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

TL;DR

This paper establishes L^p estimates for solutions to the wave equation associated with the Grushin operator, demonstrating how initial data smoothness and support influence solution regularity in Sobolev spaces.

Contribution

It proves boundedness of the wave propagator related to the Grushin operator on L^p spaces with initial data supported in a fixed strip, extending understanding of wave equations in subelliptic settings.

Findings

Solutions are in L_p^{-eta} if initial data are supported in a strip and eta > |1/p - 1/2|.

The operator ext{exp}(itG^{1/2})(1+G)^{-eta/2} extends to a bounded operator from Lp(S_C) to Lp(R^2).

Results hold for all p between 1 and infinity.

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|.