A counterexample in nonlinear interpolation

Abstract

If ( A 0 , A 1 ) ({A_0},{A_1}) is an interpolation pair with A 0 ⊂ A 1 {A_0} \subset {A_1} and T is a possibly nonlinear operator which maps A 0 {A_0} into A 0 {A_0} and A 1 {A_1} into A 1 {A_1} and satisfies ‖ T a ‖ A 0 ⩽ C ‖ a ‖ A 0 {\left \| {Ta} \right \|_{{A_0}}} \leqslant C{\left \| a \right \|_{{A_0}}} and ‖ T b − T b ′ ‖ A 1 ⩽ C ‖ b − b ′ ‖ A 1 {\left \| {Tb - Tb’} \right \|_{{A_1}}} \leqslant C{\left \| {b - b’} \right \|_{{A_1}}} for all a ∈ A 0 a \in {A_0} and b, b , b ′ ∈ A 1 b,b’ \in {A_1} and for some constant C, then it is known that T also maps the real interpolation spaces ( A 0 , A 1 ) θ , p {({A_0},{A_1})_{\theta ,p}} into themselves. We give an example showing that T need not map the complex interpolation spaces [ A 0 , A 1 ] θ {[{A_0},{A_1}]_\theta } into themselves. It is also seen that quasilinear operators may fail to preserve complex interpolation spaces.