Pointwise multipliers on weighted BMO spaces

Nakai, Eiichi

Nakai, Eiichi

Studia Mathematica, Tome 122 (1997), p. 35-56 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $ϕ : X \times ℝ_{+} \to ℝ_{+}$ , we denote by $b m o_{ϕ, p} (X)$ the set of all functions $f \in L_{l o c}^{p} (X)$ such that $s u p_{a \in X, r > 0} 1 / ϕ (a, r) (1 / μ (B (a, r)) ʃ_{B (a, r)} | f (x) - f_{B (a, r)} {|^{p} d μ)}^{1 / p} < \infty$ , 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.

Publié le : 1997-01-01

Zbl 0874.42009

EUDML-ID : urn:eudml:doc:216420

@article{bwmeta1.element.bwnjournal-article-smv125i1p35bwm,
     author = {Eiichi Nakai},
     title = {Pointwise multipliers on weighted BMO spaces},
     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {35-56},
     zbl = {0874.42009},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv125i1p35bwm}
}

Nakai, Eiichi. Pointwise multipliers on weighted BMO spaces. Studia Mathematica, Tome 122 (1997) pp. 35-56. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv125i1p35bwm/

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