## Some generalizations of the Strichartz-Brenner inequality.(English. Russian original)Zbl 0732.35118

Leningr. Math. J. 1, No. 3, 693-726 (1990); translation from Algebra Anal. 1, No. 3, 127-159 (1989).
The work is devoted to some generalizations of the Strichartz-Brenner inequalities for the Besov spaces. The classical Strichartz-Brenner inequality gives a decay estimate of the Fourier multiplier $$| \xi |^{-\nu} \exp (it| \xi |)$$ acting from the Besov space $$B^ r_{p',q}({\mathbb R}^ n)$$ into $$B^ r_{p,q}({\mathbb R}^ n)$$. Here $$p$$ and $$p'$$ are related by $$1/p+1/p'=1$$. More precisely, we have the inequality $(1)\quad \| {\mathcal F}^{-1}(| \xi |^{-\nu} \exp (it| \xi |)\hat f(\xi))\|_{B^ r_{p,q}({\mathbb R}^ n)}\leq C | t|^{-\lambda (n,\nu,p)}\| f\|_{B^ r_{p',q}({\mathbb R}^ n)},$ where $$\lambda (n,\nu,p)=(n(p-2)/p)-\nu$$ and $$((n+1)/2)((p-2)/p)\leq \nu \leq n(p-2)/p.$$
A typical application of this estimate is connected with the evolution equation $$\partial_ tu=i\sqrt{-\Delta}u$$, where $$\Delta$$ is the Laplace operator in $$\mathbb R^ n$$.
The author studies the more general case of evolution equation $$\partial_ tu=A(t)u$$, where $$A(t)$$ is a classical pseudodifferential operator with real principal symbol. Under suitable assumption on this principal symbol estimates similar to (1) are established.
Reviewer: V.Georgiev (Sofia)

### MSC:

 35S10 Initial value problems for PDEs with pseudodifferential operators 35S30 Fourier integral operators applied to PDEs 35L45 Initial value problems for first-order hyperbolic systems