Smoothing And Dispersive Estimates For 1d Schr\"odinger Equations With
BV Coefficients And Applications
Fabrice Planchon
Paris 13
Abstract:
We prove smoothing estimates for Schr\"odinger equations $i\partial_t
\phi+\partial_x (a(x) \partial_x \phi) =0$ with $a(x)\in \mathrm{BV}$,
the space of functions with bounded total variation, real, positive
and bounded from below. We then bootstrap these estimates to obtain
optimal Strichartz and maximal function estimates, all of which turn
out to be identical to the constant coefficient case. We also provide
counterexamples showing $a\in \mathrm{BV}$ to be a minimal
requirement.
http://arxiv.org/abs/math.AP/0409379