**Memoirs of the American Mathematical Society**

2015;
80 pp;
Softcover

MSC: Primary 35; 42;
Secondary 53

**Print ISBN: 978-1-4704-1873-1
Product Code: MEMO/241/1142**

List Price: $73.00

AMS Member Price: $43.80

MAA Member Price: $65.70

**Electronic ISBN: 978-1-4704-2877-8
Product Code: MEMO/241/1142.E**

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AMS Member Price: $43.80

MAA Member Price: $65.70

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

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*Roland Donninger; Joachim Krieger*

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 “distorted” 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, “Codimension one stability of the catenoid under the vanishing mean curvature flow in Minkowski space”, preprint arXiv:1310.5606 (2013).

#### Table of Contents

# Table of Contents

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

- Cover Cover11
- Title page i2
- Chapter 1. Introduction 18
- Chapter 2. Weyl-Titchmarsh Theory for 𝐴 916
- 2.1. Zero energy solutions 916
- 2.2. Perturbative solutions for small energies 1017
- 2.3. The Jost function at small energies 1118
- 2.4. The Jost function at large energies 1219
- 2.5. The Wronskians 1320
- 2.6. Computation of the spectral measure 1522
- 2.7. Global representations for 𝜑(⋅,𝜆) 1623
- 2.8. The distorted Fourier transform 1724

- Chapter 3. Dispersive Bounds 1926
- Chapter 4. Energy Bounds 2734
- Chapter 5. Vector Field Bounds 3340
- 5.1. The operator 𝐵 3340
- 5.2. Preliminaries from distribution theory 3441
- 5.3. The kernel of 𝐵 away from the diagonal 3542
- 5.4. Bounds for 𝐹 3643
- 5.5. Representation as a singular integral operator 3845
- 5.6. The diagonal part 4047
- 5.7. Boundedness on weighted spaces 4350
- 5.8. Basic vector field bounds 4552
- 5.9. Bounds involving the ordinary derivative 4754

- Chapter 6. Higher Order Vector Field Bounds 5360
- Chapter 7. Local Energy Decay 6168
- Chapter 8. Bounds for Data in Divergence Form 7178
- Bibliography 7784
- Back Cover Back Cover192