Quadratic estimates and functional calculi of perturbed Dirac operators

Axelsson, Andreas; Keith, Stephen; McIntosh, Alan

doi:10.1007/s00222-005-0464-x

Cited by 97 publications

(306 citation statements)

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“…Fortunately, there is a weaker notion of Gaussian decay, which holds on any complete Riemannian manifold, namely the notion of L 2 off-diagonal estimates, as introduced by Gaffney [27]. This notion has already proved to be a good substitute of Gaussian estimates for such questions as the Kato square root problem or L p -bounds for Riesz transforms when dealing with elliptic operators (even in the Euclidean setting) for which Gaussian estimates do not hold (see [1,4,9] in the Euclidean setting, and [2] in a complete Riemannian manifold). We show in the present work that a theory of Hardy spaces of differential forms can be developed under such a notion.…”

Section: 2)mentioning

confidence: 99%

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Hardy Spaces of Differential Forms on Riemannian Manifolds

2007

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Abstract. Let M be a complete connected Riemannian manifold. Assuming that the Riemannian measure is doubling, we define Hardy spaces H p of differential forms on M and give various characterizations of them, including an atomic decomposition. As a consequence, we derive the H p -boundedness for Riesz transforms on M, generalizing previously known results. Further applications, in particular to H ∞ functional calculus and Hodge decomposition, are given.

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Section: 2)mentioning

confidence: 99%

“…for the commutator of two operators T and S. The proof is done by induction on k and relies on a commutator argument, as in [9], Proposition 5.2. For k = 0, the conclusion is given by Lemma 3.3.…”

Section: Remark 34 Note That With the Same Notations Ifmentioning

confidence: 99%

Hardy Spaces of Differential Forms on Riemannian Manifolds

2007

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“…The example that motivated the study of perturbed Dirac operators it the following setup, introduced in [7] and exploited in [8] to reprove the Kato square root theorem obtained in [5] for second order operators and in [6] for systems. Let A ∈ L ∞ (R n ; L(C m ⊗ C n )) satisfy …”

Section: Self-adjoint D and Accretive Bmentioning

confidence: 99%

Remarks on Functional Calculus for Perturbed First-order Dirac Operators

Auscher

Stahlhut

2014

Operator Theory in Harmonic and Non-Commutative Analysis

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Abstract. We make some remarks on earlier works on R−bisectoriality in L p of perturbed first order differential operators by Hytönen, McIntosh and Portal. They have shown that this is equivalent to bounded holomorphic functional calculus in L p for p in any open interval when suitable hypotheses are made. Hytönen and McIntosh then showed that R-bisectoriality in L p at one value of p can be extrapolated in a neighborhood of p. We give a different proof of this extrapolation and observe that the first proof has impact on the splitting of the space by the kernel and range.

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“…It remains only to verify that there exists η > 0 such that 14) where E Q := Q\( j Q j ). By (iii ) we have that…”

Section: Theorem 47 Let T F (X) := ψ T (X Y) F (Y)dymentioning

confidence: 99%