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1
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On multilinear fractional integrals

100%
Grafakos L.
Studia Mathematica
|
1992
|
tom 102
|
nr 1
49-56
EN
In $ℝ^n$, we prove $L^{p₁} ×...× L^{p_{K}}$ boundedness for the multilinear fractional integrals $I_α(f₁,...,f_K)(x) = ʃ f₁(x-θ₁ y)...f_K(x-θ_K y)|y|^{α-n} dy$ where the $θ_j$'s are nonzero and distinct. We also prove multilinear versions of two inequalities for fractional integrals and a multilinear Lebesgue differentiation theorem.
2
Content available remote

Estimates for maximal singular integrals

100%
Grafakos L.
Colloquium Mathematicum
|
2003
|
tom 96
|
nr 2
167-177
EN
It is shown that maximal truncations of nonconvolution L²-bounded singular integral operators with kernels satisfying Hörmander's condition are weak type (1,1) and $L^{p}$-bounded for 1 < p< ∞. Under stronger smoothness conditions, such estimates can be obtained using a generalization of Cotlar's inequality. This inequality is not applicable here and the point of this article is to treat the boundedness of such maximal singular integral operators in an alternative way.
3
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Corrigendum to "Carleson measures associated with families of multilinear operators" (Studia Math. 211 (2012), 71-94)

64%
Grafakos L., Oliveira L.
Studia Mathematica
|
2014
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tom 221
|
nr 2
193-196
EN
We provide a modification for part of the proof of Theorem 1.2 of our article, pages 85-89, under the multivariable T(1) cancellation condition.
4
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Multilinear Fourier multipliers with minimal Sobolev regularity, I

64%
Grafakos L., Nguyen H.
Colloquium Mathematicum
|
2016
|
tom 144
|
nr 1
1-30
EN
We find optimal conditions on m-linear Fourier multipliers that give rise to bounded operators from products of Hardy spaces $H^{p_{k}}$, $0 < p_{k} ≤ 1$, to Lebesgue spaces $L^{p}$. These conditions are expressed in terms of L²-based Sobolev spaces with sharp indices within the classes of multipliers we consider. Our results extend those obtained in the linear case (m = 1) by Calderón and Torchinsky (1977) and in the bilinear case (m = 2) by Miyachi and Tomita (2013). We also prove a coordinate-type Hörmander integral condition which we use to obtain certain endpoint cases.
5
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The Marcinkiewicz multiplier condition for bilinear operators

64%
Grafakos L., Kalton N.
Studia Mathematica
|
2001
|
tom 146
|
nr 2
115-156
EN
This article is concerned with the question of whether Marcinkiewicz multipliers on $ℝ^{2n}$ give rise to bilinear multipliers on ℝⁿ × ℝⁿ. We show that this is not always the case. Moreover, we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces.
6
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Carleson measures associated with families of multilinear operators

64%
Grafakos L., Oliveira L.
Studia Mathematica
|
2012
|
tom 211
|
nr 1
71-94
EN
We investigate the construction of Carleson measures from families of multilinear integral operators applied to tuples of $L^{∞}$ and BMO functions. We show that if the family $R_{t}$ of multilinear operators has cancellation in each variable, then for BMO functions b₁, ..., bₘ, the measure $|R_{t}(b₁, ..., bₘ)(x)|² dxdt/t$ is Carleson. However, if the family of multilinear operators has cancellation in all variables combined, this result is still valid if $b_{j}$ are $L^{∞}$ functions, but it may fail if $b_{j}$ are unbounded BMO functions, as we indicate via an example. As an application of our results we obtain a multilinear quadratic T(1) type theorem and a multilinear version of a quadratic T(b) theorem analogous to those by Semmes [Proc. Amer. Math. Soc. 110 (1990), 721-726].
7
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Off-diagonal multilinear interpolation between adjoint operators

64%
Grafakos L., Lynch R. G.
Commentationes Mathematicae
|
2015
|
tom 55
|
nr 1
EN
We extend a theorem by Grafakos and Tao on multilinear interpolation between adjoint operators [Multilinear interpolation between adjoint operators, J. Funct. Anal. 199 (2003), 379-385] to an off-diagonal situation. We provide an application.
8
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Multilinear analysis on metric spaces

56%
Grafakos L., Liu L., Maldonado D., Yang D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
The multilinear Calderón-Zygmund theory is developed in the setting of RD-spaces which are spaces of homogeneous type equipped with measures satisfying a reverse doubling condition. The multiple-weight multilinear Calderón-Zygmund theory in this context is also developed in this work. The bilinear T1-theorems for Besov and Triebel-Lizorkin spaces in the full range of exponents are among the main results obtained. Multilinear vector-valued T1 type theorems on Lebesgue spaces, Besov spaces, and Triebel-Lizorkin spaces are also proved. Applications include the boundedness of paraproducts and bilinear multiplier operators on products of Besov and Triebel-Lizorkin spaces.
9
Content available remote

A sharp estimate for the Hardy-Littlewood maximal function

51%
Grafakos L., Montgomery-Smith S., Motrunich O.
Studia Mathematica
|
1999
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tom 134
|
nr 1
57-67
EN
The best constant in the usual $L^p$ norm inequality for the centered Hardy-Littlewood maximal function on $ℝ^1$ is obtained for the class of all "peak-shaped" functions. A function on the line is called peak-shaped if it is positive and convex except at one point. The techniques we use include variational methods.
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