Bulletin of the Australian Mathematical Society

Research Article

A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function

Hitoshi Tanakaa1

a1 Department of Mathematics, Gakushuin University, 1–5–1 Mejiro, Toshima-Ku, Tokyo 171–8588, Japan, e-mail: hitoshi.tanaka@gakushuin.ac.jp

Abstract

Dedicated to Professor Kôzô Yabuta on the occasion of his 60th birthday

J. Kinnunen proved that of P > 1, d ≤ 1 and f is a function in the Sobolev space W1,P(Rd), then the first order weak partial derivatives of the Hardy-Littlewood maximal function xs2133f belong to LP(Rd). We shall show that, when d = 1, Kinnunen's result can be extended to the case where P = 1.

(Received August 14 2001)