Besov spaces with non-doubling measures

Suppose that mu is a Radon measure on R-d, which may be non-doubling. The only condition on mu is the growth condition, namely, there is a constant C-0 > 0 such that for all x is an element of supp (mu) and r > 0, m(B(x,r)) <= C(0)r(n), where 0 < n <= d. In this paper, the authors establish a theory of Besov spaces. B-pq(s)(mu) for 1 <= p, q = <= infinity and vertical bar s vertical bar < theta where theta > 0 is a real number which depends on the non-doubling measure mu, C-0, n and d. The method used to de. ne these spaces is new even for the classical case. As applications, the lifting properties of these spaces by using the Riesz potential operators and the dual spaces are obtained.