Here $K_R(\cdot,\cdot;h)\in C^\infty(\mathbb{R}^n\times\mathbb{R}^n)$ is complex-valued. For $r\in\mathbb{R}$, define $H_h^r(\mathbb{R}^n)$ by the norm
where $\widehat{v}$ denotes the Fourier transform of $v$.
paragraph
admin
Assume that $K_R$ is rapidly smoothing locally in the following sense: for every compact set $L\subset\mathbb{R}^n\times\mathbb{R}^n$, every pair of multi-indices $\alpha,\beta\in\mathbb{N}_0^n$, and every $N\in\mathbb{N}$, there exists a constant $C_{L,\alpha,\beta,N}>0$ such that
Then for every $s,t\in\mathbb{R}$, every $N\in\mathbb{N}$, and every $\chi,\psi\in C_c^\infty(\mathbb{R}^n)$, there exists a constant $C_{s,t,N,\chi,\psi}>0$ such that