denote the Bochner-Riesz operator of order . We investigate the almost everywhere convergence of as
Theorem 1. Let and . For all with , almost everywhere.
Theorem 2. For all with , a.e. if is a lacunary sequence.
The proofs proceed via domination of the maximal function by square functions; however instead of seeking bounds for the square function we examine weighted inequalities, which, by duality, correspond to studying pointwise multipliers of Sobolev spaces.