Diagonal operators on Janson-Sobolev and Janson-Sobolev-Hardy spaces

Abstract

We study the Banach space and operator factorization structure of Janson-Sobolev and Janson-Sobolev-Hardy spaces. This new class of martingale spaces is determined by a $q$-adic filtration, a subspace $V\subset {\mathbb{R}}_0^{l\times q}$, and a rearrangement invariant function space $X$. Our main result shows that, for every bounded diagonal operator $D$, the operator $S={\sum }_{t=1}^s{\lambda }_{\mathcal{U}}^{k_t}(D)Q_{{\mathcal{K}}_t}^{\mathcal{B}}$ determined by the linear functionals ${\lambda }_{\mathcal{U}}^{k_t}(D)$ and the canonical projections $Q_{{\mathcal{K}}_t}^{\mathcal{B}}$, almost projectionally factors through $D$ with constant $1^{+}$. As consequences, we obtain factorization results for diagonal operators both under a natural boundedness condition on the canonical projections and for all spaces equipped with the $L^1$-norm.

62 pages, 6 figures