Commutators of multiplier operators
Given functions a and G on \({\mathbb{K}}^ K\) and m in \({\mathbb{N}}\), define the operator \(S^{(m)}\) on \(S({\mathbb{R}}^ K)\) by the formula \[ S^{(m)}f=\sum_{| \beta | <m}(\beta !)^{- 1}(X^{\beta}G)*((D^{\beta}a).f), \] where. and * denotes pointwise multiplication and convolution respectively, \((X^{\beta}G)(x)=x^{\beta}G(x)\), and \((D^{\beta}a)(x)=(\partial /\partial x)^{\beta}a(x)\). Formally, a.(G*f) is equal to \(S^{(\infty)}f.\) The paper under review obtains estimates for \[ \| a.(G*f)- S^{(m)}f\|_ p, \] in terms of \(\| f\|_ p\) and norms of derivatives of a, as well as estimates for iterates and derivatives of this difference. The techniques are generalizations of the work of \textit{R. R. Coifman} and \textit{Y. Meyer} [Au delà des opérateurs pseudodifferentiels, Astérisque 57, 185 p. (1978; Zbl 0483.35082)].
scientific article; zbMATH DE number 4000823