Pointwise multipliers on martingale Campanato spaces (Q2872220)

Let \(\mathcal{L}_{p,\phi}\) be the martingale Campanato space on a regular (i.e., in a sense: doubling) atomic filtration of a probability space \(\Omega\). If \(\phi\) is doubling and satisfies another technical hypothesis, the space of pointwise multipliers on \(\mathcal{L}_{p,\phi}\) is characterized as \(\mathcal{L}_{p,\tilde\phi}\cap L_\infty\) for a certain explicit \(\tilde\phi\). The computations rely on a ``reverse doubling'' type dichotomy established by the authors in [J. Funct. Spaces Appl. 2012, Article ID 673929, 29 p. (2012; Zbl 1254.46035)]: for nested atoms \(A_i\in\mathcal{F}_i\) \((i=n-1,n)\), it holds that either \(P(A_n)=P(A_{n-1})\) or \((1+1/R)P(A_n)\leq P(A_{n-1})\leq R\cdot P(A_n)\).