Abstract
We prove an almost everywhere convergence result for Bochner-Riesz means of $L^p$ functions on Heisenberg-type groups, yielding the existence of a $p>2$ for which convergence holds for means of arbitrarily small order. The proof hinges on a reduction of weighted $L^2$ estimates for the maximal Bochner-Riesz operator to corresponding estimates for the non-maximal operator, and a `dual Sobolev trace lemma', whose proof is based on refined estimates for Jacobi polynomials.
Highlights
The proof hinges on a reduction of weighted L2 estimates for the maximal Bochner–Riesz operator to corresponding estimates for the non-maximal operator, and a ‘dual Sobolev trace lemma’, whose proof is based on refined estimates for Jacobi polynomials
The study of Bochner–Riesz means is a classical topic in harmonic analysis
In the case of the Heisenberg groups, no nontrivial restriction estimates of this kind hold for the subLaplacian L [61]; for H-type groups with higher-dimensional centre, some estimates of Tomas– Stein type do hold [12], but the corresponding Lp boundedness results for T∗λ given by [14] are strictly included in those given by Theorem 1.2
Summary
The study of Bochner–Riesz means is a classical topic in harmonic analysis. Recall that the Bochner–Riesz means of order λ 0 of any function f ∈ L2(Rd) are defined by. In the case of the Heisenberg groups, no nontrivial restriction estimates of this kind hold for the subLaplacian L [61]; for H-type groups with higher-dimensional centre, some estimates of Tomas– Stein type do hold [12], but the corresponding Lp boundedness results for T∗λ given by [14] are strictly included in those given by Theorem 1.2 This seems to indicate once more that the investigation of Bochner–Riesz means for sub-Laplacians requires substantially new ideas and methods compared to the Euclidean case.
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