Hyperbolic operators in spaces of generalized distributions (Q1089583)

The main result of the paper is a characterization of hyperbolic operators in the space \({\mathcal D}'_{\omega}\) of Beurling generalized distributions. The notion of hyperbolicity with respect to \(t>0\) and \(t<0\) has been extended to convolution operators in \({\mathcal D}'_{\omega}\), and the following theorem which generalizes a result of Ehrenpreis has been proved. Let S be a convolution operator in \({\mathcal D}'_{\omega}\), the following are equivalent: (i) \(S\) is \(\omega\)-hyperbolic with respect to \(t>0.\) (ii) \(S\) is invertible and there exist constants \(C\), \(A\), \(M\) and \(A_ 1\) so that \[ \Im \tau \leq A(| \Im z| +\frac{1}{2(3+2MA_ 1)}\omega (\Re z,\Re \tau)) \] for all \((z,\tau)\in {\mathbb{C}}^ n\times {\mathbb{C}}\); \(\hat S(z,\tau)=0.\) (iii) There exist positive constants \(C\), \(a\) and \(A\) so that \[ | \hat S(z,\tau)| \geq C\exp (-a[| Im z| +| Im \tau | +\omega (Re z,Re \tau)], \] whenever Im \(\tau > A[| \Im z| +\omega(z,\tau)]\); \((z,\tau)\in {\mathbb{C}}^ n\times {\mathbb{C}}.\)