In this paper, we introduce a new scale of tent spaces which covers, the (weighted) tent spaces of Coifman-Meyer-Stein and of Hofmann-Mayboroda-McIntosh, and some other tent spaces considered by Dahlberg, Kenig-Pipher and Auscher-Axelsson in elliptic equations. The strong factorizations within our tent spaces, with applications to quasi-Banach complex interpolation and to multiplierduality theory, are established. This way, we unify and extend the corresponding results obtained by Coifman-Meyer-Stein, Cohn-Verbitsky and Hytönen-Rosén.
Basic notations and article structureLet R n+1Here and below, the capital letter B denotes an open ball in R n , and | · | denotes the Euclidean distance on R n .Given α > 0, we shall denote the cone, of aperture α and with vertex x ∈ R n , byand shall denote the tent, of aperture α and with base B ⊂ R n , byIf α = 1, we write the two standard objects simply as Γ(x) and B. Surrounding a point (y, t) ∈ R n+1 + , we construct its Whitney box asHere, the two numbers (α 1 , α 2 ) with α 1 > 0 and α 2 > 1, are called the Whitney parameters. They are said to be consistent if 0 < α 1 < α −1 2 < 1. Throughout this article, the set of Vinogradov notations { , ≃, } will be used. For two quantities a and b, which can be function values, set volumes, function norms or anything else, the term a b means that there exists a constant C > 0, which depends on parameters at hand, such that a ≤ Cb. In a similar way, a b means b a, and, a ≃ b means both a b and a b.This paper is organized as follows.