Boundedness for commutators of Bochner-Riesz operators below a critical index (Q2454655)

Let \(T\) be the Bochner-Riesz operator, \(\widehat{T(f)}=(1-| \xi| ^2)^\alpha_+\widehat{f}(\xi)\), and \(b\) be a function belonging to a homogeneous Lipschitz class, \(f\in\dot{\Lambda}_\beta\). The authors are interested in necessary conditions and sufficient conditions of the \(L^p\) boundedness of the commutators \([b,T]\) generated by the Bochner-Riesz operators below the critical index, \(\alpha=\frac{n-1}{2}\), and the Lipschitz functions. The necessary and sufficient condition is proved in the two-dimensional case. Some necessary conditions and sufficient conditions have also been found in higher dimensions.