Optimal forward and reverse estimates of Morawetz and Kato-Yajima type with angular smoothing index

2014/09/20 by Neal Bez, Mitsuru Sugimoto, Bez, Neal +1
Mathematics · #35B45 #35B65 #35P10 #Advanced Harmonic Analysis Research #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #math.AP #msc:35B45 #msc:35B65 #msc:35P10

paper · pdf · doi:10.48550/arxiv.1409.5843

19 pages

arxiv created 2014/09/20 · openalex publication_date 2014/09/20 · arxiv updated 2014/09/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For the solution of the free Schrödinger equation, we obtain the optimal constants and characterise extremisers for forward and reverse smoothing estimates which are global in space and time, contain a homogeneous and radial weight in the space variable, and incorporate a certain angular regularity. This will follow from a more general result which permits analogous sharp forward and reverse smoothing estimates and a characterisation of extremisers for the solution of the free Klein-Gordon and wave equations. The nature of extremisers is shown to be sensitive to both the dimension and the size of the smoothing index relative to the dimension. Furthermore, in four spatial dimensions and certain special values of the smoothing index, we obtain an exact identity for each of these evolution equations.

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