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

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 W 1, P ( R d ), then the first order weak partial derivatives of the Hardy-Littlewood maximal function ℳ f belong to L P ( R d ). We shall show that, when d = 1, Kinnunen's result can be extended to the case where P = 1.