The Hausdorff operators on the real Hardy spaces H^p(R)

Abstract

Abstract. We prove that the Hausdorff operator generated by a function φ is bounded on the real Hardy space H p (R), 0 < p ≤ 1, if the Fourier transform φ of φ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on H p (R), 2/(2α + 1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of H p (R).

Hausdorff operators on the power weighted Hardy spaces

“…Additionally, many classical operators in analysis are special cases of the Hausdorff operator if one chooses suitable kernel functions Φ (see [2,3,4,7,22]). These operators include the Hardy operator, the adjoint Hardy operator [5,8,9] and the Cesàro operator [13,23], among many others. The Hardy-Littlewood-Pólya operator and the Riemann-Liouville fractional integral can also be derived from the Hausdorff operator.…”

The WeightedLpandBMOEstimates for Fractional Hausdorff Operators on the Heisenberg Group

“…It is well known that many classical operators in harmonic analysis are special cases of Hausdorff operators if one chooses suitable Φ (see, for example, [1,[6][7][8][9][10]), including Hardy operators and Cesàro operator [11][12][13][14].…”

Hausdorff type operators on the power weighted Hardy spaces H|·|αp(Rn)

“…, , , , ), while there is no any research about the boundedness of on the real Hardy spaces for . On the other hand, when , the boundedness of on is well studied for all (see , , , ). However, there exists an essential difference between the cases and , which prevents that the methods used in the one dimensional case are adoptable in the case .…”