Several rather general sufficient conditions for the extrapolation of the calculus of generalized Dirac operators from t j 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.

Dedicated to Academician S. M. Nikol'skiȋ on the occasion of his 100th anniversary A b s t r a c t . The boundedness of anisotropic singular integral operators with the domains of definition and ranges in various anisotropic spaces of Banach-valued functions is analyzed from a unified point of view. A number of parameterized classes of sufficient conditions are obtained that are expressed in terms of the approximation D-functional. Our sufficient conditions are weaker then their known counterparts in the same settings. The inhomogeneity of the dependence on certain parameters is revealed. The results obtained are also applicable to nonsingular (in the ordinary sense) integral operators, for example, to potential-type operators. The main results are presented in the style of the Calderón-Zygmund theory. The approach is based on the study of decompositions of operators and some properties of the related function spaces.

We obtain explicit estimates of the constants related to the concentrations of measures and distance, deviation and Dvoretzky's theorem for the finite-dimensional subspaces of a wide class of function and other spaces including, in particular, various anisotropic spaces of Besov, LizorkinTriebel and Sobolev types endowed with geometrically friendly norms defined in terms of averaged differences, local polynomial approximations, functional calculus, wavelets and other means. New approaches are shown to be providing better estimates in the abstract setting as well.

Dedicated to Academician S. M. Nikol'skiȋ on the occasion of his 100th anniversary A b s t r a c t . The boundedness of anisotropic quasi-subadditive singular operators with the domains in Bochner-Lebesgue and local approximation spaces of LizorkinTriebel-type of Banach-valued functions and the ranges in Bochner-Marcinkiewicz spaces is analyzed from the point of view of extrapolation. Parameterized classes of sufficient conditions generalizing author's AAD-conditions studied earlier are obtained that are expressed in terms of the approximation D-functional and ones being close to it. Our sufficient conditions are weaker then their known counterparts in the same settings. They are new in the isotropic case of linear scalar-valued operators too. The main results are presented in the style of the Calderón-Zygmund theory. The approach is based on decompositions of operators and the usage of counterparts of Calderón-Zygmund representations and some interpolation properties of the related function spaces established by the author.

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