A resolvent estimate and a smoothing property of inhomogeneous Schrödinger equations (Q1281915)

Let \(p(\xi)> 0\) be of the class \(C^\infty(\mathbb{R}^n\setminus 0)\) and positively homogeneous of degree 1, and \(P= p(D_x)= {\mathcal F}^{-1}_\xi p(\xi){\mathcal F}_x\) the corresponding Fourier multiplier. Suppose that \(\sum= \{\xi; p(\xi)= 1\}\) has nonvanishing Gaussian curvature. The objective of this brief article is to show the following smoothing effect of inhomogeneous generalized Schrödinger equations: Theorem: Suppose \(1-n/2< s< 1/2\), \(1- n/2< \alpha< 1/2\) and let \(| x|^{1-s}f(t, x)\in L^2(\mathbb{R}_t\times \mathbb{R}^n_x)\). Then there exists a unique solution \(u(t,x)\) to \[ (\partial_t+ iP^2)u= f,\quad u|_{t= 0}= 0 \] which satisfies \(| x|^{\alpha- 1}| D_x|^{s+\alpha} u(t,x)\in L^2(\mathbb{R}_t\times \mathbb{R}^n_x)\).