Remarks on Functional Calculus for Perturbed First-order Dirac Operators

Auscher, Pascal; Stahlhut, Sebastian

doi:10.1007/978-3-319-06266-2_2

Cited by 10 publications

(7 citation statements)

References 14 publications

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“…In the case of R n , results in the same spirit (extending the range of p for which a bounded holomorphic functional calculus holds outside the Hodge range) have been recently obtained in [15] and [5]. The methods used there are different, and specific to R n .…”

Section: Introductionmentioning

confidence: 89%

Hodge–Dirac, Hodge-Laplacian and Hodge–Stokes operators in $L^p$ spaces on Lipschitz domains

McIntosh

Monniaux

2018

Rev. Mat. Iberoam.

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This paper concerns Hodge-Dirac operators D = d + δ acting in L p (Ω, Λ) where Ω is a bounded open subset of R n satisfying some kind of Lipschitz condition, Λ is the exterior algebra of R n , d is the exterior derivative acting on the de Rham complex of differential forms on Ω, and δ is the interior derivative with tangential boundary conditions. In L 2 (Ω, Λ), δ = d * and D is self-adjoint, thus having bounded resolvents (I + itD ) −1 t∈R as well as a bounded functional calculus in L 2 (Ω, Λ). We investigate the range of values p H < p < p H about p = 2 for which D has bounded resolvents and a bounded holomorphic functional calculus in L p (Ω, Λ). On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which L p (Ω, Λ) has a Hodge (or Helmholz) decomposition, being an open interval that includes 2.The Hodge-Laplacian ∆ is the square of the Hodge-Dirac operator, i.e. −∆ = D 2 , so it also has a bounded functional calculus in L p A well-known property of the differential operator d is that it commutes with a change of variables as stated below, see, e.g., [6, Definition 1.2.1 and Proposition 1.2.2].Definition 2.18. Let Ω be an open set in R n and ρ : Ω → ρ(Ω) a bilipschitz transformation. Denote by J ρ (y) the Jacobian matrix of ρ at a point y ∈ Ω and extend it to an isomorphism J ρ (y) : Λ → Λ such that J ρ (y)(e i 1 ∧ · · · ∧ e i k ) = (J ρ (y)e i 1 ) ∧ · · · ∧ (J ρ (y)e i k ), {i 1 , . . . , i k } ⊂ {1, . . . , n}.

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Section: Introductionmentioning

confidence: 89%

Hodge–Dirac, Hodge-Laplacian and Hodge–Stokes operators in $L^p$ spaces on Lipschitz domains

McIntosh

Monniaux

2018

Rev. Mat. Iberoam.

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“…The new proof of the Kato conjecture by Axelsson, Keith, and McIntosh [2006] using first-order methods was extended to the setting of L p (R d ; X) by Hytönen, McIntosh, and Portal [2008a]; here X is required to be a UMD space and in addition to satisfy the RMF property, discussed in Section 3.6.b, which was in fact first invented for the purposes of this paper. Besides the X-valued extension, this paper offered an alternative approach to the L p theory of the Kato conjecture, which has been further explored by Auscher and Stahlhut [2013], Frey, McIntosh, and Portal [2014], Hytönen and McIntosh [2010], and Hytönen, McIntosh, and Portal [2011].…”

Section: Conical Square Functionsmentioning

confidence: 99%

R-boundedness

Hytönen

Neerven

Veraar

et al. 2017

Analysis in Banach Spaces

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Analysis and Dynamics, of which T.H. is a member), the Netherlands Organisation for Scientific research (NWO) (VIDI grant 639.032.201 and VICI grant 639.033.604 to J.v.N. and VENI grant 639.031.930 and VIDI grant 639.032.427 to M.V.), and the German Research Foundation (DFG) (Research Training group 1254 and Collaborative Research Center 1173 of which L.W. is a member). We also wish to thank the Banach Center in Będlewo and Mathematisches Forschungsinstitut Oberwolfach for allowing us to spend two highly productive weeks in both wonderful locations.

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“…The analogue of Theorem 1.1 for the Hodge-Dirac operator associated with the Ornstein-Uhlenbeck operator has been established, in a more general formulation, in [48]. The related problem of the L p -boundedness of the H ∞ -calculus of Hodge-Dirac operators associated with the Kato square root problem was initiated by the influential paper [8] and has been studied by many authors [7,24,[32][33][34]51].…”

Section: Introductionmentioning

confidence: 99%