Convolution with measures on flat curves in low dimensions

Abstract

We prove L p → L q convolution estimates for the affine arclength measure on certain flat curves in R d when d ∈ {2, 3, 4}. For d = 2, 3, we also establish certain related Lorentz space estimates.

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.…”

“…We denote by Δ the closed convex hull of {(0, 0), [5]. In this article, we establish uniform endpoint estimates on T σ γ for a wider class of curves g. Before we state our main result, we introduce certain conditions on functions defined on intervals.…”

Endpoint Lp→Lq bounds for integration along certain polynomial curves

“…Other results in a similar vein are due to Choi in [4,3] and by Pan in [24]. A recent preprint of Oberlin [19] establishes bounds for T P in certain non-polynomial cases for d = 2, 3, 4.…”

“…We have also made a minor change in the algorithm of Dendrinos and Wright by using the real numbers b j determined in previous steps to perform the decomposition in Case II of step n + 1. This alteration is miniscule, and moreover, the modified algorithm is needed only to establish the upper bound on the error terms in (19). Having bounded those error terms, one can use the original algorithm in [10] to prove Theorem 2, so this change does not cause any technical issues to arise.…”