On some \(L^ p\)-estimates for hyperbolic equations
The author sets for \(k\in \mathbb{R}^ 1\), \(\xi\in \mathbb{R}^ n\): \(m_ k(\xi)=e^{i\varphi(\xi)} a_ k(\xi)\), where \(\varphi(\xi)>0\) is homogeneous of degree 1 and real analytic at \(\xi\neq 0\), \(a(\xi)\in S^{-k}\), \(0\not\in\text{supp }a_ k\), and \(S^{-k}\) is the class of smooth functions which satisfy the estimates \(| D^ \alpha a(\xi)|\leq C_ \alpha(1+|\xi|)^{-k-|\alpha|}\) for all derivations \(D^ \alpha\). Further \(M_ k(D)=F^{-1}m_ k F\) and \(\Sigma=\{\xi\in\mathbb{R}^ n\); \(\varphi(\xi)=1\}\); \(F\) denotes the Fourier transform, and \(F^{-1}\) its inverse. It is proved: If the hypersurface \(\Sigma\) is strictly convex and \(k\geq(n-1)|(1/p)-(1/2)|\), the operator \(M_ k(D)\) is bounded on the Sobolev spaces \(H_ p^ s(\mathbb{R}^ n)\) (\(1<p<+\infty\), \(s\in\mathbb{R}^ 1\)) and the Besov spaces \(B_{p,q}^ s(\mathbb{R}^ n)\) (\(1\leq p,q\leq+\infty\), \(s\in\mathbb{R}^ 1\)).
scientific article; zbMATH DE number 91596
scientific article