# Pointwise multipliers on weighted BMO spaces

Studia Mathematica (1997)

• Volume: 125, Issue: 1, page 35-56
• ISSN: 0039-3223

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## Abstract

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Let E and F be spaces of real- or complex-valued functions defined on a set X. A real- or complex-valued function g defined on X is called a pointwise multiplier from E to F if the pointwise product fg belongs to F for each f ∈ E. We denote by PWM(E,F) the set of all pointwise multipliers from E to F. Let X be a space of homogeneous type in the sense of Coifman-Weiss. For 1 ≤ p < ∞ and for $\varphi :X×{ℝ}_{+}\to {ℝ}_{+}$, we denote by $bm{o}_{\varphi ,p}\left(X\right)$ the set of all functions $f\in {L}_{loc}^{p}\left(X\right)$ such that $su{p}_{a\in X,r>0}1/\varphi \left(a,r\right)\left(1/\mu \left(B\left(a,r\right)\right){ʃ}_{B\left(a,r\right)}|f\left(x\right)-{f}_{B\left(a,r\right)}{{|}^{p}d\mu \right)}^{1/p}<\infty$, where B(a,r) is the ball centered at a and of radius r, and ${f}_{B\left(a,r\right)}$ is the integral mean of f on B(a,r). Let $bm{o}_{\varphi }\left(X\right)=bm{o}_{\varphi ,1}\left(X\right)$ and $bmo\left(X\right)=bm{o}_{1,1}\left(X\right)$. In this paper, we characterize $PWM\left(bm{o}_{\varphi 1,{p}_{1}}\left(X\right),bm{o}_{\varphi 2,{p}_{2}}\left(X\right)\right)$. The following are examples of our results. $PWM\left(bm{o}_{{\left(log\left(1/r\right)\right)}^{-\alpha }}{\left(}^{n}\right),bm{o}_{{\left(log\left(1/r\right)\right)}^{-\beta }}{\left(}^{n}\right)\right)=bm{o}_{{\left(log\left(1/r\right)\right)}^{\alpha -\beta -1}}{\left(}^{n}\right)$, 0≤β < α < 1, $PWM\left(bm{o}_{{\left(log\left(1/r\right)\right)}^{-1}}{\left(}^{n}\right),bmo{\left(}^{n}\right)\right)=bm{o}_{{\left(loglog\left(1/r\right)\right)}^{-1}}{\left(}^{n}\right),$$PWM\left(bmo\left({ℝ}^{n}\right),bm{o}_{log\left(|a|+r+1/r\right),p}\left({ℝ}^{n}\right)\right)=bmo\left({ℝ}^{n}\right)$, 1 < p < ∞, etc.

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