The Polynomial Carleson Operator

Lie, Victor

doi:10.48550/arxiv.1105.4504

Cited by 20 publications

(37 citation statements)

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“…Here though we will make use of a refinement of it as appearing in [14] when treating the exceptional sets in the Polynomial Carleson operator discretization. This refined decomposition is also carefully described in [15]. For this reason we will skip the details of this decomposition (the interested reader should consult Section 5 in [15]) and only mention that -heuristically -as an output of this procedure we will be able to write 8 P = n∈N P n ,…”

Section: Discretization Of the Families Of Tiles Pmentioning

confidence: 99%

“…This refined decomposition is also carefully described in [15]. For this reason we will skip the details of this decomposition (the interested reader should consult Section 5 in [15]) and only mention that -heuristically -as an output of this procedure we will be able to write 8 P = n∈N P n ,…”

Section: Discretization Of the Families Of Tiles Pmentioning

confidence: 99%

“…8 At a more precise level, each family Pn can be reduced to a BM O−forest of n th generation -again, for the definition see Section 4 in [15]. 9 Thus, notice that this second decomposition is dependent on f hence each function will involve different partitions.…”

Section: Discretization Of the Families Of Tiles Pmentioning

confidence: 99%

“…The proof of Proposition 1 is trivial if one uses the key observation that P cluster is just an α−dilation of a tree 15 . Indeed, heuristically, the proof reduces to the fact that the (maximal) Hilbert transform is bounded from L 1 to L 1,∞ .…”

Section: The Casementioning

confidence: 99%

“…• in our paper we embrace the approach introduced by Fefferman in [6], and further developed in [15], [14]. Among others, this offers us more flexibility in treating the forest estimates and the advantage of having a tile decomposition which is directly adapted to the exceptional sets of the Carleson operator and to the size of f .…”

Section: Remarksmentioning

confidence: 99%

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On the pointwise convergence of the sequence of partial Fourier sums along lacunary subsequences

Lie

2012

Journal of Functional Analysis

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In his 2006 ICM invited address, Konyagin mentioned the following conjecture: if S n f stands for the n-th partial Fourier sum of f and {n j } j ⊂ N is a lacunary sequence, then S n j f is a.e. pointwise convergent for any f ∈ L log log L. In this paper we will show that sup j |S n j (f )| 1,∞ C f 1 log log(10 + f ∞ f 1 ).As a direct consequence we obtain that S n j f → f a.e. for f ∈ L log log L log log log L. The (discrete) Walsh model version of this last fact was proved by Do and Lacey but their methods do not (re)cover the (continuous) Fourier setting. The key ingredient for our proof is a tile decomposition of the operator sup j |S n j (f )| which depends on both the function f and on the lacunary structure of the frequencies. This tile decomposition, called (f, λ)-lacunary, is directly adapted to the context of our problem, and, combined with a canonical mass decomposition of the tiles, provides the natural environment to which the methods developed by the author in "On the boundedness of the Carleson operator near L 1 " apply.

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Section: Discretization Of the Families Of Tiles Pmentioning

confidence: 99%

Section: Discretization Of the Families Of Tiles Pmentioning

confidence: 99%

Section: Discretization Of the Families Of Tiles Pmentioning

confidence: 99%

Section: The Casementioning

confidence: 99%

Section: Remarksmentioning

confidence: 99%

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On the pointwise convergence of the sequence of partial Fourier sums along lacunary subsequences

Lie

2012

Journal of Functional Analysis

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Polynomial Carleson Operators Along Monomial Curves in the Plane

et al. 2017

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Abstract. We prove L p bounds for partial polynomial Carleson operators along monomial curves (t, t m ) in the plane R 2 with a phase polynomial consisting of a single monomial. These operators are "partial" in the sense that we consider linearizing stoppingtime functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, L 2 bounds for partial operators along curves imply the full strength of the L 2 bound for a one-dimensional Carleson operator, and for a quadratic Carleson operator. Our methods, which can at present only treat certain combinations of curves and phases, in some cases adapt a T T * method to treat phases involving fractional monomials, and in other cases use a known vector-valued variant of the Carleson-Hunt theorem.

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Dyadic Triangular Hilbert Transform of Two General Functions and One Not Too General Function

2015

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The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well-studied objects of harmonic analysis. We investigate L p bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.

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The Polynomial Carleson Operator

Cited by 20 publications

References 40 publications

On the pointwise convergence of the sequence of partial Fourier sums along lacunary subsequences

On the pointwise convergence of the sequence of partial Fourier sums along lacunary subsequences

Polynomial Carleson Operators Along Monomial Curves in the Plane

Dyadic Triangular Hilbert Transform of Two General Functions and One Not Too General Function

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