The L^p-L^p mapping properties of convolution operators with the affine arclength measure on space curves

Abstract

The L p -improving properties of convolution operators with measures supported on space curves have been studied by various authors. If the underlying curve is non-degenerate, the convolution with the (Euclidean) arclength measure is a bounded operator from L 3/2 (R 3 ) into L 2 (R 3 ). Drury suggested that in case the underlying curve has degeneracies the appropriate measure to consider should be the affine arclength measure and he obtained a similar result for homogeneous curves t i-> (/, t 2 , t k ), / > 0 for k > 4. This was further generalized by Pan to curves t i->-(/, t k ,«'), / > 0 for 1 < Jt < /, it + / > 5. In this article, we will extend Pan's result to (smooth) compact curves of finite type whose tangents never vanish. In addition, we give an example of a flat curve with the same mapping properties.2000 Mathematics subject classification: primary 42B15; secondary 42B10.

Convolution estimates related to space curves

“…The theorem generalizes many results previously known for convolution estimates related to space curves, namely [1][2][3][4][5][6]. This article is organized as follows: in the following section, a uniform estimate for convolution operators with measures supported on plane curves.…”

“…have been studied by many authors [1][2][3][4][5][6][7][8]. The use of the affine arclength measure was suggested by Drury [2] to mitigate the effect of degeneracy and has been helpful to obtain uniform estimates.…”

Convolution with Measures on Polynomial Curves

“…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.…”

“…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.…”

Two estimates for curves in the plane

“…Drury ([3]) pointed out that the damping factor φ (x) 1/3 could compensate for flatness in such estimates. But his and subsequent results (see, e.g., [1]) gave bounds depending on certain ancillary constants. The (quite simple) proofs of Theorems 1 and 2 have no such dependence.…”