A Vector Field Method on the Distorted Fourier Side and Decay for Wave Equations with Potentials.

Cover -- Title page -- Chapter 1. Introduction -- 1.1. Main results -- 1.2. Related work -- 1.3. Method of proof -- 1.4. Further discussion -- 1.5. Notations and conventions -- Chapter 2. Weyl-Titchmarsh Theory for -- 2.1. Zero energy solutions -- 2.2. Perturbative solutions for small energies -- 2.3. The Jost function at small energies -- 2.4. The Jost function at large energies -- 2.5. The Wronskians -- 2.6. Computation of the spectral measure -- 2.7. Global representations for (⋅, ) -- 2.8. The distorted Fourier transform -- Chapter 3. Dispersive Bounds -- 3.1. Fundamental dispersive estimate -- 3.2. Improved decay -- 3.3. Comparison with the free case -- Chapter 4. Energy Bounds -- 4.1. Properties of the distorted Fourier transform -- 4.2. Generalized energy bounds -- Chapter 5. Vector Field Bounds -- 5.1. The operator -- 5.2. Preliminaries from distribution theory -- 5.3. The kernel of away from the diagonal -- 5.4. Bounds for -- 5.5. Representation as a singular integral operator -- 5.6. The diagonal part -- 5.7. Boundedness on weighted spaces -- 5.8. Basic vector field bounds -- 5.9. Bounds involving the ordinary derivative -- Chapter 6. Higher Order Vector Field Bounds -- 6.1. More commutator estimates -- 6.2. Higher order vector field bounds -- 6.3. The inhomogeneous problem -- Chapter 7. Local Energy Decay -- 7.1. Basic local energy decay -- 7.2. Bounds involving the scaling vector field -- 7.3. The inhomogeneous case -- Chapter 8. Bounds for Data in Divergence Form -- 8.1. Bounds for the sine evolution -- 8.2. Bounds involving the scaling vector field -- 8.3. Bounds for the inhomogeneous problem -- Bibliography -- Back Cover.

The authors study the Cauchy problem for the one-dimensional wave equation \partial_t^2 u(t,x)-\partial_x^2 u(t,x)+V(x)u(t,x)=0. The potential V is assumed to be smooth with asymptotic behavior V(x)\sim -\tfrac14 |x|^{-2}\mbox{ as } |x|\to \infty. They derive dispersive estimates, energy estimates, and estimates involving the scaling vector field t\partial_t+x\partial_x, where the latter are obtained by employing a vector field method on the âeoedistortedâe Fourier side. In addition, they prove local energy decay estimates. Their results have immediate applications in the context of geometric evolution problems. The theory developed in this paper is fundamental for the proof of the co-dimension 1 stability of the catenoid under the vanishing mean curvature flow in Minkowski space; see Donninger, Krieger, Szeftel, and Wong, âeoeCodimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski spaceâe, preprint arXiv:1310.5606 (2013).

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