The Hausdorff operator is bounded on the real Hardy space 𝐻¹(ℝ)

Abstract

We prove that the Hausdorff operator generated by a function φ ∈ L 1 ( R ) \varphi \in L^{1} ({\mathbb {R}}) is bounded on the real Hardy space H 1 ( R ) H^{1} ({\mathbb {R}}) . The proof is based on the closed graph theorem and on the fact that if a function f f in L 1 ( R ) L^{1} ({\mathbb {R}}) is such that its Fourier transform f ^ ( t ) \widehat {f} (t) equals 0 0 for t > 0 t>0 (or for t > 0 t>0 ), then f ∈ H 1 ( R ) f\in H^{1} ({\mathbb {R}}) .