2020/11/08 by Huang, Xiaoqi, Sogge, Christopher D.
#35 #58 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics
paper · doi:10.48550/arxiv.2011.04007
We generalize the Strichartz estimates for Schrödinger operators on compact manifolds of Burq, Gérard and Tzvetkov [10] by allowing critically singular potentials V. Specifically, we show that their 1/p--loss LptLqx(I× M)-Strichartz estimates hold for e-itHV when HV=-Δg+V(x) with V∈ Ln/2(M) if n≥3 or V∈ L1+δ(M), δ>0, if n=2, with (p,q) being as in the Keel-Tao theorem and I⊂ \mathbb R a bounded interval. We do this by formulating and proving new "quasimode" estimates for scaled dyadic unperturbed Schrödinger operators and taking advantage of the the fact that 1/q'-1/q=2/n for the endpoint Strichartz estimates when (p,q)=(2,2n/(n-2)). We also show that the universal quasimode estimates that we obtain are saturated on \em any compact manifolds; however, we suggest that they may lend themselves to improved Strichartz estimates in certain geometries using recently developed "Kakeya-Nikodym" techniques developed to obtain improved eigenfunction estimates assuming, say, negative curvatures.