COMPLEX INTERPOLATION ON BESOV-TYPE AND TRIEBEL-LIZORKIN-TYPE SPACES

Let theta is an element of (0, 1), s(0), s(1) is an element of R, tau(0), tau(1) is an element of [0, infinity), p(0), p(1) is an element of (0, infinity), q(0), q(1) is an element of (0, infinity], s = s(0)(1 - theta) + s(1)theta, tau = tau(0)(1 - theta) + tau(1)theta, 1/p = 1-theta/p(0) + theta/p(1) and 1/q = 1-theta/q(0) + theta/q(1). In this paper, under the restriction tau(0)/p(1) = tau(1)/p(0), the authors establish the complex interpolation, on Triebel-Lizorkin-type spaces, that [(F) over circle (s0,tau 0)(p0,q0) (R-n), (F) over circle (s1,tau 1)(p1,q1) (R-n)](theta) = (F) over circle (s,tau)(p,q)(R-n) = [F-p0,q0(s0,tau 0) (R-n), (F) over circle (s1,tau 1)(p1,q1) (R-n)](theta) = [(F) over circle (s0,tau 0)(p0,q0) (R-n), F-p1,q1(s1 tau 1) (R-n)](theta), where (F) over circle (s,tau)(p,q) (R-n) denotes the closure of the Schwartz functions in F-p,q(s,tau) (R-n). Similar results on Besov-type spaces and Besov-Morrey spaces are also presented. As a corollary, the authors obtain the complex interpolation for Morrey spaces that, for all 1 < p(0) <= u(0) < infinity, 1 < p(1) <= u(1) < infinity and 1 < p <= u < infinity such that 1/u = 1-theta/u(0) + theta/u(1), 1/p = 1-theta/p(0) + theta/p(1) and p(0)u(1) = p(1)u(0), [(M) over circle (u0)(p0)(R-n), (M) over circle (u1)(p1)(R-n)](theta) = (M) over circle (u)(p)(R-n) = [M-p0(u0)(R-n), (M) over circle (u1)(p1)(R-n)](theta) = [(M) over circle (u0)(p0)(R-n), M-p1(u1)(R-n)](theta), where (M) over circle (p)(u)(R-n) denotes the closure of the Schwartz space in M-u(p)(R-n). It is known that, if p(0)u(1) not equal p(1)u(0), these conclusions on Morrey spaces may not be true.