Hausdorff operators on function spaces (Q692451)

The authors start with the \(H^1\)-boundedness of a general Hausdorff operator. We remark that atomic decomposition for proving \(H^1\) boundedness of the Hausdorff operator has already been used in the reviewer's paper [Acta Sci. Math. 74, No. 3--4, 845--851 (2008; Zbl 1199.47155)]. More interesting is the use of the same approach to the boundedness of the Hausdorff operator in the local Hardy space \(h^1\). This type of spaces is rather important, and the results of the paper under review are apparently the first attempt to develop a theory of Hausdorff operators in \(h^1\). The authors also deal with the Herz-type spaces.