Weighted Inequalities for Bochner-riesz Means in the Plane

for some fixed large N0; we shall call such weights admissible. Rubio de Francia [11] showed that for every w ∈ L(R) there is a nonnegative W ∈ L(R) such that ‖W‖2 ≤ Cλ‖w‖2, Cλ < ∞ if λ > 0, and the analogous weighted norm inequality for S t holds uniformly in t. He used methods related to factorization theory of operators and the proof gave no information on how to construct w from W . In [3] the first author explicitly constructed for every q ≥ 2 an operator Wq,λ, bounded on L(R), such that (1.1) holds for w ∈ L(R) and W = Wq,λw; in fact given W2,λ one choses Wq,λw to be (W2,λ(w)). See also Córdoba [8] for a related result concerning S t . In [3] it was observed that the operator Wq,λ was bounded on L(R) for q ≤ r ≤ 2q and the question arose whether Wq,λ can be chosen to be independent of q. We shall show that this is indeed the case; for each λ > 0 we construct an operator Wλ such that (1.1) holds with W = Wλ and Wλ is bounded on L r if 2 ≤ r ≤ ∞. Moreover this operator is pointwise bounded by a positive operator (involving a Besicovich-type maximal function acting on w) which itself is bounded on L for 4 ≤ s ≤ ∞.