Mixed-Norm Inequalities and Operator Space $L_p$ Embedding Theory

Let $f_1, f_2, \ldots, f_n$ be a family of independent copies of a given random variable $f$ in a probability space $(\Omega, \mathcal{F}, \mu)$. Then, the following equivalence of norms holds whenever $1 \le q \le p < \infty$, $$\left( \int_{\Omega} \left[ \sum_{k=1}^n |f_k|^q \right]^{p/q} d \mu \right)^{1/p} \sim \max_{r \in \{p,q\}} \left\{ n^{1/r} \left( \int_\Omega |f|^r d\mu \right)^{1/r} \right\}.$$ The authors prove a noncommutative analogue of this inequality for sums of free random variables over a given von Neumann subalgebra. This formulation leads to new classes of noncommutative function spaces which appear in quantum probability as square functions, conditioned square functions and maximal functions.

Let $f_1, f_2, \ldots, f_n$ be a family of independent copies of a given random variable $f$ in a probability space $(\Omega, \mathcal{F}, \mu)$. Then, the following equivalence of norms holds whenever $1 \le q \le p < \infty$, $( \int_{\Omega}[ \sum_{k=1}^n |f_k|^q ]^{p/q} d \mu )^{1/p} \sim \max_{r \in \{p,q\}} \{ n^{1/r}( \int_\Omega |f|^r d\mu)^{1/r} \}$. The authors prove a noncommutative analogue of this inequality for sums of free random variables over a given von Neumann subalgebra. This formulation leads to new classes of noncommutative function spaces which appear in quantum probability as square functions, conditioned square functions and maximal functions.