Addition of Lower Order Terms to Weakly Hyperbolic Operators with Preservation of Their Type of Hyperbolicit

Abstract

For an m-homogeneous hyperbolic (with respect to the vector N) operator P_m, and a weight function g: 1) we find the conditions on the lower order terms {Q}, operators {P_m(D) + Q(D)} to become g-hyperbolic with respect to any vector N¹ from a neighborhood O(N) of the vector N, 2) we show that the operators obtained by adding lower order terms have fundamental solutions whose supports are in the cone from upper half-space \(\overline {{H_N}} : = \{ (x,N) \ge 0\} \), 3) we show that if P(D):= (P_m + Q)(D), f ∈ G^ℜ (where G^ℜ is some Gevrey type space) and supp f ⊂ H_N:= {(x, N) > 0}, the equation P(D)u = f has a solution u ∈ G^ℜ such that supp \(u \subset \overline {{H_N}} \).