J. Aust. Math. Soc. 83 (2007), no. 3, pp. 297–326.

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

Sergey S. Ajiev
School of Mathematics and Statistics
UNSW Sydney, NSW 2052
Received 6 August 2004; revised 6 June 2005
Communicated by A. H. Dooley


Several rather general sufficient conditions for the extrapolation of the calculus of generalized Dirac operators from L_2 to L_{p} 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.

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2000 Mathematics Subject Classification: primary 46E15, 46E30, 47A60, 47A65;
secondary 47A05, 47A55, 46B20, 46C99, 42C99
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