Θ-type Calderón-Zygmund operators with non-doubling measures

Let µ be a Radon measure on ℝd which may be non-doubling. The only condition that µ must satisfy is µ(B(x, r)) ≤ Crn for all x∈ℝd, r > 0 and for some fixed 0 < n ≤ d. In this paper, under this assumption, we prove that θ-type Calderón-Zygmund operator which is bounded on L2(µ) is also bounded from L∞(µ) into RBMO(µ) and from Hatb1,∞ (µ) into L1(µ). According to the interpolation theorem introduced by Tolsa, the Lp(µ)-boundedness (1 < p < ∞) is established for θ-type Calderón-Zygmund operators. Via a sharp maximal operator, it is shown that commutators and multilinear commutators of θ-type Calderón-Zygmund operator with RBMO(µ) function are bounded on Lp(µ) (1 < p < ∞).