Strong correctness criterion for non-local two-point boundary value problem in the band (Q1311159)

The problem \[ u_t(x,t) = P(- iD_x) u(x,t),\;x \subset \mathbb{R}^n,\;t \in [0,T] \] is considered where \(P\) is an arbitrary polynomial with constant complex coefficients and \(u\) satisfies the nonlocal boundary condition \(Au(x,0) + Bu(x,T) = u_0(x)\) for a given function \(u_0\). Necessary and sufficient conditions are obtained for the problem to possess a smoothing property, i.e. \[ \bigl |u(t) \bigr |_m \leq C(m) |u_0 |_0 \quad \text{for} \quad t \in (0,T) \] where \(|f |_m\) stands for the norm in \(C^m (\mathbb{R}^n)\). Moreover, analogous estimates are deduced in certain weighted norms.