# Lp-estimates for the wave equation associated to the Grusin operator

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 $|f|_{L_p^\alpha}:=|(1+G)^{\alpha/2}f|_{L_p}$, 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 $L_p$-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 $L_p(S_C)$ to $L_p(R^2)$ for all $\alpha>|1/p-1/2|$.

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Meyer

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