Damping oscillatory integrals with polynomial phase.

On X-ray transforms for rigid line complexes and integrals over curves in $\mathbb {R}^4$

“…The necessity of this condition is known and attributed to A. Carbery and M. Christ in [10]; one tests A on small balls and on small rectangles with dimensions δ × · · · × δ d . The conjecture is known in dimensions d = 2, 3; see Strichartz [20], Littman [9] for the case d = 2, and Oberlin [13] for the case d = 3.…”

A convolution estimate for a measure on a curve in $\mathbb {R}^4$

“…The method there was later used in [2], [8], [9], and [10] to treat more general curves in R 3 . The only nontrivial result for n > 3 is the fact, proved in [4], that the midpoint of S lies in T . The purpose of this paper is to prove a partial result if n = 4.…”

“…Then (see §4 of [4]) T is contained in the closed convex hull of S and {(0, 0), (1, 1)} if I is bounded, and in S if I is unbounded. If n = 2 or 3 these inclusions are equalities.…”