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

Kapitanskiĭ, L. V.

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

Keywords:

interpolation spaces; Besov spaces; decay estimate; Fourier multiplier; evolution equation; pseudodifferential operator; real principal symbol