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Almost Everywhere Convergence of Bochner–Riesz Means On The Heisenberg Group and Fractional Integration On The Dual
Author(s) -
Gorges Dirk,
Müller Detlef
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/85.1.139
Subject(s) - mathematics , heisenberg group , dimension (graph theory) , operator (biology) , almost everywhere , laplace operator , combinatorics , fractional laplacian , lambda , homogeneous , pure mathematics , mathematical analysis , physics , quantum mechanics , biochemistry , chemistry , repressor , transcription factor , gene
Let L denote the sub‐Laplacian on the Heisenberg group H n 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 H n . We prove convergence a.e. of the Bochner‐Riesz means T r λ f as r → 0 for λ > 0 and for all f ∈ L p (H n ), 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 awith a ∈ C, defined on the dual of H n by ∂ ω af ^: =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 ) ∈ H n . 2000 Mathematical Subject Classification : 22E30, 43A80.