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 StatisticsUNSW Sydney, NSW 2052Australiaajievss@unsw.edu.au
Received 6 August 2004; revised 6 June 2005
Communicated by A. H. Dooley

## Abstract

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.

 2000 Mathematics Subject Classification: primary 46E15, 46E30, 47A60, 47A65;secondary 47A05, 47A55, 46B20, 46C99, 42C99 (Metadata: XML, RSS, BibTeX)

## References

1. S. S. Ajiev, ‘Anisotropic supersingular integral operators and approximation formula’, to appear in Math. Inequal. Appl.
2. S. S. Ajiev, Singular and supersingular operators on function spaces, approximation and extrapolation (Ph.D. Thesis, Australian National University: Canberra, 2003). MR1994511
3. S. S. Ajiev, ‘On the boundedness of singular integral operators from some classes II’, Analysis Mathematica 32 (2006), 81–112. MR2248066
4. D. Albrecht, P. Auscher, X. Duong and A. McIntosh, ‘Operator theory and harmonic analysis’, in: Instructional workshop on Analysis and Geometry, Part III (Canberra, 1995), volume 34 of Proc. Centre Math. Appl. Austral. Nat. Univ. (1996) pp. 77–136. MR1394696
5. P. Auscher, ‘On L^{p} estimates for square roots of second order elliptic operators on \mathbb{R}^{n}\,’, Publ. Mat. 48 (2004), 159–186. MR2044643
6. P. Auscher, T. Coulhon, X. T. Duong and S. Hofmann, ‘Riesz transform on manifolds and heat kernel regularity’, Ann. Sci École. Norm Sup (4) 37 (2004), 911–957. MR2119242
7. P. Auscher, S. Hofmann, M. Lacey, A. McIntosh and Ph. Tchamitchian, ‘The solution of Kato square root problem for second order elliptic operators on \mathbb{R}^{n}\,’, Ann. of Math. (2) 156 (2002), 633–654. MR1933726
8. P. Auscher, S. Hofmann, A. McIntosh and Ph. Tchamitchian, ‘The Kato square root problem for high order elliptic operators and systems on \mathbb{R}^{n}\,’, J. Evol. Equ. 1 (2002), 361–385. MR1877264
9. P. Auscher and Ph. Tchamitchian, ‘Square root problem for divergence operators and related topics’, Astérisque (1998), viii+172. MR1651262
10. A. Axelsson, S. Keith and A. McIntosh, ‘Functional calculus of Dirac operators with measurable coefficients’, Invent. Math. 163 (2006), 455–497. MR2207232
11. M. S. Baouendi and G. Goulaouic, ‘Commutation de l'intersection et des founcteurs d'interpolation’, C. R. Acad. Sci. Paris Sér A-B 265 (1967), 313–315. MR225132
12. S. Blunck and P. Kunstmann, ‘Calderón–Zygmund theory for non-integral operators and the H^{∞}-functional calculus’, Rev. Mat. Iberoamericana 19 (2003), 919–942. MR2053568
13. R. Coifman, A. McIntosh and Y. Meyer, ‘L'intégrale de Cauchy définit un opérateur borné sur L_{2}(\mathbb{R}^{n}) pour les courbes lipschitziennes’, Ann. of Math. (2) 116 (1982), 361–387. MR672839
14. M. Cowling, I. Doust, A. McIntosh and A. Yagi, ‘Banach space operators with a bounded H^{∞}-calculus’, J. Aust. Math. Soc. 60 (1996), 51–89. MR1364554
15. E. B. Davies, ‘Uniformly elliptic operators with measurable coefficients’, J. Funct. Anal. 132 (1995), 141–169. MR1346221
16. X. T. Duong and A. McIntosh, ‘Singular integral operators with non-smooth kernels on irregular domains’, Rev. Mat. Iberoamericana 15 (1999), 233–265. MR1715407
17. X. T. Duong and D. W. Robinson, ‘Semigroup kernels, poisson bounds, and holomorphic functional calculus’, J. Funct. Anal. 142 (1996), 89–128. MR1419418
18. W. Hebisch, ‘A multiplier theorem for Schrödinger operators’, Colloq. Math. 60–61 (1990), 659–664. MR1096404
19. S. Hofmann and J. M. Martell, ‘L_{p}-bounds for Reiesz transforms and square roots associated to second order elliptic operators’, Publ. Mat. 47 (2003), 497–515. MR2006497
20. S. Hofmann and A. McIntosh, ‘The solution of the Kato problem in two dimensions’, in: Proceedings of the conference on harmonic analysis and PDE (El Escorial, 2000), volume extra (2002) pp. 143–160. MR1964818
21. J. L. Lions, Equations différentielles operationelles et problèmes aux limites (Springer-Verlag, Berlin, 1961). MR153974
22. J. M. Martell, ‘Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications’, Studia Math 161 (2004), 113–145. MR2033231
23. A. McIntosh, ‘On representing closed accretive sesquilinear forms as (a^{1/2}u, a^{*1/2}v)\,’, in: Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. III (Paris, 1980/1981), Res. Notes in Math. 70 (1982) pp. 252–267. MR670278
24. A. McIntosh, ‘Operators which have an H_{∞} functional calculus’, in: Miniconference on operator theory and partial differential equations (North Ryde, 1986), Proc. Centre Math. Anal. Austral. Nat. Univ. 14 (1986) pp. 210–231. MR912940
25. K. Moszynski, ‘Concerning some version of the Lax–Milgram Lemma in normed spaces’, Studia Math. LXVII (1980), 65–78. MR583402
26. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland Publishing Company, Amsterdam–Oxford, 1995). MR1328645
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