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Extrapolation of the functional calculus of generalized Dirac operators and related embedding and Littlewood-Paley-type theorems. I

Part of: General theory of linear operators Nontrigonometric harmonic analysis Normed linear spaces and Banach spaces; Banach lattices Linear function spaces and their duals

Published online by Cambridge University Press:  09 April 2009

Sergey S. Ajiev
Sergey S. Ajiev
Affiliation:
School of Mathematics and StatisticsUNSWSydney NSW 2052Australiae-mail: ajievss@unsw.edu.au
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Abstract

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Several rather general sufficient conditions for the extrapolation of the calculus of generalized Dirac operators from L2 to Lp are established. As consequences, we obtain some embedding theorems, quadratic estimates and Littlewood–Paley theorems in terms of this calculus in Lebesgue spaces. Some further generalizations, utilised in Part II devoted to applications, which include the Kato square root model, are discussed. We use resolvent approach and show the irrelevance of the semigroup one. Auxiliary results include a high order counterpart of the Hilbert identity, the derivation of new forms of ‘off-diagonal’ estimates, and the study of the structure of the model in Lebesgue spaces and its interpolation properties. In particular, some coercivity conditions for forms in Banach spaces are used as a substitution of the ellipticity ones. Attention is devoted to the relations between the properties of perturbed and unperturbed generalized Dirac operators. We do not use any stability results.

Keywords

functional calculusgeneralized embeddingLittlewood-Paley theoremextrapolation of singular operatorsgeneralized Dirac operatorHilbert identityoff-diagonal estimatesKato square root model.

MSC classification

Secondary: 46E15: Banach spaces of continuous, differentiable or analytic functions 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47A60: Functional calculus 47A65: Structure theory 47A05: General (adjoints, conjugates, products, inverses, domains, ranges, etc.) 47A55: Perturbation theory 46B20: Geometry and structure of normed linear spaces 46C99: None of the above, but in this section 42C99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 3 , December 2007 , pp. 297 - 326
Copyright
Copyright © Australian Mathematical Society 2007

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