XIII: 01, 1-3, LNM 721 (1979)
BORELL, Christer
On the integrability of Banach space valued Walsh polynomials (Banach space valued random variables)
The $L^2$ space over the standard Bernoulli measure on $\{-1,1\}^N$ has a well-known orthogonal basis $(e_\alpha)$ indexed by the finite subsets of $N$. The Walsh polynomials of order $d$ with values in a Banach space $E$ are linear combinations $\sum_\alpha c_\alpha e_\alpha$ where $c_\alpha\in E$ and $\alpha$ is a finite subset with $d$ elements. It is shown that on this space (as on the Wiener chaos spaces) all $L^p$ norms are equivalent with precise bounds, for $1<p<\infty$. The proof uses the discrete version of hypercontractivity
Keywords: Walsh polynomials, Hypercontractivity
Nature: Original
Retrieve article from Numdam

---