Bases of splashes and linear operators in anisotropic Lizorkin-Triebel spaces. (English. Russian original) Zbl 0909.35161
Mishchenko, E. F. (ed.) et al., The theory of functions and differential equations. Collected papers. In honor of the ninetieth birthday of Academician Sergej Mikhailovich Nikol’skij. Birmingham, AL: MAIK Nauka/Interperiodica Publishing, Proc. Steklov Inst. Math. 210, 2-21 (1995); translation from Tr. Mat. Inst. Steklova 210, 5-30 (1995).
The authors consider Besov-Lizorkin spaces \(F^{(s)}_{pq}\) of anisotropic type, defined as in H. Triebel [Theory of function spaces, Birkhäuser/Basel (1983; Zbl 0546.46027)]. Continuity on spaces \(F^{(s)}_{pq}\) is studied for pseudo-differential operators \(p(x,D)\), with symbol \(p(x,\xi)\) satisfying anisotropic estimates of the form:
\[
| D^\alpha_x D^\beta_\xi p(x,\xi)|\leq c_{\alpha\beta}(1+[\xi])^{m-\langle\beta, M\rangle+ \delta\langle\alpha, M\rangle},
\]
where \(M= (M_1,\dots, M_n)\), \(\langle\gamma, M\rangle= \gamma_1M_1+\cdots+ \gamma_nM_n\), \(0\leq\delta< 1\), and
\[
[\xi]= \sum^n_{j= 1}|\xi_j|^{1/M_j}.
\]
Essential technical tool in the proof is the use of suitable bases of splashes.
For the entire collection see [Zbl 0838.00008].
For the entire collection see [Zbl 0838.00008].
Reviewer: L.Rodino (Torino)
MSC:
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
