The sharp range of L p -estimates for the class of Hörmander-type oscillatory integral operators is established in all dimensions under a positivedefinite assumption on the phase. This is achieved by generalising a recent approach of the first author for studying the Fourier extension operator, which utilises polynomial partitioning arguments. The main result implies improved bounds for the Bochner-Riesz conjecture in dimensions n ě 4.3 Strictly speaking, in [9] weaker L 8´Lp bounds are proven, but the methods can be used to establish the L p´Lp strengthening: see, for instance, [14, §9] where the L p´Lp argument appears (although in a slightly disguised form). 4 In particular, Lee [19] proved that for positive-definite phases (1.4) holds for p ě 2¨n`2 n in all dimensions, extending the range in Theorem 1.1 when n is odd.

The sharp Wolff-type decoupling estimates of Bourgain-Demeter are extended to the variable coefficient setting. These results are applied to obtain new sharp local smoothing estimates for wave equations on compact Riemannian manifolds, away from the endpoint regularity exponent. More generally, local smoothing estimates are established for a natural class of Fourier integral operators; at this level of generality the results are sharp in odd dimensions, both in terms of the regularity exponent and the Lebesgue exponent.2010 Mathematics Subject Classification. Primary: 35S30, Secondary: 35L05. 1 3 Such inequalities are also conjectured to hold at the endpoint (that is, the case σ = 1/p) and endpoint estimates have been obtained for a further restricted range of p in high-dimensional cases: see [24] and [29].4 The examples in [32] concern certain oscillatory integral operators of Carleson-Sjölin type, defined with respect to the geodesic distance on M . Their results lead to counterexamples for local smoothing estimates via a variant of the well-known implication "local smoothing ⇒ Bochner-Riesz". Implications of this kind will be discussed in detail in §4.

We consider Guth's approach to the Fourier restriction problem via polynomial partitioning. By writing out his induction argument as a recursive algorithm and introducing new geometric information, known as the polynomial Wolff axioms, we obtain improved bounds for the restriction conjecture, particularly in high dimensions. Consequences for the Kakeya conjecture are also considered.