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.

We prove that the maximal operator associated with variable homogeneous planar curves (t, ut α ) t∈R , α = 1 positive, is bounded on L p (R 2 ) for each p > 1, under the assumption that u : R 2 → R is a Lipschitz function. Furthermore, we prove that the Hilbert transform associated with (t, ut α ) t∈R , α = 1 positive, is bounded on L p (R 2 ) for each p > 1, under the assumption that u : R 2 → R is a measurable function and is constant in the second variable. Our proofs rely on stationary phase methods, T T * arguments, local smoothing estimates and a pointwise estimate for taking averages along curves.

The theory of Fourier integral operators is surveyed, with an emphasis on local smoothing estimates and their applications. After reviewing the classical background, we describe some recent work of the authors which established sharp local smoothing estimates for a natural class of Fourier integral operators. We also show how local smoothing estimates imply oscillatory integral estimates and obtain a maximal variant of an oscillatory integral estimate of Stein. Together with an oscillatory integral counterexample of Bourgain, this shows that our local smoothing estimates are sharp in odd spatial dimensions. Motivated by related counterexamples, we formulate local smoothing conjectures which take into account natural geometric assumptions arising from the structure of the Fourier integrals.

The multigene family which codes for the mouse major urinary proteins (MUPs) consists of approximately 35 genes. Most of these are members of two different groups, Group 1 and Group 2, which can be distinguished by nucleic acid hybridisation. By screening a Charon 4A library of mouse DNA with probes from the 5′‐flanking region of a MUP gene, we have isolated clones that contain both a Group 1 and a Group 2 gene, orientated in a divergent fashion, with 15 kb of DNA between the 5′ ends of the genes. We show that this pairwise arrangement is the predominant organisation of MUP genes in the BALB/c genome. We argue that the head‐to‐head gene pair is the unit both of DNA organisation and of evolution. Taking into account the genes themselves, the intervening 15 kb and the homologous 3′‐flanking regions, this unit is approximately 45 kb long. We also show that some MUP genes may be linked in a tail‐to‐tail fashion with 26‐28 kb between the 3′ ends of two genes. This suggests that the minimum distance between successive 45‐kb units is approximately 7 kb.

We consider a higher dimensional version of the Benjamin-Ono equation, ∂tu−R 1 ∆u+u∂x 1 u = 0, where R 1 denotes the Riesz transform with respect to the first coordinate. We first establish sharp space-time estimates for the associated linear equation. These estimates enable us to show that the initial value problem for the nonlinear equation is locally well-posed in L 2 -Sobolev spaces H s (R d ), with s > 5/3 if d = 2 and s > d/2 + 1/2 if d 3. We also provide ill-posedness results.With d = 1, the available local well-posedness theory has been based on compactness methods. Indeed, Molinet, Saut and Tzvetkov [19] proved that the problem cannot be solved in L 2 -Sobolev spaces H s by Picard iteration. We will show that

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