Boundedness of commutators of high-dimensional Hausdorff operators (Q416331)

The one-dimensional Hausdorff operator is defined as NEWLINE\[NEWLINEh_{\Phi}f(x)=\int_{0}^{\infty}\frac{\Phi(t)}{t}f\left(\frac{x}{t}\right)dt,NEWLINE\]NEWLINE where \(\Phi\) is a locally integrable function on \((0,\infty)\). Recently, several multidimensional extensions have been introduced and studied.NEWLINENEWLINEOne of them is the operator NEWLINE\[NEWLINEH_{\Phi}f(x)=\int_{\mathbb{R}^n}\frac{\Phi\left(|y|^{-1}x\right)}{|y|^n}f(y)\, dy,NEWLINE\]NEWLINE where \(\Phi\) is a radial function defined on \(\mathbb{R}^+\). Replacing \(\Phi\) with NEWLINE\[NEWLINE\Phi_1(t)=t^{-n}\chi_{(1,\infty)}(t),\;\;\;\;\;\Phi_2(t)=\chi_{(0,1)}(t),NEWLINE\]NEWLINE \(H_{\Phi}f\) becomes the high dimensional Hardy operator NEWLINE\[NEWLINEHf(x)=\frac{1}{|x|^n}\int_{|y|<|x|}f(y)\, dy,\;\;\;x\in\mathbb{R}^n\setminus\{0\},NEWLINE\]NEWLINE and its adjoint operator NEWLINE\[NEWLINEH^*f(x)=\int_{|y|\geq|x|}\frac{f(y)}{|y|^n}\, dy,NEWLINE\]NEWLINE respectively, which arise in harmonic analysis. The authors prove the boundedness of the commutators \(H_{\Phi,b}=b\cdot H_{\Phi}-H_{\Phi}\cdot b\) on Lebesgue space, where \(b\) is a central BMO function or a Lipschitz function. Furthermore, the boundedness on Herz spaces and Morrey-Herz spaces is obtained.