Smoothing and dispersive estimates for 1D Schrödinger equations with BV coefficients and applications

We prove smoothing estimates for Schrödinger equations i∂tϕ+∂x(a(x)∂xϕ)=0 with a(x)∈BV, real 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∈BV to be in a sense a minimal requirement. Finally, we provide an application to sharp well-posedness for a generalized Benjamin–Ono equation.