Weighted weak type inequalities for maximal commutators of Bochner-Riesz operator
Let \((B^a_r(f))^\land(\xi)=(1-r^2| \xi| ^2)^a_+\hat{f}(\xi)\), where \(r>0\) and \(a\in{\mathbb R}\). The maximal Bochner-Riesz operator \(B_{\ast}^a(f)\) is defined by \[ B_{\ast}^a(f)(x)=\sup_{r>0}\left| B_r^a(f)(x)\right| . \] Let \(b\in \text{BMO}({\mathbb R}^n)\), \(m\in{\mathbb N}\), \[ B^{m,\,a}_{r,\,b}(f)(x)=\int_{{\mathbb R}^n}B^a_r(x-y)f(y) [b(x)- b(y)]^m\,dy \] and \[ B^{m,\,a}_{\ast,\,b}(f)(x)=\sup_{r>0}\left| B^{m,\,a}_{r,\,b}(f)(x)\right| . \] In this paper, the authors prove that if \(a>(n-1)/2\), then the operator \(B^{m,\,a}_{\ast,\,b}\) satisfies some weighted endpoint estimate of \(L\,(\log L)\) type. Moreover, the authors also prove that if \(p\in (1,\,\infty)\) and \(a>n/p-(n+1)/2\), \(B^a_{\ast}\) and \(B^{m,\,a}_{\ast,\,b}\) are of weak weighted \((p,\,p)\)-type with two different weights.
scientific article; zbMATH DE number 1990099