Boundedness of vector-valued Calderón-Zygmund operators on non-homogeneous metric measure spaces

Let ( X , d , μ ) be a non-homogeneous metric measure space and satisfies non-atomic condition that μ ( { x } ) = 0 for all x ∈ X . B 1 and B 2 are Banach spaces. In this paper, we show that the boundedness of the vector-valued Calderón-Zygmund operator T → from L 2 ( X , B 1 ) to L 2 ( X , B 2 ) is equivalent to T → from L p ( X , B 1 ) to L p ( X , B 2 ) for p ∈ ( 1 , ∞ ) , and from L 1 ( X , B 1 ) to L 1 , ∞ ( X , B 2 ) . As an application, we prove that if T → is bounded from L 2 ( X , B 1 ) to L 2 ( X , B 2 ) , then its maximal operator is bounded from L p ( X , B 1 ) to L p ( X , B 2 ) for p ∈ ( 1 , ∞ ) and from L 1 ( X , B 1 ) to L 1 , ∞ ( X , B 2 ) .