By Detlef Muller, Marco M. Peloso, Fulvio Ricci
The authors contemplate the Hodge Laplacian ? at the Heisenberg crew H n , endowed with a left-invariant and U(n) -invariant Riemannian metric. For 0=k=2n 1 , allow ? ok denote the Hodge Laplacian constrained to ok -forms. during this paper they deal with 3 major, comparable questions: (1) no matter if the L 2 and L p -Hodge decompositions, 1 (2) no matter if the Riesz transforms d? -12 okay are L p -bounded, for 1<<8 ; (3) how you can turn out a pointy Mihilin-Hormander multiplier theorem for ? ok , 0=k=2n 1
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Extra info for Analysis of the Hodge Laplacian on the Heisenberg group
Thus, C is the orthogonal projection onto ker ∂ when acting on (0, q)-forms as well as onto ker ∂ ∗ when acting on (n, q)-forms, and C¯ is the orthogonal projection onto ker ∂¯ when acting on (p, 0)-forms as well as onto ker ∂¯∗ when acting on (p, n)forms. Regard ∂ as a closed operator from L2 Λp,q to L2 Λp+1,q . 5) Rp = ∂ 0 (I − C) = 1 ∂ for p = 0 , ⎪ ⎩ − 12 −1 ∂ n−1 = n 2∂ for p = n − 1 . 4. 6) 1 2 p = 1 2 p+1 Rp =∂ . 7) R∗p = ∂ = ∂∗ 1 2 0 ⎪ ⎩ − 12 ∗ − 12 ∗ ¯ n (I − C) n−1 ∂ = ∂ S0 Λp,q are for 1 ≤ p ≤ n − 2 , for p = 0 , for p = n − 1 .
3. 5) S1 |A|2 ξ = |A|2 S1 ξ for all ξ ∈ D1 . 1) and (i) imply that ξ, S1 A∗ η = ξ, A∗ S2 η , hence S1 A ∗ η = A ∗ S2 η for all η ∈ D2 . 5). 5) by means of general spectral theory. However, this hope is destroyed by a classical example due to Nelson (cf. 5) will in general not suﬃce to conclude that the operators S1 and |A|2 commute, in the sense that their respective spectral resolutions commute. 2). 7) for the square root of such a matrix. In the sequel, by PH1 : H → H1 we shall denote the orthogonal projection from the Hilbert space H onto its closed subspace H1 .
5. 9) N= ( − i T − T 2) ( + i T − T 2) . 6. Let R = −T −2 Q∗ N Q on (W0p,q )2 . Then R= R11 0 0 R22 , where R11 = (Γ + m)2 ΔH + 2m(2m − ) + 2(Γ + m) (2m − )ΔH − 2mT 2 + ΔH (ΔH − T 2 ) + 2m T 2 , maps W0p,q bijectively onto itself, and R22 = (Γ − m)2 ΔH + 2m(2m − ) − 2(Γ − m) (2m − )ΔH − 2mT 2 + ΔH (ΔH − T 2 ) + 2m T 2 , maps Ξp,q bijectively onto itself, and is zero on Cp W0p,q ⊕ C¯q W0p,q , the orthogonal complement of Ξp,q in W0p,q . Proof. The proof of the formulas for the components of R is postponed to the Appendix.