REAL INTERPOLATION FOR GRAND BESOV AND TRIEBEL-LIZORKIN SPACES ON RD-SPACES

Let X be an RD-space, namely, a metric space enjoying both doubling and reverse doubling properties. In this paper, for all s is an element of [-1, 1] and p, q is an element of (0,infinity] the authors introduce the grand Besov spaces ABoverdot(p,q)(s)(X) and grand Triebel-Lizorkin spaces AFoverdot(p,q)(s)(X), and prove that when epsilon is an element of (0,1), vertical bar s vertical bar < epsilon and p is an element of (max{n/(n + epsilon), n/(n + epsilon +s)}, infinity], ABoverdot(p,q)(s)(X) boolean AND (Goverdot(0)(epsilon)(beta, gamma))' = Boverdot(p,q)(s)(X) with q is an element of (0, infinity] and AFoverdot(p,q)(s) (X) boolean AND (Goverdot(0)(epsilon)(beta, gamma)' =Foverdot(p,q)(s) (X) with q is an element of (max{n/(n + epsilon), n/(n+epsilon+s)},infinity] for all admissible beta and gamma, where (Goverdot(0)(epsilon) (beta, gamma) is the space of test functions. As applications, the authors obtain sonic real interpolation results on these grand Besov and Triebel-Lizorkin spaces. The corresponding results for inhomogeneous spaces are also presented.