The Hausdorff operators on the real Hardy spaces $H^p({\Bbb R})$
Volume 148 / 2001
Studia Mathematica 148 (2001), 37-45
MSC: Primary 47B38; Secondary 42B30.
DOI: 10.4064/sm148-1-4
Abstract
We prove that the Hausdorff operator generated by a function $\phi $ is bounded on the real Hardy space $H^p({\mathbb R})$, $0 < p \le 1,$ if the Fourier transform $\widehat {\phi }$ of $\phi $ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Ces\accent18 aro operator of order $\alpha $ on $H^p({\mathbb R})$, $2/(2\alpha +1) < p \le 1$. Our proof is based on the atomic decomposition and molecular characterization of $H^p({\mathbb R})$.
