Norm estimates in besov and Lizorkin-Triebel spaces for the solutions of second-order linear hyperbolic equations

Abstract

In the paper one considers the nonhomogeneous hyperbolic equation

$$\partial _t^2 u + iB\left( t \right)\partial _t u + A\left( t \right)u = h$$

on$\left[ {0, T} \right] \times \mathfrak{M}$, where$\mathfrak{M}$=R ⁿ or$\mathfrak{M}$ is a smooth closed manifold, A(t) and B(t) are pseudodifferential operators on$\mathfrak{M}$, depending on t ε [0, T], of orders 2 and 1, respectively. For the solutions of equation (1) for small t one establishes estimates of the form

With arbitrary r εR and integer ℓ≥0, where for G:,. and E:,. one can take the Besov space B:,.$\left( \mathfrak{M} \right)$ or the Lizorkin-Triebet space F:,.$\left( \mathfrak{M} \right)$, depending on the values n, v, p, q₁ q₂, and the “Brenner number” m, which are determined from the principal symbols of the operators A(0) and B(0); also the actual form of the scalar function σ_{v,p, n} (t) depends on n, v, p, q₁ q₂, and m: it may be power-like ∣t∣^v−n+2n/p, or logarithmic ¦log¦t¦∥, or a constant. In addition, one obtains estimates of the form

characterizing the integrability properties over timespace and the smoothing (for t>0) of the solutions of equation (1).