L^p-estimates for the wave equation associated to the Grushin operator

Abstract

Let G:=-((d/dx)^2+x^2(d/du)^2) denote the Grusin operator on R^2. Consider the Cauchy problem for the associated wave equation on R x R^2, 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|_Lp^\alpha:=|(1+G)^{\alpha/2}f|_Lp, defined in terms of the differential operator G. Let S_C denote the strip S_C:={(x,u) in R^2, |x|<=C} in R^2. We prove that for 1<=p<=\infty the solution v is in L_p^{-\alpha} if our initial data f and g are Lp-functions supported in a fixed strip S_C, C>0, and if \alpha>|1/p-1/2| holds. In fact, we show that for every C>0 the operator \exp(itG^{1/2})(1+G)^{-\alpha/2}, defined for Schwartz functions, extends to a bounded operator from Lp(S_C) to Lp(R^2) for all \alpha>|1/p-1/2|.