Abstract
We prove weighted \begin{document}$ L^2 $\end{document} estimates for the Klein-Gordon equation perturbed with singular potentials such as the inverse-square potential. We then deduce the well-posedness of the Cauchy problem for this equation with small perturbations, and go on to discuss local smoothing and Strichartz estimates which improve previously known ones.
Highlights
IntroductionOne of the key ingredients in the proof of Theorem 1.1 is weighted L2 estimates for solutions of the inhomogeneous Klein-Gordon equation ∂t2u − ∆u + u = F (x, t), which are obtained in Sections 2 and 3
Consider the Cauchy problem for the Klein-Gordon equation with a potential V : ∂t2u − ∆u + V (x)u + u = 0, u∂t(ux(,x0,)0=) f (x), = g(x), (1.1)where (x, t) ∈ Rn × R and ∆ is the n dimensional Laplacian.In [7], D’Ancona established the following smoothing estimate for the Klein-Gordon flow with |V | ∼ |x|−2 √|x|−1eit −∆+V +1f L2x,t f H1/2 (1.2)by extending Kato’s H-smoothing theory developed in [10, 11] to the flow
One of the key ingredients in the proof of Theorem 1.1 is weighted L2 estimates for solutions of the inhomogeneous Klein-Gordon equation ∂t2u − ∆u + u = F (x, t), which are obtained in Sections 2 and 3
Summary
One of the key ingredients in the proof of Theorem 1.1 is weighted L2 estimates for solutions of the inhomogeneous Klein-Gordon equation ∂t2u − ∆u + u = F (x, t), which are obtained in Sections 2 and 3. In [8] this is extended to the wave admissible pairs but with the stronger assumption (1.4) on the potential Compared with these previous results, our theorem improves the perturbation by the inverse-square potential c/|x|−2 and the pairs (q, r) for which the estimate holds. We denote A B to mean A ≤ CB with unspecified constants C > 0
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