Abstract
For any interpolation pair (A 0 A 1), Peetre’sK-functional is defined by:
It is known that for several important interpolation pairs (A 0,A 1), all the interpolation spacesA of the pair can be characterised by the property ofK-monotonicity, that is, ifa∈A andK(t, b; A0, A1)≦K(t, a; A0, A1) for all positivet thenb∈A also.
We give a necessary condition for an interpolation pair to have its interpolation spaces characterized byK-monotonicity. We describe a weaker form ofK-monotonicity which holds for all the interpolation spaces of any interpolation pair and show that in a certain sense it is the strongest form of monotonicity which holds in such generality. On the other hand there exist pairs whose interpolation spaces exhibit properties lying somewhere betweenK-monotonicity and weakK-monotonicity. Finally we give an alternative proof of a result of Gunnar Sparr, that all the interpolation spaces for (L pv , L qw ) areK-monotone.
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Research supported by the C. N. R. S.
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Cwikel, M. Monotonicity properties of interpolation spaces. Ark. Mat. 14, 213–236 (1976). https://doi.org/10.1007/BF02385836
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DOI: https://doi.org/10.1007/BF02385836