Endpoint bounds for a generalized Radon transform

Stovall, Betsy

doi:10.1112/jlms/jdp033

Cited by 22 publications

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“…It is a remarkable result of Christ [9] that (up to the endpoints) these restrictions on p and q are in fact sufficient for (1) to hold in this non-degenerate situation. Stovall [30], building on an argument of Christ [10], has converted Christ's restricted weak-type estimates at the endpoints into strong type estimates.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Universal Lp improving for averages along polynomial curves in low dimensions

Dendrinos

Laghi

Wright

2009

Journal of Functional Analysis

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We prove sharp L p → L q estimates for averaging operators along general polynomial curves in two and three dimensions. These operators are translation-invariant, given by convolution with the so-called affine arclength measure of the curve and we obtain universal bounds over the class of curves given by polynomials of bounded degree. Our method relies on a geometric inequality for general vector polynomials together with a combinatorial argument due to M. Christ. Almost sharp Lorentz space estimates are obtained as well.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Universal Lp improving for averages along polynomial curves in low dimensions

Dendrinos

Laghi

Wright

2009

Journal of Functional Analysis

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“…This method was applied in [9] to prove strong-type estimates for the operator in (1) in the cases d = 2, 3 and in [27] to prove strong-type estimates for convolution with affine arclength measure along the moment curve. Using Christ's techniques, one can show that the next two lemmas imply Theorem 1.…”

Section: Reduction Of Theorem 1 To Two Lemmasmentioning

confidence: 99%

“…Using Christ's techniques, one can show that the next two lemmas imply Theorem 1. See [27] for details. For notational convenience, we define the quantity…”

Section: Reduction Of Theorem 1 To Two Lemmasmentioning

confidence: 99%

“…The relative simplicity of Lemma 3 is a matter of luck more than anything else. See [27] for more explanation. We remark that Lemma 4 is slightly different from the corresponding lemma in [27], but implies the strong-type bound (and accompanying Lorentz space improvement) by exactly the same proof.…”

Section: Reduction Of Theorem 1 To Two Lemmasmentioning

confidence: 99%

“…In [7], Christ showed that it was possible to use arguments similar to those in [6] to prove strong-type estimates. We mention three subsequent applications of this technique (this article being a fourth), namely [9] wherein Dendrinos, Laghi and Wright consider T P in dimension 3, [17] in which Laghi proves endpoint bounds for a restricted X-ray transform, and the author's [27] which establishes strong-type bounds for convolution with affine arclength measure along the moment curve in dimensions d 4 (as mentioned above, low-dimensional results were already known). In this article, we will use techniques from [6,7,9,27], mentioned above, as well as from [14], in which Gressman established restricted weak-type bounds for convolution with certain measures along polynomial curves whose entries are monomials.…”

Section: Introductionmentioning

confidence: 99%

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Endpoint Lp→Lq bounds for integration along certain polynomial curves

Stovall

2010

Journal of Functional Analysis

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Existence of Extremizers for a Model Convolution Operator

Biswas

2019

J Fourier Anal Appl

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The operator T , defined by convolution with the affine arc length measure on the moment curve parametrized by h(t) = (t, t 2 , ..., t d ) is a bounded operator from L p to L q if ( 1 p , 1 q ) lies on a line segment. In this article we prove that at non-end points there exist functions which extremize the associated inequality and any extremizing sequence is pre compact modulo the action of the symmetry of T . We also establish a relation between extremizers for T at the end points and the extremizers of an X-ray transform restricted to directions along the moment curve. Our proof is based on the ideas of Michael Christ on convolution with the surface measure on the paraboloid.The above theorem was proved for d = 2 by Littman in [20]. Oberlin [21] showed that it is sufficient to satisfy the above condition when d = 3. The theorem was proved up to the end point for any dimension by Christ in [2]. Extending ideas from [2], Stovall proved the strong type end point bound [22].By using methods based on several observations and results of Christ (see [4,2,3,22]), together with some refinements by Stovall and Dendrinos [12] we have been able to refine the associated inequality for this operator. To state our results, first we need to introduce some definitions.Let (p, q) be as above. Let A be the operator norm of T . That isDefinition 1.4. δ-Quasiextremizer pair. Let δ>0. We say that an ordered pair (f, g) of measurable functions on R d is an δ-quasiextremizer pair ifDefinition 1.5. Extremizing sequence. An extremizing sequence is any sequence f n ∈ L p such that f n L p = 1, T f n L q → A.Let us denote the moment curve by h(t) = (t, t 2 , ..., t d ). Our main theorems are as follows.

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Endpoint bounds for a generalized Radon transform

Cited by 22 publications

References 10 publications

Universal Lp improving for averages along polynomial curves in low dimensions

Universal Lp improving for averages along polynomial curves in low dimensions

Endpoint Lp→Lq bounds for integration along certain polynomial curves

Existence of Extremizers for a Model Convolution Operator

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