Analysis of the Hodge Laplacian on the Heisenberg Group

Analysis of the Hodge Laplacian on the Heisenberg Group Synopsis

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-Hörmander multiplier theorem for ? k , 0?k?2n 1 .