2007/09/14 by Ralf Meyer, Meyer, Ralf · 2 citations
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.AP #math.SP
paper · pdf · doi:10.48550/arxiv.0709.2188
105 pages, 7 figures
arxiv created 2007/09/14 · openalex publication_date 2007/09/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G:=-((d/dx)2+x2(d/du)2) denote the Grusin operator on R2. Consider the Cauchy problem for the associated wave equation on R x R2, given by ((d/dt)2+G)v =0, v(0,.)=f, d/dt v(0,.)=g, where t denotes time and f, g are suitable functions. The focus of this thesis lies on smoothness properties of the solution v for fixed time t with respect to the initial data. Smoothness can be measured in terms of Sobolev norms |f|Lpα:=|(1+G)α/2f|Lp, defined in terms of the differential operator G. Let SC denote the strip SC:=(x,u) in R2, |x|<=C in R2. We prove that for 1<=p<=∞ the solution v is in Lp-α if our initial data f and g are Lp-functions supported in a fixed strip SC, C>0, and if α>|1/p-1/2| holds. In fact, we show that for every C>0 the operator exp(itG1/2)(1+G)-α/2, defined for Schwartz functions, extends to a bounded operator from Lp(SC) to Lp(R2) for all α>|1/p-1/2|.