Monotonicity properties of interpolation spaces

Abstract

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.