Boundedness of Calderón–Zygmund operators on ball Campanato-type function spaces

Abstract

Let X be a ball quasi-Banach function space on R n satisfying some mild assumptions. In this article, the authors first find a reasonable version T of the Calderón-Zygmund operator T on the ball Campanato-type function space L X,q,s,d (R n ) with q ∈ [1, ∞), s ∈ Z n + , and d ∈ (0, ∞). Then the authors prove that T is bounded on L X,q,s,d (R n ) if and only if, for any γ ∈ Z n + with |γ| ≤ s, T * (x γ ) = 0, which is hence sharp. Moreover, T is proved to be the adjoint operator of T , which further strengthens the rationality of the definition of T . All these results have a wide range of applications. In particular, even when they are applied, respectively, to weighted Lebesgue spaces, variable Lebesgue spaces, Orlicz spaces, Orliczslice spaces, Morrey spaces, mixed-norm Lebesgue spaces, local generalized Herz spaces, and mixed-norm Herz spaces, all the obtained results are new. The proofs of these results strongly depend on the properties of the kernel of T under consideration and also on the dual theorem on L X,q,s,d (R n ).