On the existence of limit values on the boundary for solutions of elliptic equations

Let $u(x)=u(x_1,\dots,x_n)$ be a solution of the elliptic equation
$$ \Delta^mu+P(i\partial/{\partial x})u=0, $$
in a strip, where $m\ge1$ and $P(\xi)$ is an arbitrary polynomial with constant coefficients of degree at most $2m-1$. Necessary and sufficient conditions are found for the existence of $W_2^k$-limit values of the function $u(x)$ on the boundary for arbitrary real $k$.