Commutators of bilinear pseudodifferential operators and Lipschitz functions
For a given pseudodifferential operator \(T\) and a function \(a\), the commutator \([T,a]\) is given by \[ [T,a] (f) = T (af) - a T(f). \] Continuity properties of such commutators are intensively studied for different symbol classes of the operator \(T\) and different function spaces as their domains (\(a\) usually being of a Lipschitz class). Some of the main references are [\textit{R. R. Coifman} and \textit{Y. Meyer}, Au dela des opérateurs pseudo-différentiels. Société Mathématique de France, Paris (1978; Zbl 0483.35082)], [\textit{R. R. Coifman} and \textit{Y. Meyer}, Ann. Inst. Fourier 28, No. 3, 177--202 (1978; Zbl 0368.47031)], Zbl [\textit{P. Auscher} and \textit{M. E. Taylor}, Commun. Partial Differ. Equations 20, No. 9--10, 1743--1775 (1995; Zbl 0844.35149)], [\textit{M. Taylor}, Proc. Am. Math. Soc. 131, No. 5, 1501--1507 (2003; Zbl 1022.35096)], [\textit{M. E. Taylor}, Proc. Am. Math. Soc. 143, No. 12, 5265--5274 (2015; Zbl 1334.42034)]. An extension of the classical results to the bilinear case can be found in [\textit{Á. Bényi} and \textit{T. Oh}, J. Fourier Anal. Appl. 20, No. 2, 282--300 (2014; Zbl 1327.35471)]. The authors introduce bilinear versions of different classes of operators considered in [\textit{P. Auscher} and \textit{M. E. Taylor}, Commun. Partial Differ. Equations 20, No. 9--10, 1743--1775 (1995; Zbl 0844.35149)], and show that, modulo smoothing operator, those classes strictly contain the corresponding Hörmander class. In such situations bilinear versions of the main results from [loc. cit.] are given. Moreover, the results from [\textit{Á. Bényi} and \textit{T. Oh}, J. Fourier Anal. Appl. 20, No. 2, 282--300 (2014; Zbl 1327.35471)] are extended to a broader class of symbols. The paper is an important contribution to the theory of commutators for pseudodifferential operators.
scientific article; zbMATH DE number 6897260
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