\(L^p-L^q\) estimates for dispersive equations and related applications (Q1025053)

The authors are interested in \(L^p-L^q\) estimates for the solutions to the following Cauchy problem \[ \partial_tu=iP(D)u,\;u(0,x)=x_0(x),\;x\in\mathbb{R}^n. \] Here \(P:\mathbb{R}^n\to\mathbb{R}\) is a real elliptic polynomial of order \(m\geq 2\) with principal part \(P_m(\xi)\), without loss of generality \(P_m(\xi)>0\) for \(\xi\neq 0\). There exist results for the non-degenerate case of \(P\), that is, the Hessian matrix \(HP_m\) of \(P_m\) satisfies \(HP_m\neq 0\) for \(\xi\neq 0\). This condition is equivalent to the condition \((H_1)\): \(1/\lambda_k(\xi)=O(|\xi|^{-(m-2)})\) as \(|\xi|\to\infty\), here \(\lambda_k=\lambda_k(\xi)\) is the \(k\)-th eigenvalue of \(HP\). In the present paper the authors devote to the degenerate case \((H_b)\), \(b\in(0,1)\): \(1/\lambda_k(\xi)=O(|\xi|^{-(m-2)b})\). By using well-known tools of harmonic analysis they study oscillatory integrals. The most interesting case is the behavior of oscillatory integrals around stationary points of the phase function, here stationary phase method is applied. The paper is completed by considerations to resolvent estimates for higher-order Schrödinger operators \(iP(D)+V(x,D)\) with a special potential \(V(x,D)\).