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GENERALIZATIONS OF THE CARLESONHUNT THEOREM I. THE CLASSICAL SINGULARITY CASE
, 2005
"... Abstract. In this article, we prove L p estimates for a general maximal operator, which extend both the classical CoifmanMeyer [2] and CarlesonHunt [1], [7] theorems in harmonic analysis. 1. ..."
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Abstract. In this article, we prove L p estimates for a general maximal operator, which extend both the classical CoifmanMeyer [2] and CarlesonHunt [1], [7] theorems in harmonic analysis. 1.
PARAPRODUCTS WITH FLAG SINGULARITIES I. A CASE STUDY
, 2006
"... In this paper we prove L p estimates for a trilinear operator, whose symbol is given by the product of two standard symbols, satisfying the well known MarcinkiewiczHÃ¶rmanderMihlin condition. Our main result contains in particular the classical CoifmanMeyer theorem. This trilinear operator is ..."
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In this paper we prove L p estimates for a trilinear operator, whose symbol is given by the product of two standard symbols, satisfying the well known MarcinkiewiczHÃ¶rmanderMihlin condition. Our main result contains in particular the classical CoifmanMeyer theorem. This trilinear operator is the simplest example of a large class of multilinear operators, which we called paraproducts with flag singularities.
On the biCarleson operator. I. The Walsh case
, 2002
"... We prove L p estimates for the Walsh model of the maximal biCarleson operator defined below. The corresponding estimates for the Fourier model will be obtained in the sequel [20] of this paper. ..."
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We prove L p estimates for the Walsh model of the maximal biCarleson operator defined below. The corresponding estimates for the Fourier model will be obtained in the sequel [20] of this paper.
FLAG PARAPRODUCTS
, 2008
"... We describe the theory of flag paraproducts and their relationship to the field of differential equations. The main goal of the present paper is to describe the theory of a new class of multilinear operators which we named “paraproducts with flag singularities ” (or in short “flag paraproducts”). ..."
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We describe the theory of flag paraproducts and their relationship to the field of differential equations. The main goal of the present paper is to describe the theory of a new class of multilinear operators which we named “paraproducts with flag singularities ” (or in short “flag paraproducts”). These objects, which have been introduced in [11] as being generalizations of the “lacunary versions“ of the “biest operators “ of [14], [15], [18], turned out in the meantime to have very
VARIATIONNORM AND FLUCTUATION ESTIMATES FOR ERGODIC BILINEAR AVERAGES
, 2015
"... For any dynamical system, we show that higher variationnorms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respe ..."
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For any dynamical system, we show that higher variationnorms for the sequence of ergodic bilinear averages of two functions satisfy a large range of bilinear Lp estimates. It follows that, with probability one, the number of fluctuations along this sequence may grow at most polynomially with respect to (the growth of) the underlying scale. These results strengthen previous works of Lacey and Bourgain where almost surely convergence of the sequence was proved (which is equivalent to the qualitative statement that the number of fluctuations is finite at each scale). Via transference, the proof reduces to establishing new bilinear Lp bounds for variationnorms of truncated bilinear operators on R, and the main new ingredient of the proof of these bounds is a variationnorm extension of maximal Bessel inequalities of Lacey and Demeter–Tao–Thiele.