A note on a maximal function over arbitrary sets of directions

Let M-S be the universal maximal operator over unit vectors of arbitrary di rections. This operator is not bounded in L-2(R-2). We consider a sequence of operators over sets of finite equidistributed directions converging to M -S. We provide a new proof of N. Katz's bound for such operators. As a coro llary, we deduce that M-S is bounded from some subsets of L-2 to L-2. These subsets are composed of positive functions whose Fourier transforms have a logarithmic decay or which are supported on a disc.