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Majoration de la transformée de Fourier de certaines mesures

Noël Lohoué, Jacques Peyrière (1983)

Annales de l'institut Fourier

Soit p une fonction polynôme de R m dans R . On considère la mesure μ p sur le graphe de p dont la projection sur R m est la mesure de Lebesgue. On étudie ici le comportement de la transformée de Fourier μ ^ p ( u , v ) lorsque v approche de 0 (de telles distributions apparaissent comme caractères de représentations de groupes de Lie nilpotents). On étend des résultats de L. Corwin et F.P. Greenleaf (Comm. on Pure and Applied Math., 31 (1975), 681–705) au cas où le gradient de la partie de p homogène de plus haut degré...

Marcinkiewicz integrals on product spaces

H. Al-Qassem, A. Al-Salman, L. C. Cheng, Y. Pan (2005)

Studia Mathematica

We prove the L p boundedness of the Marcinkiewicz integral operators μ Ω on ℝ n ₁ × ⋯ × ℝ n k under the condition that Ω ∈ L ( l o g L ) k / 2 ( n ₁ - 1 × ⋯ × n k - 1 ) . The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials

Pascal Auscher, Besma Ben Ali (2007)

Annales de l’institut Fourier

We show various L p estimates for Schrödinger operators - Δ + V on ℝ n and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen. Our main tools are improved Fefferman-Phong inequalities and reverse Hölder estimates for weak solutions of - Δ + V and their gradients.

Maximal singular integrals on product homogeneous groups

Yong Ding, Shuichi Sato (2014)

Studia Mathematica

We prove L p boundedness for p ∈ (1,∞) of maximal singular integral operators with rough kernels on product homogeneous groups under a sharp integrability condition of the kernels.

Mean Oscillation and Boundedness of Multilinear Integral Operators with General Kernels

Liu Lanzhe (2014)

Archivum Mathematicum

In this paper, the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces are proved. The integral operators include singular integral operator with general kernel, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

Measure-preserving quality within mappings.

Stephen Semmes (2000)

Revista Matemática Iberoamericana

In [6], Guy David introduced some methods for finding controlled behavior in Lipschitz mappings with substantial images (in terms of measure). Under suitable conditions, David produces subsets on which the given mapping is bilipschitz, with uniform bounds for the bilipschitz constant and the size of the subset. This has applications for boundedness of singular integral operators and uniform rectifiability of sets, as in [6], [7], [11], [13]. Some special cases of David's results, concerning projections...

Mixed A p - A ∞ estimates with one supremum

Andrei K. Lerner, Kabe Moen (2013)

Studia Mathematica

We establish several mixed A p - A ∞ bounds for Calderón-Zygmund operators that only involve one supremum. We address both cases when the A ∞ part of the constant is measured using the exponential-logarithmic definition and using the Fujii-Wilson definition. In particular, we answer a question of the first author and provide an answer, up to a logarithmic factor, to a conjecture of Hytönen and Lacey. Moreover, we give an example to show that our bounds with the logarithmic factors can be arbitrarily smaller...

Modulation invariant and multilinear singular integral operators

Michael Christ (2005/2006)

Séminaire Bourbaki

In a series of papers beginning in the late 1990s, Michael Lacey and Christoph Thiele have resolved a longstanding conjecture of Calderón regarding certain very singular integral operators, given a transparent proof of Carleson’s theorem on the almost everywhere convergence of Fourier series, and initiated a slew of further developments. The hallmarks of these problems are multilinearity as opposed to mere linearity, and especially modulation symmetry. By modulation is meant multiplication by characters...

Muckenhoupt-Wheeden conjectures in higher dimensions

Alberto Criado, Fernando Soria (2016)

Studia Mathematica

In recent work by Reguera and Thiele (2012) and by Reguera and Scurry (2013), two conjectures about joint weighted estimates for Calderón-Zygmund operators and the Hardy-Littlewood maximal function were refuted in the one-dimensional case. One of the key ingredients for these results is the construction of weights for which the value of the Hilbert transform is substantially bigger than that of the maximal function. In this work, we show that a similar construction is possible for classical Calderón-Zygmund...

Multidimensional decay in the van der Corput lemma

Michael Ruzhansky (2012)

Studia Mathematica

We establish a multidimensional decay of oscillatory integrals with degenerate stationary points, gaining the decay with respect to all space variables. This bridges the gap between the one-dimensional decay for degenerate stationary points given by the classical van der Corput lemma and the multidimensional decay for non-degenerate stationary points given by the stationary phase method. Complex-valued phase functions as well as phases and amplitudes of limited regularity are considered. Conditions...

Multilinear almost diagonal estimates and applications

Árpád Bényi, Nikolaos Tzirakis (2004)

Studia Mathematica

We prove that an almost diagonal condition on the (m + 1)-linear tensor associated to an m-linear operator implies boundedness of the operator on products of classical function spaces. We then provide applications to the study of certain singular integral operators.

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