Extensions of the Stein–Tomas Theorem

Abstract

Abstract. We prove an endpoint version of the Stein-Tomas restriction theorem, for a general class of measures, and with a strengthened Lorentz space estimate. A similar improvement is obtained for Stein's estimate on oscillatory integrals of Carleson-Sjölin-Hörmander type and some spectral projection operators on compact manifolds, and for classes of oscillatory integral operators with one-sided fold singularities.

The Fourier restriction and Kakeya problems over rings of integers modulo N

“…In order to understand the rôle of the scaling structure in both of these problems, it is useful to examine necessary conditions for the estimates (3) or (4) to hold when Σ is the paraboloid, as defined in (2). In this case (4), for instance, can be written as 4 1…”

“…Theorem 6.1 is, in fact, a special case of a more general result concerning L 2 Fourier restriction on locally compact abelian (LCA) groups. In particular, in [27] it is observed that an argument of Bak and Seeger [3] can be extended to a class of LCA groups which admit a primitive form of Littlewood-Paley theory. Unfortunately, the full details of the hypotheses of the main result in [27] are somewhat involved and are therefore not reproduced here.…”

Sharpness of the Mockenhaupt–Mitsis–Bak–Seeger restriction theorem in higher dimensions

“…In general terms, the Cantor construction proceeds as follows. Start with the interval [1,2]. Divide it into n 1 equal subintervals, select t 1 of them, and discard the rest.…”

$$(L^{r}, L^{s})$$(Lr,Ls) resolvent estimate for the sphere off the line $$\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$$1r-1s=2n

“…See also Bak-Seeger [1], and Carbery-Seeger-Waigner-Wright [4] which contains an abstract version of this result. Bourgain's interpolation technique will be used to prove restricted weak type inequality at the two endpoints of a fixed line segment in the proposition.…”

“…For the finite sum T (1) , we apply the first estimate (5), the positive exponential bound, in the lemma, and for the remaining infinite T (2) , we apply (6), the negative exponential bound. These give…”