Tensor Gaussian measures on Lp( E)

Abstract

Let ( i, H, E) and ( j, K, F) be abstract Wiener spaces and let α be a reasonable norm on E ⊗ F. We are interested in the following problem: is ( i ⊗ j, H \\ ̂ bo 2 K, E \\ ̂ bo αF ) an abstract Wiener space ? The first thing we do is to prove that the setting of the problem is meaningfull: namely, i ⊗ j is always a continuous one to one map from H \\ ̂ bo 2 K into E \\ ̂ bo αF . Then we exhibit an example which shows that the answer cannot be positive in full generality. Finally we prove that if F= L p (X, X ,λ) for some σ-finite measure λ ⩾ 0 then (i⊗j, H ⊗ 2K,L p (X, X ,λ) is an abstract Wiener space. By-products are some new results on γ-radonifying operators, and new examples of Banach spaces and cross norms for which the answer is affirmative (in particular α = π the projective norm, and F= L 1(X, X ,λ)).