Pointwise multipliers on martingale Campanato spaces

TL;DR

This paper introduces generalized Campanato spaces on probability spaces and characterizes the set of all pointwise multipliers on these spaces, extending classical results related to BMO.

Contribution

It defines new generalized Campanato spaces $\\mathcal{L}_{p,\phi}$ and provides a characterization of their pointwise multipliers, broadening the understanding of function space multipliers.

Findings

Characterization of pointwise multipliers on generalized Campanato spaces.

Extension of BMO space results to broader Campanato spaces.

New insights into the structure of multipliers on these spaces.

Abstract

We introduce generalized Campanato spaces $\mathcal{L}_{p,\phi}$ on a probability space $(\Omega,\mathcal{F},P)$, where $p\in[1,\infty)$ and $\phi:(0,1]\to(0,\infty)$. If $p=1$ and $\phi\equiv1$, then $\mathcal{L}_{p,\phi}=\mathrm{BMO}$. We give a characterization of the set of all pointwise multipliers on $\mathcal{L}_{p,\phi}$.