In this paper,by introducing the space with weak mixed norms,weak type estimates of two kinds of multilinear fractional Hausdorff operators RΦ,β and SΦ,β on Lebesgue spaces are shown.By virtue of Marcinkiewicz interpolation,strong type estimates of these two operators on Lebesgue spaces are also obtained.Our methods shed some new light on dealing with the case of non-radial function Φ.
Hausdorff operator is an important operator raised from the dilation on Euclidean space and rooted in the classical summability of number series and Fourier series. It is also connected to many well known operators in real and complex analysis. This article is a survey of some recent developments and extensions on the Hausdorff operator. Particularly, various boundedness properties of the Hausdorff operators, studied recently by our research group, are addressed.
In this paper, we will prove the Triebel-Lizorkin boundedness for some oscillatory singular integrals with the kernel （x） satisfying a condition introduced by Grafakos and Stefanov. Our theorems will be proved under various conditions on the phase function, radial and nonradial. Since the L p boundedness of these operators is not complete yet, the theorems extend many known results.
