Directional operators and radial functions on the plane

Abstract

Let EçS1 be a set with Minkowski dimension d(E)1. We consider the Hardy-Littlewood maximal function, the Hilbert transform and the maximal Hilbert transform along the directions of E. The main result of this paper shows that these operators are bounded on Lradp (R2) for p>1+d(E) and unbounded when p<1+d(E). We also give some end-point results. © 1995 Institut Mittag-Leffler.