By Detlef Muller, Marco M. Peloso, Fulvio Ricci
The authors ponder the Hodge Laplacian ? at the Heisenberg team 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 okay -forms. during this paper they tackle 3 major, similar 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) the right way to end up a pointy Mihilin-Hormander multiplier theorem for ? ok , 0=k=2n 1
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D. From the previous results we immediately get an explicit formula for U2, , at least when p = 0 and q = 0. However, if p = 0 or q = 0, our formulas, when properly interpreted, persist, and we obtain the following result: Recall that if p = 0, then X p,q = (I − C)X p,q , and if q = 0, then Y p,q = (I − C)Y p,q . Let us correspondingly put r so that r if p ≥ 1, if p = 0, = r is always invertible on X p,q , and if q ≥ 1, if q = 0, = r on Y p,q . 12. 10. 9. 13. 12 maps the space Z p,q onto V2,p,q and intertwines D := Δ0 + i(q − p)T + ( + 1)(n − k + + 1) with Δk p,q on the core.
M. PELOSO, and F. RICCI 50 In particular, if we know that A∗ A maps D1 into H1 , then A∗1 A1 ξ = A∗ Aξ for every ξ ∈ D1 . Proof. Since A(D1 ) ⊂ E1 , it suﬃces to prove that A∗1 = PH1 A∗ on E1 . But, if x ∈ D1 ⊂ dom A1 , ξ ∈ E1 ⊂ dom A∗1 , then x, A∗1 ξ = A1 x, ξ = Ax, ξ = x, A∗ ξ = x, PH1 A∗ ξ . This implies that A∗1 ξ = PH1 A∗ ξ, since D1 is dense in H1 . D. 1. A unitary intertwining operator for V0p,q We recall from the preceding discussion that the intertwining operator on V0p,q is Φ, which reduces to the identity on this space.
D. 9. 7) maps W0p,q , respectively Ξp,q , onto and intertwines D± with Δk on the core. + − Moreover, U1, : W0p,q → L2 Λk and U1, : Ξp,q → L2 Λk are linear isometries p,q,+ p,q,− onto their ranges V1, and V1, , respectively, which intertwine D+ resp. 17) Δk V p,q,± 1, ± ± −1 = U1, D± (U1, ) on dom Δk . V p,q,± 1, ± −1 ± ) denotes the inverse of U1, when viewed as an operator into its Here, (U1, p,q,± range V1, . ± Finally, if we regard of U1, as an operator mapping into L2 Λk , then P1,± := p,q,± ± ± ∗ P1,p,q,± := U1, (U1, ) is the orthogonal projection from L2 Λk onto V1, .
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