Consider the one-dimensional Schrödinger equation (1) i ∂u ∂t = ∂ 2 u ∂x 2 , with solution operator e −it∆. If u(t, x) is defined on R × S 1 , where S 1 = R/2πZ is the unit circle, then e −it∆ has a very complicated behavior. It turns out that, if t is a rational multiple of 2π, then (2) S(t, x) = e −it∆ δ(x) can be written as a finite linear combination of… (More)
CONTENTS Introduction. 0. Pseudodifferential operators and linear PDE. §0.1 The Fourier integral representation and symbol classes §0.2 Schwartz kernels of pseudodifferential operators §0.3 Adjoints and products §0.4 Elliptic operators and parametrices §0.5 L 2 estimates §0.6 Gårding's inequality §0.7 The sharp Gårding inequality §0.8 Hyperbolic evolution… (More)
This book is designed to provide graduate students and other researchers in dynamical systems theory with an introduction to the ergodic theory of Lebesgue spaces. The author's aim is to present a technically complete account which offers an in-depth understanding of the techniques of the field, both classical and modern. Thus, the basic structure theorems… (More)
We continue the work of Lopes Filho, Mazzucato and Nussenzveig Lopes  on the vanishing viscosity limit of circularly symmetric vis-cous flow in a disk with rotating boundary, shown there to converge to the inviscid limit in L 2-norm as long as the prescribed angular velocity α(t) of the boundary has bounded total variation. Here we establish convergence… (More)
This paper explores and ties together three themes. The first is to establish regularity of a metric tensor, on a manifold with boundary, on which there are given Ricci curvature bounds, on the manifold and its boundary, and a Lipschitz bound on the mean curvature of the boundary. The second is to establish geometric convergence of a (sub)sequence of… (More)
We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation ∆u−e 2u = K 0 (z) on general open surfaces. A few other topics are discussed, including boundary… (More)
In this paper we shall examine reflection of singularities of solutions of first-order equations of the form in a region 9 with boundary given by y=O; say 9 = X a9. Here G = G (y) = G (y , x , 0,) is a smooth one-parameter family of pseudo-differential operators of order one on a 9, G (y) E PS(1); u takes on values in a vector space, @, and G is a k X k… (More)
We show that, if U is the representation of SOe(n+1, 1) on L 2 (S n) given by (2.11), and P is a bounded operator on L 2 (S n), then P belongs to OP S 0 1,0 (S n) if and only if P (g) = U(g)P U(g) −1 is a C ∞ function on SOe(n + 1, 1) with values in the Banach space L(L 2 (S n)).