Monotonicity properties of interpolation spaces

For any interpolation pair (A ₀ A ₁), Peetre’sK-functional is defined by: $$Kleft( {t,a;A_0 ,A_1 } right) = mathop {inf }limits_{a = a_0 + a_1 } left( {left| {a_0 } right|_{A_0 } + tleft| {a_1 } right|_{A_1 } } right).$$ 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 _v ^p , L _w ^q ) areK-monotone.