Complex Interpolation of Besov-Type Spaces on Domains

Let Omega subset of R-d (d >= 2) be a bounded Lipschitz domain. In this article, the author mainly studies complex interpolation of Besov-type spaces on the domain Omega, namely, we investigate the interpolation [B-p0,q0(s0 tau 0) (Omega), B-p1,q1(s1 tau 1)(Omega)](Theta) = B-p,q(lozenge s,tau)(Omega) under certain conditions on the parameters, where B-p,q(lozenge s,tau)(Omega) denotes the so-called diamond space associated with the Besov-type space. To this end, we first establish the equivalent characterization of the diamond space B-p,q(lozenge s,tau)(R-d) in terms of Littlewood-Paley decomposition and differences. Via some examples, we also show that this interpolation result does not hold under some other assumptions on the parameters or when Omega = Rd.