Currently displaying 1 – 20 of 21

Showing per page

Order by Relevance | Title | Year of publication

On multilinear fractional integrals

Loukas Grafakos — 1992

Studia Mathematica

In n , we prove L p × . . . × L p K boundedness for the multilinear fractional integrals I α ( f , . . . , f K ) ( x ) = ʃ f ( x - θ y ) . . . f K ( x - θ K y ) | y | α - n d y 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.

Estimates for maximal singular integrals

Loukas Grafakos — 2003

Colloquium Mathematicae

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.

Hardy space estimates for multilinear operators (II).

Loukas Grafakos — 1992

Revista Matemática Iberoamericana

We continue the study of multilinear operators given by products of finite vectors of Calderón-Zygmund operators. We determine the set of all r ≤ 1 for which these operators map products of Lebesgue spaces L(R) into the Hardy spaces H(R). At the endpoint case r = n/(n + m + 1), where m is the highest vanishing moment of the multilinear operator, we prove a weak type result.

Carleson measures associated with families of multilinear operators

Loukas GrafakosLucas Oliveira — 2012

Studia Mathematica

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 ( b , . . . , b ) ( x ) | ² d x d t / 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...

A new of looking at distributional estimates; applications for the bilinear Hilbert transform.

Dimitriy BilykLoukas Grafakos — 2006

Collectanea Mathematica

Distributional estimates for the Carleson operator acting on characteristic functions of measurable sets of finite measure were obtained by Hunt. In this article we describe a simple method that yields such estimates for general operators acting on one or more functions. As an application we discuss how distributional estimates are obtained for the linear and bilinear Hilbert transform. These distributional estimates show that the square root of the bilinear Hilbert transform is exponentially lntegrable...

Radial maximal function characterizations for Hardy spaces on RD-spaces

Loukas GrafakosLiguang LiuDachun Yang — 2009

Bulletin de la Société Mathématique de France

An RD-space 𝒳 is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type 𝒳 having “dimension” n , there exists a p 0 ( n / ( n + 1 ) , 1 ) such that for certain classes of distributions, the L p ( 𝒳 ) quasi-norms of their radial maximal functions and grand maximal functions are equivalent when p ( p 0 , ] . This result yields a radial maximal function characterization for Hardy spaces on 𝒳 .

Multilinear Fourier multipliers with minimal Sobolev regularity, I

Loukas GrafakosHanh Van Nguyen — 2016

Colloquium Mathematicae

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...

The Marcinkiewicz multiplier condition for bilinear operators

Loukas GrafakosNigel J. Kalton — 2001

Studia Mathematica

This article is concerned with the question of whether Marcinkiewicz multipliers on 2 n 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.

Hardy space estimates for multilinear operators (I).

Ronald R. CoifmanLoukas Grafakos — 1992

Revista Matemática Iberoamericana

In this article we study bilinear operators given by inner products of finite vectors of Calderón-Zygmund operators. We find that necessary and sufficient condition for these operators to map products of Hardy spaces into Hardy spaces is to have a certain number of moments vanishing and under this assumption we prove a Hölder-type inequality in the H space context.

On multilinear singular integrals of Calderón-Zygmund type.

Loukas GrafakosRodolfo H. Torres — 2002

Publicacions Matemàtiques

A variety of results regarding multilinear singular Calderón-Zygmund integral operators is systematically presented. Several tools and techniques for the study of such operators are discussed. These include new multilinear endpoint weak type estimates, multilinear interpolation, appropriate discrete decompositions, a multilinear version of Schur's test, and a multilinear version of the T1 Theorem suitable for the study of multilinear pseudodifferential and translation invariant operators. A maximal...

Multilinear analysis on metric spaces

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...

Page 1 Next

Download Results (CSV)