PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 5, Pages 1391–1396 S 0002-9939(99)05159-X Article electronically published on August 5, 1999 THE HAUSDORFF OPERATOR IS BOUNDED ON THE REAL HARDY SPACE H 1 (R) ELIJAH LIFLYAND AND FERENC M ´ ORICZ (Communicated by Christopher D. Sogge) Abstract. We prove that the Hausdorff operator generated by a function ϕ ∈ L 1 (R) is bounded on the real Hardy space H 1 (R). The proof is based on the closed graph theorem and on the fact that if a function f in L 1 (R) is such that its Fourier transform  f (t) equals 0 for t< 0 (or for t> 0), then f ∈ H 1 (R). 1. Preliminaries We recall that the Fourier transform  f of a function f in L 1 (R) is defined by letting  f (t) := 1 √ 2π  R f (x)e −itx dx, t ∈ R; while its Hilbert transform  f is defined by letting  f (x) := 1 π (P.V.)  ∞ 0 f (x − u) − f (x + u) u du, x ∈ R, where the principal value integral is defined to be lim ε↓0  ∞ ε . It is well known that this limit exists for almost every x ∈ R. The real Hardy space H 1 (R) is defined to be H 1 (R) := {f ∈ L 1 (R):  f ∈ L 1 (R)}, endowed with the norm ‖f ‖ H 1 := ‖f ‖ L 1 + ‖  f ‖ L 1 , (1.1) where ‖f ‖ L 1 :=  R |f (x)| dx. Received by the editors June 25, 1998. 1991 Mathematics Subject Classification. Primary 47B38; Secondary 46A30. Key words and phrases. Fourier transform, Hilbert transform, real Hardy space H 1 (R), Haus- dorff operator, Ces`aro operator, closed graph theorem. This research was partially supported by the Minerva Foundation through the Emmy Noether Institute at the Bar-Ilan University and by the Hungarian National Foundation for Scientific Research under Grant T 016 393. c 2000 American Mathematical Society 1391 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use