Θ-type Calderón-Zygmund Operators with Non-doubling Measures

Let μ be a Radon measure on R d 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 H1,∞atb ( μ ) into L 1 ( 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 < ∞ ).