Energy Estimates for a Class of High-Order Hyperbolic Equations

The present paper is the first part of an investigation of some problems related to the solvability of high-order hyperbolic equations in spaces of functions bounded or almost-periodic with respect to time variable t. High-order operators are treated under the additional condition on lower terms: the full symbol of the operator has no zeros in a strip $\delta$_-< lm$\tau$ <$\delta$₊, where $\tau$ and t are dual variables, and $\delta$_\pm can assign the value \pm \infty . In this context Leray's separating operator method is developed and two-sided energy estimates in the case of constant coefficients are obtained. These estimates are extended to the operators with variable coefficients if the derivatives of the coefficients are sufficiently 'small".