Notes on an inequality by pisier for functions on the discrete cube
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Notes on an inequality by pisier for functions on the discrete cube
Notes on an Inequality by Pisier for Functions on the Discrete Cube. R.. Wagner. School of Mathematical Sciences, Tel Av
Notes on an Inequality on the Discrete Cube
b y P i s i e r for F u n c t i o n s
R.. Wagner School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
1
Introduction
We study functions from the discrete cube { - 1 , 1} '~ to a linear normed space B. For such functions f define D i f ( z ) to be the vector 1(=)-l(z~) where z i 2 has the same coordinates as z except at t h e / - t h place. Define
i=1
In [P] Pisier proved that for every 1 < p < oo and every f : { - 1 , 1} '~ -~ B one has
tli(~) - EIIIL,(8) _
Clog'~lllVflp(~)llL,(~)"
In [T] Talagrand showed that for B = ~ the logarithmic factor can be removed, whereas in general it is sharp with dependence on p < c¢. We will close here the final gap in this inequality, by showing that the logarithmic factor can be dropped for p = o0. The argument we employ is an elementary counting argument. We will then turn to revise Pisier's proof of his inequality above. The proof we present uses the same mechanism, but applies it differently.
2
The Main Result
T h e o r e m 1. Let f : { - 1 , 1}" --~ B , where B is a normed linear space. Then
