Abstract
We obtain weak-type (p,p) endpoint bounds for Bochner–Riesz means for the Hermite operator H=-Δ+|x| 2 in ℝ n ,n≥2 and for other related operators, for 1≤p≤2n/(n+2), extending earlier results of Thangavelu and of Karadzhov.
Highlights
The Bochner–Riesz conjecture is that, for n ≥ 2 and 1 ≤ p ≤ 2n/(n + 1), SRδ is bounded on Lp (Rn) if and only if δ > δ(p) = n 1 − 1 − 1 . (2) p2 2
The main goal of this paper is to extend the results of [15,33] to weak-type endpoint results for the range 1 ≤ p ≤ 2n/(n + 2)
Theorem 1 is proved by using the a priori estimate (8), along with the Lp eigenfunction bounds (7) for the Hermite operator, and the approach in the work of Christ [6, 7] and Tao [27]
Summary
Convergence of the Bochner–Riesz means on Lebesgue Lp spaces is one of the classical problems in harmonic analysis. Theorem 1 is proved by using the a priori estimate (8), along with the Lp eigenfunction bounds (7) for the Hermite operator, and the approach in the work of Christ [6, 7] and Tao [27]. Their approach is based on L2 Calderón–Zygmund techniques (as used in Fefferman [11]), a spatial decomposition of the Bochner–Riesz summation, and the fact that if the inverse.
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