Oscillatory integrals and multiplier problem for the disc

A bilinear approach to the restriction and Kakeya conjectures

“…[26]). This conjecture has been verified for n = 2 [7], but remains open in higher dimensions. The main difficulty lies in making the q exponent as low as possible; the estimate is trivial for q = ∞, Hölder's inequality can be used to raise p, and in certain cases factorization theory can be used to lower p. When (p, q) lie on the sharp line…”

Bochner-Riesz means with respect to a rough distance function

“…Note that E has as its multiplier a nice Schwartz function, so E is trivially bounded for all p. Comparing the definition of E with (2.72), one sees that the composed operator F −1 [µ 3,3 Ef ] differs from C only by a simple operator, which we will denote I:…”

“…Specifically, we have (2.15) for all p 0 when d = 2 (see [3], [4]) and for d ≥ 3 we have it for p 0 > 2 + 4 d due to Lee [10]. For the bound (2.16), we use a trivial bound in R 2 × R and R 3 × R, and the bound of Laba and Wolff [9] for d ≥ 4.…”

“…In 1972, Carleson and Sjölin verified the conjecture in R 2 [3]. In 1973, Fefferman showed a connection between the restriction conjecture for the sphere and the Bochner-Riesz conjecture [7].…”

Lebesgue space estimates for a class of Fourier integral operators associated with wave propagation

“…We formulate a curvature hypothesis which appeared in [3], [10] for classes of oscillatory integral operators (see [13], [2] for current results on these classes). We follow the exposition in [14] and impose conditions on the following projection maps.…”