Estimates of solutions of the Cauchy problem for Schrödinger type equations in the spaces \(L_ p^{\alpha}\) and \(B^{\alpha}_{pq}\). IV (Q1124035)

The following Cauchy problem for the Schrödinger type equations: \(i(du/dt)=P(D_ 1,D_ 2)u,\) \(t>0\), \(u(0)=u_ 0\in S(R^ 2)\), where P is a polynom with real constant coefficients, \(D_ k=i(\partial /\partial x_ k),\) \(k=1,2\); is considered. For a certain class of differential operators \(P(D_ 1,D_ 2)\) necessary and sufficient conditions for the existence of the bounds \(\| u(t)\|_{\Lambda^ 0_ p}\leq C_{\alpha}(t)\| u_ 0\|_{\Lambda_ p^{\alpha}},\) \(t\geq 0\), \(\forall u_ 0\in S(R^ 2)\), where \(\Lambda_ p^{\alpha}\) is either a Liouville space \(L_ p^{\alpha}(R^ 2)\) or a Besov space \(B^{\alpha}_{pq}(R^ 2)\), are received. The asymptotic behavior of the function \(C_{\alpha}(t)\), as \(t\to \infty\), is studied.