Higher order estimates in complex interpolation theory

Richard Rochberg

Vol. 174 (1996), No. 1, 247–267

DOI: 10.2140/pjm.1996.174.247

Abstract

Suppose {A_𝜃}_0≤𝜃≤1 is a scale of Banach spaces generated from the couple (A₀,A₁) by complex interploation. If T is a linear operator which is bounded on the couple then T is bounded on the entire scale. Also, associated to the scale is an operator Ω₁ which is generally nonlinear and unbounded on A_1∕2 such that the commutator [T,Ω₁] is bounded on A_1∕2. Here we extend the construction and produce a sequence Ω₂,Ω₃,… which are increasingly nonlinear and unbounded but such that certain combinations with T are bounded. The first example is [T,Ω₂] − Ω₁[T,Ω₁].

Mathematical Subject Classification 2000

Primary: 46M35

Secondary: 47B38

Milestones

Received: 1 November 1993

Published: 1 May 1996