Parametrix for a hyperbolic initial value problem with dissipation in some region

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We consider the initial value problem for a pseudodifferential equation with first order hyperbolic part, and an order γ> 0 dissipative term. Under an assumption, depending on an integer parameter L ≥ 2 such that 2 γ< L, we construct for this initial value problem a parametrix that is a Fourier integral operator of type ρ= 1 - γ/L. The assumption implies that where the principal symbol of the dissipative term is zero, the terms of order up to L-1 in its Taylor series also vanish.