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We consider a semiclassical matrix Schrödinger operator of the form
where , are real-analytic, admits a nondegenerate minimum at 0 with , is nontrapping at energy , and is a symmetric matrix of first-order pseudodifferential operators with analytic symbols. We also assume that . Then, denoting by the first eigenvalue of , and under some ellipticity condition on and additional generic geometric assumptions, we show that the unique resonance of such that (as ) satisfies
where is a symbol with , is the so-called Agmon distance associated with the degenerate metric , between 0 and , and , are integers that depend on the geometry.
We construct quasimodes for the Klein–Gordon equation on the black hole exterior of Kerr–AdS (anti- de Sitter) spacetimes. Such quasimodes are associated with time-periodic approximate solutions of the Klein–Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semiclassical nonlinear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semiclassical parameter. Our construction results in exponentially small errors in the semiclassical parameter. This implies that general solutions to the Klein Gordon equation on Kerr–AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound.
We prove a complete family of cylindrical estimates for solutions of a class of fully nonlinear curvature flows, generalising the cylindrical estimate of Huisken and Sinestrari [Invent. Math.175:1 (2009), 1–14, §5] for the mean curvature flow. More precisely, we show, for the class of flows considered, that, at points where the curvature is becoming large, an -convex () solution either becomes strictly -convex or its Weingarten map becomes that of a cylinder . This result complements the convexity estimate we proved with McCoy [Anal. PDE 7:2 (2014), 407–433] for the same class of flows.
We define variable parameter analogues of the affine arclength measure on curves and prove near-optimal -improving estimates for associated multilinear generalized Radon transforms. Some of our results are new even in the convolution case.
We use a wave packet transform and weighted norm estimates in phase space to establish propagation of singularities for solutions to time-dependent scalar hyperbolic equations that have coefficients of limited regularity. It is assumed that the second order derivatives of the principal coefficients belong to , and that is a solution to the homogeneous equation of global Sobolev regularity or 1. It is then proven that the wavefront set of is a union of maximally extended null bicharacteristic curves.
We establish, in a rather general setting, an analogue of DiPerna–Lions theory on well-posedness of flows of ODEs associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into .
When specialized to the setting of Euclidean or infinite-dimensional (e.g., Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of metric measure spaces, introduced by Ambrosio, Gigli and Savaré [DukeMath. J.163:7 (2014) 1405–1490] and the object of extensive recent research, fits into our framework. Therefore we provide, for the first time, well-posedness results for ODEs under low regularity assumptions on the velocity and in a nonsmooth context.