Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces.
type: Item
instance of: scholarly article
title: Bilinear pseudodifferential operators with forbidden symbols on Lipschitz and Besov spaces.
zbMATH Open document ID: 1037.35113
DOI: 10.1016/S0022-247X(03)00245-2
author: Árpád Bényi
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-09-04T00:00:00Z
review text: The author considers bilinear pseudodifferential operators of the form \[ T_\sigma(f,g)(x)= \int_{{\mathbb R}^n} \int_{{\mathbb R}^n} \sigma(x,\xi,\eta) \widehat f(\xi)\widehat g(\eta)\,e^{ix\cdot(\xi+\eta)}\,d\xi\,d\eta \] with (Friedrichs-Kumano-go double) symbols in the class \(BS_{\rho,\delta}^m\) satisfying the inequalities \[ | \partial_x^\alpha \partial_\xi^\beta\partial_\eta^\gamma \sigma(x,\xi,\eta)| \leq C_{\alpha\beta\gamma} (1+| \xi| +| \eta| )^{m+\delta| \sigma| -\rho(| \beta| +| \gamma| )} \] for all \((x,\xi,\eta)\in{\mathbb R}^{3n}\). It is shown that \(T_\sigma\) with symbols in the (so-called exotic or forbidden) class \(BS_{1,1}^0\) are bounded on products of Lipschitz and Besov spaces. The proof relies on decomposition techniques related to Littlewood-Paley theory and Coifman-Mayer time-frequency analysis, thus reducing the problem to the study of bilinear pseudodifferential operators with elementary double symbols.
reviewed by: Rémi Vaillancourt
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47G30
Mathematics Subject Classification ID: 42B25
zbMATH DE Number: 1974960
zbMATH Keywords: pseudodifferential operators
zbMATH Keywords: bilinear operators
zbMATH Keywords: forbidden symbols
zbMATH Keywords: elementary symbols
zbMATH Keywords: Besov spaces
zbMATH Keywords: Lipschitz spaces
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1974960