Concerning some version of the Lax-Milgram Lemma in normed spaces

Moszyński, Krzysztof

doi:10.4064/sm-68-1-67-78

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“…This is a weaker hypothesis than the ellipticity assumption used in [10] to establish L2-calculus. Despite the fact that our form coercivity condition for an operator B : L p -*• L p is stronger than that considered in [25] (which is equivalent to the boundedness of B' 1 ), it is still weaker for p = 2 than the restriction…”

Section: Introductionmentioning

confidence: 92%

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

Ajiev¹

2007

J. Aust. Math. Soc.

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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.

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Section: Introductionmentioning

confidence: 92%