The weak-type (1, 1) of L log L homogeneous convolution operator

Abstract

We show that a homogeneous convolution kernel on an arbitrary homogeneous group which is L log L on the unit annulus is bounded on Lp for 1 < p < ∞ and is of weak-type (1, 1), generalizing the result of Seeger [9]. The proof is in a similar spirit to that of Christ and Rubio de Francia [3].