Abstract. We prove an abstract Strichartz estimate, which implies previously unknown endpoint Strichartz estimates for the wave equation (in dimension n 4) and the Schrödinger equation (in dimension n 3). Three other applications are discussed: local existence for a nonlinear wave equation; and Strichartz-type estimates for more general dispersive equations and for the kinetic transport equation. 1. Introduction. In

A systematic treatment of multilinear Calderón-Zygmundoperators is presented. The theory developed includes strong type and endpoint weak type estimates, interpolation, the multilinear T1 theorem, anda variety of results regarding multilinear multiplier operators.

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

Let (X, dµ) be a measure space, H a Hilbert space and σ> 0. Consider a family of linear operators U(t) : H → L2 X defined for each t ∈ R. Let U ∗ (t) : L2 X → H be the adjoint of U(t). We assume that the family U(t) satisfies

We show that any L1 embedding of the transportation cost (a.k.a. Earthmover) metric on probability measures supported on the grid {0, 1,..., n} 2 ⊆ R 2 incurs distortion Ω � � log n �. We also use Fourier analytic techniques to construct a simple L1 embedding of this space which has distortion O(log n). 1

We study general spectral multiplier theorems for self-adjoint positive definite operators on L²(X, µ), where X is any open subset of a space of homogeneous type. We show that the sharp Hörmander-type spectral multiplier theorems follow from the appropriate estimates of the L² norm of the kernel of spectral multipliers and the Gaussian bounds for the corresponding heat kernel. The sharp Hörmander-type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the Riesz means summability, which we also study here. We discuss several examples, which include sharp spectral multiplier theorems for a class of scattering operators on R³ and new spectral multiplier theorems for the Laguerre and Hermite expansions.

this paper we continue the study of functional calculus for the Laplace-- Beltrami operator on symmetric spaces of the noncompact type begun in [3]; this paper is dedicated to a study of the Poisson semigroup, which we define shortly. Let G and K be a connected noncompact semisimple Lie group with finite center and a maximal compact subgroup thereof, and consider the symmetric space G=K; also denoted by X: We denote by n the dimension of X; by

Abstract. Given a fixed p � = 2 we prove a simple and effective characterization of all radial multipliers of FL p (R d) provided that the dimension d is sufficiently large. The method also yields new L q space-time regularity results for solutions of the wave equation in high dimensions.

Abstract. In this paper, we provide a version of the Mihlin-Hörmander multiplier theorem for multilinear operators in the case where the target space is L p for p ≤ 1. This extends a recent result of Tomita [15] who proved an analogous result for p> 1. 1.

Abstract. We show that if L is a second-order uniformly elliptic operator in divergence form on R d, then C1(1 + |α|) d/2 ≤ ‖Liα‖L1→L1,∞ ≤ C2(1 + |α|) d/2. We also prove that the upper bounds remain true for any operator with the finite speed propagation property. 1. Introduction. Assume that aij ∈ C ∞ (R d), aij = aji for 1 ≤ i, j ≤ d and that κI ≤ (aij) ≤ τI for some positive constants κ and τ. We define a positive self-adjoint operator L on L 2 (R d) by the formula (1) L = − ∑ ∂iaij∂j. We refer readers to [8] for the precise definition and basic properties of L. In particular, L