Besov spaces with non-doubling measures

Abstract

Abstract. Suppose that µ is a Radon measure on R d , which may be nondoubling. The only condition on µ is the growth condition, namely, there is a constant C 0 > 0 such that for all x ∈ supp (µ) and r > 0,where 0 < n ≤ d. In this paper, the authors establish a theory of Besov spaceṡ B s pq (µ) for 1 ≤ p, q ≤ ∞ and |s| < θ, where θ > 0 is a real number which depends on the non-doubling measure µ, C 0 , n and d. The method used to define 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.

Some characterizations of upper doubling conditions on metric measure spaces

“…Calderón–Zygmund operator theory has been developed on such non‐homogeneous spaces; see for examples –, and . See also – and –, –, , for related function spaces, the Littlewood–Paley theory, vector‐valued inequalities, and weighted norm inequalities on such non‐homogeneous spaces.…”