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\(\mathfrak{M}\)=Rn or\(\mathfrak{M}\) is a smooth closed manifold, A(t) and B(t) are pseudodifferential operators on\(\mathfrak{M}\), depending on t e [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 eR 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 σv,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).