# 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.

## How to cite

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Nakai, Eiichi. "Pointwise multipliers on weighted BMO spaces." Studia Mathematica 125.1 (1997): 35-56. <http://eudml.org/doc/216420>.

@article{Nakai1997,
abstract = {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 $ϕ: X×ℝ_\{+\} → ℝ_\{+\}$, we denote by $bmo_\{ϕ,p\}(X)$ the set of all functions $f ∈ L^\{p\}_\{loc\}(X)$ such that $sup_\{a ∈ X, r>0\} 1/ϕ(a,r) (1/μ(B(a,r)) ʃ_\{B(a,r)\} |f(x) -f_\{B(a,r)\}|^p dμ)^\{1/p\} < ∞$, where B(a,r) is the ball centered at a and of radius r, and $f_\{B(a,r)\}$ is the integral mean of f on B(a,r). Let $bmo_\{ϕ\}(X) = bmo_\{ϕ,1\}(X)$ and $bmo(X) = bmo_\{1,1\}(X)$. In this paper, we characterize $PWM(bmo_\{ϕ1,p_1\}(X), bmo_\{ϕ2,p_2\}(X))$. The following are examples of our results. $PWM(bmo_\{(log(1/r))^\{-α\}\}(^n),bmo_\{(log(1/r))^\{-β\}\}(^n)) = bmo_\{(log(1/r))^\{α-β-1\}\}(^n)$, 0≤β < α < 1, $PWM (bmo_\{(log(1/r))^\{-1\}\}(^n),bmo(^n)) = bmo_\{(log log(1/r))^\{-1\}\}(^n),$$PWM (bmo(ℝ^n),bmo_\{log(|a|+r+1/r),p\}(ℝ^n)) = bmo(ℝ^n), 1 < p < ∞, etc.}, author = {Nakai, Eiichi}, journal = {Studia Mathematica}, keywords = {multiplier; pointwise multiplier; bounded mean oscillation; space of homogeneous type; functions of bounded mean oscillation; weighted BMO spaces; pointwise multipliers}, language = {eng}, number = {1}, pages = {35-56}, title = {Pointwise multipliers on weighted BMO spaces}, url = {http://eudml.org/doc/216420}, volume = {125}, year = {1997}, } TY - JOUR AU - Nakai, Eiichi TI - Pointwise multipliers on weighted BMO spaces JO - Studia Mathematica PY - 1997 VL - 125 IS - 1 SP - 35 EP - 56 AB - Let E and F be spaces of real- or complex-valued functions defined on a set X. A real- or complex-valued function g defined on X is called a pointwise multiplier from E to F if the pointwise product fg belongs to F for each f ∈ E. We denote by PWM(E,F) the set of all pointwise multipliers from E to F. Let X be a space of homogeneous type in the sense of Coifman-Weiss. For 1 ≤ p < ∞ and for ϕ: X×ℝ_{+} → ℝ_{+}, we denote by bmo_{ϕ,p}(X) the set of all functions f ∈ L^{p}_{loc}(X) such that sup_{a ∈ X, r>0} 1/ϕ(a,r) (1/μ(B(a,r)) ʃ_{B(a,r)} |f(x) -f_{B(a,r)}|^p dμ)^{1/p} < ∞, where B(a,r) is the ball centered at a and of radius r, and f_{B(a,r)} is the integral mean of f on B(a,r). Let bmo_{ϕ}(X) = bmo_{ϕ,1}(X) and bmo(X) = bmo_{1,1}(X). In this paper, we characterize PWM(bmo_{ϕ1,p_1}(X), bmo_{ϕ2,p_2}(X)). The following are examples of our results. PWM(bmo_{(log(1/r))^{-α}}(^n),bmo_{(log(1/r))^{-β}}(^n)) = bmo_{(log(1/r))^{α-β-1}}(^n), 0≤β < α < 1, PWM (bmo_{(log(1/r))^{-1}}(^n),bmo(^n)) = bmo_{(log log(1/r))^{-1}}(^n),$$PWM (bmo(ℝ^n),bmo_{log(|a|+r+1/r),p}(ℝ^n)) = bmo(ℝ^n)$, 1 < p < ∞, etc.
LA - eng
KW - multiplier; pointwise multiplier; bounded mean oscillation; space of homogeneous type; functions of bounded mean oscillation; weighted BMO spaces; pointwise multipliers
UR - http://eudml.org/doc/216420
ER -

## References

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