Arkiv för Matematik

  • Ark. Mat.
  • Volume 14, Number 1-2 (1976), 213-236.

Monotonicity properties of interpolation spaces

Michael Cwikel

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For any interpolation pair (A0A1), Peetre’s K-functional is defined by: $K\left( {t,a;A_0 ,A_1 } \right) = \mathop {\inf }\limits_{a = a_0 + a_1 } \left( {\left\| {a_0 } \right\|_{A_0 } + t\left\| {a_1 } \right\|_{A_1 } } \right).$

It is known that for several important interpolation pairs (A0, A1), all the interpolation spaces A of the pair can be characterised by the property of K-monotonicity, that is, if a∈A and K(t, b; A0, A1)≦K(t, a; A0, A1) for all positive t then b∈A also.

We give a necessary condition for an interpolation pair to have its interpolation spaces characterized by K-monotonicity. We describe a weaker form of K-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 between K-monotonicity and weak K-monotonicity. Finally we give an alternative proof of a result of Gunnar Sparr, that all the interpolation spaces for (L ${}_{v}^{p}$ , L ${}_{w}^{q}$ ) are K-monotone.


Research supported by the C. N. R. S.

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Ark. Mat., Volume 14, Number 1-2 (1976), 213-236.

Received: 5 January 1976
First available in Project Euclid: 31 January 2017

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1976 © Institut Mittag-Leffler


Cwikel, Michael. Monotonicity properties of interpolation spaces. Ark. Mat. 14 (1976), no. 1-2, 213--236. doi:10.1007/BF02385836.

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