On conditional expectations in L^p(mu;L^q(nu;X))

TL;DR

This paper characterizes when the conditional expectation operator acts as a bounded linear map between specific Banach space-valued Lebesgue spaces, providing necessary and sufficient conditions.

Contribution

It establishes precise criteria for the boundedness of conditional expectations in mixed $L^p$-$L^q$ spaces with Banach space values, extending previous understanding.

Findings

Necessary and sufficient conditions for boundedness of conditional expectation

Characterization of the image as a strongly $$-measurable subspace

Extension of classical results to Banach space-valued functions

Abstract

Let $(A,\mathscr{A},\mu)$ and $(B,\mathscr{B},\nu)$ be probability spaces, let $\mathscr{F}$ be a sub-$\sigma$-algebra of the product $\sigma$-algebra $\mathscr{A}\times\mathscr{B}$, let $X$ be a Banach space, and let $1< p,q< \infty$. We obtain necessary and sufficient conditions in order that the conditional expectation with respect to $\mathscr{F}$ defines a bounded linear operator from $L^p(\mu;L^q(\nu;X))$ onto $L^p_{\mathscr{F}}(\mu;L^q(\nu;X))$, the closed subspace in $L^p(\mu;L^q(\nu;X))$ of all functions having a strongly $\mathscr{F}$-measurable representative.