Almost Everywhere Convergence of Bochner–Riesz Means On The Heisenberg Group and Fractional Integration On The Dual
2002; Wiley; Volume: 85; Issue: 1 Linguagem: Inglês
10.1112/plms/85.1.139
ISSN1460-244X
Autores Tópico(s)Advanced Mathematical Modeling in Engineering
ResumoLet L denote the sub-Laplacian on the Heisenberg group Hn and T r L a m b d a : = ( 1 − r L ) + λ the corresponding Bochner-Riesz operator. Let Q denote the homogeneous dimension and D the Euclidean dimension of Hn. We prove convergence a.e. of the Bochner-Riesz means T r λ f as r → 0 for λ > 0 and for all f ∈ Lp(Hn), provided that \frac{Q-1}{Q} \Big(\frac{1}{2} - \frac{\lambda}{D-1} \Big) < 1/p \le 1/2. Our proof is based on explicit formulas for the operators ∂ ω a with a ∈ C, defined on the dual of Hn by ∂ ω a f ^ : = ω a f ^ , which may be of independent interest. Here ω is given by ω ( z , u ) : = | z | 2 − 4 i u for all (z,u) ∈ Hn. 2000 Mathematical Subject Classification: 22E30, 43A80.
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