Hausdorff operators on modulation and Wiener amalgam spaces

Abstract

We give the sharp conditions for boundedness of Hausdorff operators on certain modulation and Wiener amalgam spaces.

Hausdorff Operators on Modulation SpacesMp,ps

“…, (see [20]). Note that in [20], only the modulation spaces without potential were taken into consideration.…”

“…, (see [20]). Note that in [20], only the modulation spaces without potential were taken into consideration. Since the Hausdorff operator is not an operator of convolution type, and the dilation properties of modulation spaces are more complicated with potential, it is quite interesting to establish the sharp conditions for the boundedness of Hausdorff operator on modulation spaces with potential.…”

“…Since the Hausdorff operator is not an operator of convolution type, and the dilation properties of modulation spaces are more complicated with potential, it is quite interesting to establish the sharp conditions for the boundedness of Hausdorff operator on modulation spaces with potential. As in the simple case in [20], the classical method used for the Lebesgue space is also not applicable here. As we will see, the potential index plays important roles not only in the dilation property of , , but also in the boundedness of Hausdorff operator on , .…”

“…As we will see, the potential index plays important roles not only in the dilation property of , , but also in the boundedness of Hausdorff operator on , . Therefore, the method in [20] must be modified to cope with this new situation.…”

Hausdorff operators on Sobolev spaces Wk,1

“…However, the boundedness of Hausdorff operator can be characterized in only few cases. We refer the reader to [7,20] for the characterization of the bounded Hausdorff operators on Lebesgue spaces, to [5,16] for the characterization of the bounded Hausdorff operators on Hardy spaces H 1 and h 1 , and to [21] for the characterization of the bounded Hausdorff operators on modulation and Wiener amalgam spaces. Until now, there is no result regarding the boundedness of Hausdorff operators on Sobolev spaces.…”