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1
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Pointwise multipliers for functions of weighted bounded mean oscillation

100%
Nakai E.
Studia Mathematica
|
1993
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tom 105
|
nr 2
105-119
EN
For $w : ℝ^{n} × ℝ_{+} → ℝ_{+}$ and 1 ≤ p < ∞, let $bmo_{{}w,p}(ℝ^n)$ be the set of locally integrable functions f on $ℝ^n$ for which $sup_{I}(1/w(I) ʃ_{I} |f(x)-f_{I}|^p dx)^{1/p} < ∞$ where I = I(a,r) is the cube with center a whose edges have length r and are parallel to the coordinate axes, w(I) = w(a,r) and $f_{I}$ is the average of f over I. If w satisfies appropriate conditions, then the following are equivalent: (1) $fg ∈ bmo_{w,p}(ℝ^n)$ whenever $f ∈ ℝ bmo_{w,p}(ℝ^n)$, (2) $g ∈ L^∞(ℝ^n)$ and $sup_{I}( 1/w*(I) ʃ_{I} |g(x)-g_{I}|^p dx)^{1/p} < ∞$, where $w* = w/Ψ, Ψ = Ψ_{1} + Ψ_{2}$ and $Ψ_{1}(a,r) = (ʃ_{1}^{max(2,|a|,r)} (w(O,t)^{1/p})/(t^{n/p+1}) dt)^p$, $Ψ_{2}(a,r) = (ʃ_{r}^{max(2,|a|,r)} (w(a,t)^{1/p})/(t^{n/p+1}} dt)^p$.
2
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Orlicz-Morrey spaces and the Hardy-Littlewood maximal function

100%
Nakai E.
Studia Mathematica
|
2008
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tom 188
|
nr 3
193-221
EN
We prove basic properties of Orlicz-Morrey spaces and give a necessary and sufficient condition for boundedness of the Hardy-Littlewood maximal operator M from one Orlicz-Morrey space to another. For example, if f ∈ L(log L)(ℝⁿ), then Mf is in a (generalized) Morrey space (Example 5.1). As an application of boundedness of M, we prove the boundedness of generalized fractional integral operators, improving earlier results of the author.
3
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Pointwise multipliers on weighted BMO spaces

100%
Nakai E.
Studia Mathematica
|
1997
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tom 125
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nr 1
35-56
EN
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. $PWM(bmo_{(log(1/r))^{-α}}(𝕋^n),bmo_{(log(1/r))^{-β}}(𝕋^n)) = bmo_{(log(1/r))^{α-β-1}}(𝕋^n)$, 0≤β < α < 1, $PWM (bmo_{(log(1/r))^{-1}}(𝕋^n),bmo(𝕋^n)) = bmo_{(log log(1/r))^{-1}}(𝕋^n),$ $PWM (bmo(ℝ^n),bmo_{log(|a|+r+1/r),p}(ℝ^n)) = bmo(ℝ^n)$, 1 < p < ∞, etc.
4
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The Campanato, Morrey and Hölder spaces on spaces of homogeneous type

100%
Nakai E.
Studia Mathematica
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2006
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tom 176
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nr 1
1-19
EN
We investigate the relations between the Campanato, Morrey and Hölder spaces on spaces of homogeneous type and extend the results of Campanato, Mayers, and Macías and Segovia. The results are new even for the ℝⁿ case. Let (X,d,μ) be a space of homogeneous type and (X,δ,μ) its normalized space in the sense of Macías and Segovia. We also study the relations of these function spaces for (X,d,μ) and for (X,δ,μ). Using these relations, we can show that theorems for the Campanato, Morrey or Hölder spaces on the normal space are valid for the function spaces on any space of homogeneous type. As an application we obtain boundedness of some operators related to partial differential equations, boundedness of fractional differential and integral operators, and give characterizations of pointwise multipliers.
5
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Fractional integral operators on $B^{p,λ}$ with Morrey-Campanato norms

64%
Matsuoka K., Nakai E.
Banach Center Publications
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2011
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tom 92
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nr 1
249-264
EN
We introduce function spaces $B^{p,λ}$ with Morrey-Campanato norms, which unify $B^{p,λ}$, $CMO^{p,λ}$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{α}$ on these spaces.
6
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Pointwise multipliers on martingale Campanato spaces

64%
Nakai E., Sadasue G.
Studia Mathematica
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2014
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tom 220
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nr 1
87-100
EN
We introduce generalized Campanato spaces $𝓛_{p,ϕ}$ on a probability space (Ω,ℱ,P), where p ∈ [1,∞) and ϕ: (0,1] → (0,∞). If p = 1 and ϕ ≡ 1, then $𝓛_{p,ϕ} = BMO$. We give a characterization of the set of all pointwise multipliers on $𝓛_{p,ϕ}$.
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