Pointwise multipliers on weighted BMO spaces

Eiichi Nakai

Studia Mathematica (1997)

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

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 ϕ : X × ℝ + → ℝ + , we denote by b m o ϕ , p ( X ) the set of all functions f ∈ L l o c p ( X ) such that s u p a ∈ X , r > 0 1 / ϕ ( a , r ) ( 1 / μ ( B ( a , r ) ) ʃ B ( a , r ) | f ( x ) - f B ( a , r ) | p d μ ) 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 b m o ϕ ( X ) = b m o ϕ , 1 ( X ) and b m o ( X ) = b m o 1 , 1 ( X ) . In this paper, we characterize P W M ( b m o ϕ 1 , p 1 ( X ) , b m o ϕ 2 , p 2 ( X ) ) . The following are examples of our results. P W M ( b m o ( l o g ( 1 / r ) ) - α ( n ) , b m o ( l o g ( 1 / r ) ) - β ( n ) ) = b m o ( l o g ( 1 / r ) ) α - β - 1 ( n ) , 0≤β < α < 1, P W M ( b m o ( l o g ( 1 / r ) ) - 1 ( n ) , b m o ( n ) ) = b m o ( l o g l o g ( 1 / r ) ) - 1 ( n ) , P W M ( b m o ( ℝ n ) , b m o l o g ( | a | + r + 1 / r ) , p ( ℝ n ) ) = b m o ( ℝ n ) , 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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