Abstract. Bilinear restriction estimates have been appeared in work of Bourgain, Klainerman, and Machedon. In this paper we develop the theory of these estimates (together with the analogues for Kakeya estimates). As a consequence we improve the (L p, L p) spherical restriction theorem of Wolff [27] from p> 42/11 to p> 34/9, and also obtain a sharp (L p, L q) spherical

Abstract. Recently Wolff [28] obtained a sharp L 2 bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of “elliptic surfaces ” such as paraboloids. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon the known restriction theory for the paraboloid and sphere.

this paper. The author also thanks the reviewer for some helpful remarks.

The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. In many cases the Euclidean arguments carry over easily to the finite setting (and are, in fact, somewhat cleaner), but there

Abstract. For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α> 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature. As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1, 2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the Laplace-Beltrami operator on Riemannian manifolds for p> 2. 1.

Abstract not found

Abstract. This is the fourth article of our series. Here, we apply the results of [AM1] to study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Poincaré inequalities. 1. Introduction and

In this note we improve the known L p-bounds for Bochner-Riesz operators and their maximal operators.

neretin mpim-bonn.mpg.de, neretin main.mccme.rssi.ru Consider a (not necessarily irreducible) unitary representation of a semisimple Lie group G in a Hilbert space H (for instance we can consider an action of the group G in some function space on some homogeneous space, a tensor product of irreducible unitary representations, a restriction of an irreducible unitary representation to a subgroup etc.). There arises a natural problem: to decompose our representation into irreducible representations, in other words, to find the spectrum of the representation. Problems of this type were widely investigated for the last 50 years (since [Kre]) and by now many such spectral problems have been solved completely or partially. In several simple cases the spectra are continuous and more or less ‘uniform’. But sometimes such spectra contain strange discrete increments which means that there exist minimal G-invariant subspaces in the Hilbert space H. First examples of discrete increments were observed in [Nai], [Puk], [H-Ch], [Mol1] (1961- 1966) and now there exists a large literature devoted to discrete spectra,

Abstract. We demonstrate the (H 1, L 1,2) or (L p, L p,2) mapping properties of several rough operators. In all cases these estimates are sharp in the sense that the Lorentz exponent 2 cannot be replaced by any lower number. 1.