Extrapolation of operators and the limits of applicability of Schur's test [Elektronisk resurs]

• Generalizations of the Schur test for integral operators are studied. The authors look at the test from the point of the modern theory of extrapolation of operators. The generalizations are of two types. First of all, instead of operators with positive kernel, a more general class of positive operators is considered. Moreover, instead of Lp spaces, Orlicz, Lorentz and Marcinkiewicz spaces are taken as underlying spaces. In particular it is stated that the Schur extrapolation theorem holds for the class of positive operators. Several negative results are formulated. For instance, the authors show that the Schur extrapolation theorem does not hold for the sublinear Hardy operator in the Lp scale and for the classical Hardy operator in some reflective Orlicz spaces. Moreover, it is claimed that for a certain class of operators one can use Lorentz and Marcinkiewicz spaces instead of L1.and L∞ Proofs are not given. 

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