Let X be an A1-regular lattice of measurable functions and let Q be a projection that is also a Calderón–Zygmund operator. In this case, it is possible to define a space XQ consisting of functions f ∈ X for which Qf = f in a certain sense. By using the Bourgain approach to interpolation, we show that the couple (L Q1 , XQ) is K-closed in (L1, X). This result is sharp in the sense that, in general, A1-regularity cannot be replaced by weaker conditions such as Ap-regularity for p > 1.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 424, 2014, pp. 186–200.
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Rutsky, D.V. Weighted Calderón–Zygmund Decomposition with Some Applications to Interpolation. J Math Sci 209, 783–791 (2015). https://doi.org/10.1007/s10958-015-2526-y
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DOI: https://doi.org/10.1007/s10958-015-2526-y