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New function spaces of BMO type, the John-Nirenberg inequality, interpolation, and applications. (English) Zbl 1153.26305
Summary: In this paper, we introduce and develop some new function spaces of BMO (bounded mean oscillation) type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type. The new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity, and they generalize the classical BMO space. We show that the John–Nirenberg inequality holds on these spaces and they interpolate with $$L^{p}$$ spaces by the complex interpolation method. We then give applications on $$L^{p}$$-boundedness of singular integrals whose kernels do not satisfy the Hörmander condition.

##### MSC:
 26A45 Functions of bounded variation, generalizations 26D15 Inequalities for sums, series and integrals 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 46B70 Interpolation between normed linear spaces 46E15 Banach spaces of continuous, differentiable or analytic functions 46M35 Abstract interpolation of topological vector spaces
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