Weak and strong type estimates for the multilinear pseudo-differential operators. (English) Zbl 1441.47058
This paper is devoted to the investigation of the boundedness of multilinear pseudodifferential operators as
\[
T_{\sigma}(\vec{f})(x) = \int_{(\mathbb{R}^n)^m} e^{2\pi i x\cdot (\xi_1 +\ldots + \xi_m)} \sigma(x,\vec{\xi}) \widehat{f_1}(\xi_1)\cdots \widehat{f_m}(\xi_m) \,d\vec{\xi},
\]
with symbols \(\sigma\) in suitable (multilinear versions of) Hörmander classes and, for locally integrable functions \(\mathbf{b}=(b_1,\ldots,b_m)\), of the commutators
\[
T_{\sigma,\mathbf{b}\Sigma}(\vec{f})(x) = \sum_{j=1}^m [b_j,T_{\sigma}]_j(\vec{f})(x),
\]
where each term is the commutator of \(b_j\) and \(T_{\sigma}\) in the \(j\)-th entry of \(T_{\sigma}\).
In more detail, the authors prove local exponential and subexponential decay estimates for \(T_{\sigma}(\vec{f})\) and \(T_{\sigma,\mathbf{b}\Sigma}(\vec{f})\), respectively. Subsequently, weighted mixed weak type inequalities for \(T_{\sigma}\) are derived, in the spirit of similar weak type estimates recently proved for Calderón-Zygmund and generalized maximal operators [A. K. Lerner et al., Adv. Math. 220, No. 4, 1222–1264 (2009; Zbl 1160.42009); C. Pérez, J. Funct. Anal. 128, No. 1, 163–185 (1995; Zbl 0831.42010)]. Techniques of sparse domination are used and also endpoint cases are encompassed. Moreover, classical weighted estimates for \(T_{\sigma}(\vec{f})\) and \(T_{\sigma,\mathbf{b}\Sigma}(\vec{f})\) are improved, namely, sharp results are obtained by means of dyadic analysis techniques.
In more detail, the authors prove local exponential and subexponential decay estimates for \(T_{\sigma}(\vec{f})\) and \(T_{\sigma,\mathbf{b}\Sigma}(\vec{f})\), respectively. Subsequently, weighted mixed weak type inequalities for \(T_{\sigma}\) are derived, in the spirit of similar weak type estimates recently proved for Calderón-Zygmund and generalized maximal operators [A. K. Lerner et al., Adv. Math. 220, No. 4, 1222–1264 (2009; Zbl 1160.42009); C. Pérez, J. Funct. Anal. 128, No. 1, 163–185 (1995; Zbl 0831.42010)]. Techniques of sparse domination are used and also endpoint cases are encompassed. Moreover, classical weighted estimates for \(T_{\sigma}(\vec{f})\) and \(T_{\sigma,\mathbf{b}\Sigma}(\vec{f})\) are improved, namely, sharp results are obtained by means of dyadic analysis techniques.
Reviewer: Salvatore Ivan Trapasso (Torino)
MSC:
| 47G30 | Pseudodifferential operators |
| 42B20 | Singular and oscillatory integrals (Calderón-Zygmund, etc.) |
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
| 47H60 | Multilinear and polynomial operators |
Keywords:
pseudo-differential operators; local decay estimates; sharp weighted inequalities; endpoint weak boundsReferences:
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