Directional maximal operators with smooth densities.

Journal article, 2009

We study directional maximal operators on Rn with smooth densities. We prove that if the classical directional maximal operator in a given set of directions is weak type (1, 1), then the corresponding smooth-density maximal operator in that set of directions will be bounded on Lq for q suitably large, depending on the order of the stationary points of the density function. In contrast to the classical case, if q is too small, the smooth density operator need not be bounded on Lq. Improving upon previously known results, we also establish that if the density function has only finitely many extreme points, each of finite order, then any maximal operator in a finite sum of diadic directions is bounded on all Lq for q > 1.