Directional operators and radial functions on the plane

Abstract

LetEçS¹ 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 L ${}_{rad}^{p}$ (R²) for p>1+d(E) and unbounded when p<1+d(E). We also give some end-point results.