  We define a scale of L^q Carleson norms, all of which characterize the
membership of a function in BMO. The phenomenon is analogous to the
John-Nirenberg inequality, but on the level of Carleson measures. The classical
Carleson condition corresponds to the L^2 case in our theory.
  The result is applied to give a new proof for the L^p-boundedness of
paraproducts with a BMO symbol. A novel feature of the argument is that all p
are covered at once in a completely interpolation-free manner. This is achieved
by using the L^1 Carleson norm, and indicates the usefulness of this notion.
Our approach is chosen so that all these results extend in a natural way to the
case of X-valued functions, where X is a Banach space with the UMD property.
