We prove imbedding and
multiplier theorems for discrete Littlewood–Paley spaces introduced by M.
Frazier and B. Jawerth in their theory of wavelet-type decompositions of
Triebel–Lizorkin spaces. The corresponding inequalities for discrete spaces
defined in terms of characteristic functions of dyadic cubes, with respect
to an arbitrary positive locally finite measure on the Euclidean space, are
useful in the theory of tent spaces, weighted inequalities, duality theorems,
interpolation by analytic and harmonic functions, etc. Our main tools are
vector-valued maximal inequalities, a dyadic version of the Carleson measure
theorem, and Pisier’s factorization lemma. We also consider more general
inequalities, with an arbitrary family of measurable functions in place of
characteristic functions of dyadic cubes, which occur in the factorization theory of
operators.
