Pseudodifferential operators on localized Besov spaces. (English) Zbl 1303.47068
The authors consider pseudodifferential operators \(\sigma(x,D)\) with symbol \(\sigma(x,\eta)\in S^m_{1,\delta}\), \(0\leq\delta< 1\), satisfying the additional condition
\[
|D^\alpha_\eta D^\beta_x\sigma(x+ h,\eta)- D^\alpha_\eta D^\beta_x \sigma(x,\eta)|\leq C_{\alpha\beta} \lambda(|h|\,|\eta|^\delta) (1+|\eta|)^{m-|\alpha|+ \delta|\beta|},
\]
where \(\lambda\) is a positive, nondecreasing and concave function on \([0,+\infty)\), called the modulus of continuity.
Sharp conditions on \(\lambda\) are given, granting the boundedness of \(\sigma(x,D)\) from \((B^{s+m}_{p,q}(\mathbb{R}^n))_{\ell^r}\) to \((B^s_{p,q} (\mathbb{R}^n))_{\ell^r}\). The case \(\delta=1\) was already treated by G. Bourdaud and M. Moussai [Bull. Sci. Math., II. Sér. 112, No. 4, 419–432 (1988; Zbl 0675.35092)].
Sharp conditions on \(\lambda\) are given, granting the boundedness of \(\sigma(x,D)\) from \((B^{s+m}_{p,q}(\mathbb{R}^n))_{\ell^r}\) to \((B^s_{p,q} (\mathbb{R}^n))_{\ell^r}\). The case \(\delta=1\) was already treated by G. Bourdaud and M. Moussai [Bull. Sci. Math., II. Sér. 112, No. 4, 419–432 (1988; Zbl 0675.35092)].
Reviewer: Luigi Rodino (Torino)
MSC:
| 47G30 | Pseudodifferential operators |
| 47B38 | Linear operators on function spaces (general) |
| 35S05 | Pseudodifferential operators as generalizations of partial differential operators |
Keywords:
pseudodifferential operators; Besov spaces; localized Besov spaces; pointwise multipliers spacesCitations:
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