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On multipliers for $$\text{BMO}_ \phi$$ on general domains. (English) Zbl 0809.42003
In a previous paper [Kodai Math. J. 16, No. 1, 79-89 (1993; Zbl 0783.42014)]the author characterized the class of pointwise multipliers on $$\text{BMO}(D)$$, for a domain $$D\subset \mathbb{R}^ d$$ [for the case $$D= \mathbb{R}^ n$$ see S. Janson’s article in Ark. Mat. 14, 189-196 (1976; Zbl 0341.43005)]. In the present paper this characterization is extended to a class $$\text{BMO}_{\phi,p}(D)$$, consisting of $$L^ p_{\text{loc}}$$-functions satisfying the condition $$\sup_{Q\in D}(\phi(l(Q)))^{-1}\bigl(m(Q)^{-1}\int_ Q | f- f_ Q|^ p dx\bigr)^{1/p}< \infty$$, for suitable $$\phi$$.