Analysis of the Hodge Laplacian on the Heisenberg Group

ISBN-13: 9781470409395 / Angielski

Detlef Muller;Marco M. Peloso;Fulvio Ricci

Opis

The authors consider the Hodge Laplacian on the Heisenberg group H n , endowed with a left-invariant and U(n) -invariant Riemannian metric. For 0 k 2n 1 , let k denote the Hodge Laplacian restricted to k -forms. In this paper they address three main, related questions: (1) whether the L 2 and L p -Hodge decompositions, 1 (2) whether the Riesz transforms d 12 k are L p -bounded, for 1<< ; (3) how to prove a sharp Mihilin-Hoermander multiplier theorem for k , 0 k 2n 1 .

Spis treści

• Introduction
• Differential forms and the Hodge Laplacian on Hn
• Bargmann representations and sections of homogeneous bundles
• Cores, domains and self-adjoint extensions
• First properties of Dk exact and closed forms
• A decomposition of L2 Lkh related to the ¶ and ¶ complexes
• Intertwining operators and different scalar forms for Dk
• Unitary intertwining operators and projections
• Decomposition of L2 Lk
• Lp-multipliers
• Decomposition of LpLk and boundedness of the Riesz transforms
• Applications
• Appendix
• Bibliography