## Estimates for the solutions of the Cauchy problem for a hyperbolic equation of higher order in spaces $$H_ p$$.(Russian)Zbl 0605.35049

Continuing former investigations of A. Miyachi on estimates for the wave equation in the spaces $$L_ p$$ and $$H_ p$$ [see J. Fac. Sci., Univ. Tokyo, Sect. I A 27, 331-354 (1980; Zbl 0437.35042)] here sufficient and necessary conditions for the existence of a-priori $$H_ p$$-estimates (resp. for $$p>1$$ $$L_ p$$-estimates) for the solutions of a Cauchy problem for a hyperbolic equation of fourth order and their derivatives by $$H_ p$$-norms (resp. for $$p>1$$ $$L_ p$$-norms) are proved.
The proof is carried out using properties of Fourier $$H_ p$$- multipliers.
Reviewer: K.Barckow

### MSC:

 35L30 Initial value problems for higher-order hyperbolic equations 35B45 A priori estimates in context of PDEs 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems

Zbl 0437.35042
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