Endpoint <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:mo>→</mml:mo><mml:msup><mml:mi>L</mml:mi><mml:mi>q</mml:mi></mml:msup></mml:math> bounds for integration along certain polynomial curves

Stovall, Betsy

doi:10.1016/j.jfa.2010.08.008

Cited by 17 publications

(2 citation statements)

References 25 publications

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“…When u = 1, T f (•, 1) is a classical operator in harmonic analysis. For some related results, we refer to [21,28,29,40,41] and the references therein. In this section, our main goal is to determine the almost optimal range of exponents ( 1p , 1 q ) such that the L p x (R 2 ) → L q x (R 2 ) estimate for T holds uniformly for u ∈ [1,2].…”

Section: The Local Smoothing Phenomenonmentioning

confidence: 99%

Permutation polynomials of the form cx + Tr q l

et al. 2017

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Section: The Local Smoothing Phenomenonmentioning

confidence: 99%

Permutation polynomials of the form cx + Tr q l

et al. 2017

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“…It is worth noting that the family of translation-invariant examples T 2 includes the moment curve, for which restricted weak-type bounds in all dimensions were first established by Christ [7]. Later work of Stovall [27], building on results of Dendrinos and Wright [9], achieved endpoint L p -improving inequalities as a byproduct of a more general theorem concerning polynomial curves equipped with affine arc length measure. Recent work of Christ, Dendrinos, Stovall, and Street [8] pushes these ideas even further.…”

Section: Examples 51 a Concrete Examplementioning

confidence: 99%

Uniform Sublevel Radon-like Inequalities

Gressman¹

2011

J Geom Anal

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This paper is concerned with establishing uniform weighted L p -L q estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances whenever a scalar inequality holds for certain associated measures (the inequality is of the sort studied by Oberlin [42], relating measures of parallelepipeds to powers of their Euclidean volumes). These ideas lead to previously unknown, weighted affine-invariant estimates for Radon-like operators as well as new L p -improving estimates for degenerate Radon-like operators with folding canonical relations which satisfy an additional curvature condition of Greenleaf and Seeger [23] for FIOs (building on the ideas of Sogge [51] and Mockenhaupt, Seeger, and Sogge [37]); these new estimates fall outside the range of estimates which are known to hold in the generality of the FIO context.

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Some Subcritical Estimates for the $$\ell ^p$$-Improving Problem for Discrete Curves

Dendrinos

Hughes

Vitturi

2022

J Fourier Anal Appl

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We apply Christ’s method of refinements to the $$\ell ^p$$ ℓ p -improving problem for discrete averages $${\mathcal {A}}_N$$ A N along polynomial curves in $${\mathbb {Z}}^d$$ Z d . Combined with certain elementary estimates for the number of solutions to certain special systems of diophantine equations, we obtain some restricted weak-type $$p \rightarrow p'$$ p → p ′ estimates for the averages $${\mathcal {A}}_N$$ A N in the subcritical regime. The dependence on N of the constants here obtained is sharp, except maybe for an $$\epsilon $$ ϵ -loss.

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Endpoint Lp→Lq bounds for integration along certain polynomial curves

Abstract: We establish strong-type endpoint L p (R d ) → L q (R d ) bounds for the operator given by convolution with affine arclength measure on polynomial curves for d 4. The bounds established depend only on the dimension d and the degree of the polynomial.

Cited by 17 publications

References 25 publications

Permutation polynomials of the form cx + Tr q l

Permutation polynomials of the form cx + Tr q l

Uniform Sublevel Radon-like Inequalities

Some Subcritical Estimates for the $$\ell ^p$$-Improving Problem for Discrete Curves

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