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Let X be an RD-space, namely, a metric space enjoying both doubling and reverse doubling properties. In this paper, for all s 2 (i1;1) and p;q 2 (0;1), the authors introduce the grand Besov spaces A _ B s(X ) and grand Triebel-Lizorkin spaces A _ F s p;q(X ), and prove that when † 2 (0;1), jsj < † and p 2 (maxfn=(n + †);n=(n + † + s)g;1), A _ B s (X ) \ ( " G † 0 (fl;∞)) 0 = _ B s (X ) with q 2 (0;1) and A _ F s p;q(X )\( " G † 0(fl;∞)) 0 = _ F s p;q(X ) with q 2 (maxfn=(n+†);n=(n+†+s)g;1) for all admissible fl and ∞, where " G † 0 (fl;∞) is the space of test functions. As applications, the authors obtain some real interpolation results on these grand Besov and Triebel-Lizorkin spaces. The corresponding results for inhomogeneous spaces are also presented.

Real interpolation for grand Besov and Triebel-Lizorkin spaces on RD-spaces