Convolution with Measures on Polynomial Curves

Abstract

This paper is concerned with convolution estimates for certain measures on degenerate curves in R 2 and R 3 . Analogous estimates in R n , n ≥ 4, were recently obtained for the (nondegenerate) curve (t, t 2 , . . . , t n ) in [4] -see also [9] and [10]. Here is some of the history of this problem. Ideas going back to [6] show, for example, that if µ is the measure given by dt on the circle (cos(t), sin(t)) or on the parabola (t, t 2 ), thenAnd it is easy to see that these estimates are optimal -see [7] for more on this. The feature, common to these two curves, which in retrospect gives rise to (1) is the fact that on both of them the measure dt is a multiple of the measure κ 1 3 (s)ds where ds is arclength and κ is curvature. Drury [5] was the first to notice the importance of the measures µ given by dµ = κ 1 3 (s)ds in the context of (1). In particular, it was Drury's idea to obtain (1) for the measure dµ = κ 1 3 (s)ds on degenerate curves. His result (Theorem 1 in [5]) applies to curves of the form (t, p(t)), so that dµ = |p (t)| 1 3 dt, where the convex function p satisfies certain regularity conditions. The paper [8] contains a similar result, valid for any real-valued polynomial p. And that estimate is uniform for polynomials of a fixed degree. Theorem 1 below generalizes this: the estimate (1) holds for curves (p 1 (t), p 2 (t)) with dµ = κ 1 3 (s)ds if p 1 and p 2 are real-valued polynomials, and the convolution bounds are uniform in p 1 and p 2 if the degree of these polynomials is fixed.Part of the motivation for the above-mentioned work of Drury stems from the fact that convolution estimates for curves in R 2 can be used to obtain convolution estimates for curves in R 3 -see [7]. The main result in [7] is the following: suppose that p 1 (t) and p 2 (t) are polynomials and that the two vectors (p (j ) 1 (t), p (j ) 2 (t)), j = 1, 2, are linearly independent for every t ∈

Universal Lp improving for averages along polynomial curves in low dimensions

“…In this paper we continue an investigation by Oberlin to establish such a result for averaging operators along general polynomial curves in R d when d = 2 or d = 3 (in [22], the d = 2 case was fully resolved and partially resolved for d = 3). More specifically, if γ : I → R d parametrises a smooth curve in R d on an interval I , set…”

“…In this paper we give an alternative approach to the results in [22] and strengthen the threedimensional result to general polynomial curves P(t) = (P 1 (t), P 2 (t), P 3 (t)); furthermore all estimates will be uniform over the class of polynomials of bounded degree. Our hope is that this approach will generalise to general polynomials curves in all dimensions.…”

“…In [22] Oberlin established (1) in two dimensions for the family of polynomial curves γ (t) = P(t) = (P 1 (t), P 2 (t)) where each P 1 and P 2 is a general real polynomial of bounded degree. Specifically he established (1) with a constant C only depending on the degrees of the polynomials defining P. This is a natural class of curves to consider as the number of sign changes of L P is controlled by the degree of the polynomials P j .…”

“…Drury initiated an investigation along these lines for the problem of achieving precise regularity results for averages along curves or surfaces, in particular determining sharp L p → L q estimates, and this has been followed up by several authors; see for example [7,8,15,19,20,22,[24][25][26][27].…”

“…The key is now to show that (22) leads to a contradiction; this implies that (21) holds, and therefore…”