A family of Bochner–Riesz type multipliers on the two-dimensional disk

Let s = (ν,μ) ∈ R2 and define ms(ξ ) .= (1 − ξ1)ν−μ(1 − |ξ |2) μ + on R2. Given p ∈ [1,+∞[, we prove some necessary and sufficient conditions on s such that ms be a Fourier multiplier for Lp. We employ two different techniques, according to ν = 0 or ν = 0. In the latter case, the results obtained are optimal.  2006 Elsevier Inc. All rights reserved.