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1
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Boundedness of Marcinkiewicz functions.

100%
Sakamoto M., Yabuta K.
Studia Mathematica
|
1999
|
tom 135
|
nr 2
103-142
EN
The $L^p$ boundedness(1 < p < ∞) of Littlewood-Paley's g-function, Lusin's S function, Littlewood-Paley's $g*_λ$-functions, and the Marcinkiewicz function is well known. In a sense, one can regard the Marcinkiewicz function as a variant of Littlewood-Paley's g-function. In this note, we treat counterparts $μ_{S}^{ϱ}$ and $μ_{λ}^{*,ϱ}$ to S and $g*_λ$. The definition of $μ_{S}^{ϱ}(f)$ is as follows: $μ_{S}^{ϱ}(f)(x) = (ʃ_{|y-x| < t}| 1/t^{ϱ} ʃ_{|z|≤ t} Ω(z)/(|z|^{n-ϱ}) f(y-z) dz|^2 (dydt)/(t^{n+1}) )^{1/2}$, where Ω(x) is a homogeneous function of degree 0 and Lipschitz continuous of order β (0 < β ≤ 1) on the unit sphere $S^{n-1}$, and $ʃ_{S^{n-1}} Ω(x')dσ(x') = 0$. We show that if σ = Reϱ > 0, then $μ_{S}^{ϱ}$ is $L^p$ bounded for max(1,2n/(n+2σ) < p < ∞, and for 0 < ϱ ≤ n/2 and 1 ≤ p ≤ 2n/(n+2ϱ), then $L^p$ boundedness does not hold in general, in contrast to the case of the S function. Similar results hold for $μ_{λ}^{*,ϱ}$. Their boundedness in the Campanato space $ε^{α,p}$ is also considered.
2
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Multilinear Calderón-Zygmund operators on weighted Hardy spaces

81%
Li W., Xue Q., Yabuta K.
Studia Mathematica
|
2010
|
tom 199
|
nr 1
1-16
EN
Grafakos-Kalton [Collect. Math. 52 (2001)] discussed the boundedness of multilinear Calderón-Zygmund operators on the product of Hardy spaces. Then Lerner et al. [Adv. Math. 220 (2009)] defined $A_{p⃗}$ weights and built a theory of weights adapted to multilinear Calderón-Zygmund operators. In this paper, we combine the above results and obtain some estimates for multilinear Calderón-Zygmund operators on weighted Hardy spaces and also obtain a weighted multilinear version of an inequality for multilinear fractional integrals, which is related to the classical Trudinger inequality.
3
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A remark to a paper of Janas "Toeplitz operators related to certain domains in $C^{n}$"

60%
Yabuta K.
Studia Mathematica
|
1978
|
tom 62
|
nr 1
73-74
4
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Generalizations of Calderón-Zygmund operators

40%
Yabuta K.
Studia Mathematica
|
1985
|
tom 82
|
nr 1
17-31
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