x n+ : we have
f(x)= A
.
J
f(y)p(x,y)dy
at a pomt x that
= (xo, t)
Jp(x, y) dy =
where
cnt p(x'Y)=(t 2 +l x o_YI2)(n+l)/2
E nn x n+. (Here,
Cn
n +1 .!!.:U = r(-2-)/7r ,such 2
1.)
r
The connection betw
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