Martingale Transforms between Martingale Hardy-amalgam Spaces

Abstract

We discuss martingale transforms between martingale Hardy-amalgam spaces $H_{p,q}^s,{\mathcal{Q}}_{p,q}$ and ${\mathcal{P}}_{p,q}.$ Let $0<p<q<\infty ,p_1<p$ and $q_1<q$ and let $f=\left(f_n,n\in \mathbb{N}\right)$ be a martingale in ${\mathcal{P}}_{p_1,q_1}$ ; then, we show that its martingale transforms are the martingales in ${\mathcal{P}}_{p,q}$ for some $p,q$ and similarly for $H_{p,q}^s$ and ${\mathcal{Q}}_{p,q}.$