WEIGHTED INEQUALITIES INVOLVING TWO HARDY OPERATORS WITH APPLICATIONS TO EMBEDDINGS OF FUNCTION SPACES

Abstract

We find necessary and sufficient conditions for the two-operator weighted inequality ¥ R 0 1 t R 0 f(s)ds q w(t)dt 1/q 6C ¥ R 0 ¥ R t f(s) s ds p v(t)dt 1/p . We use this inequality to study embedding properties between the function spaces S p (u) equipped with the norm kfkSp(u) = ¥ R 0 (f (t) f (t)) p u(t)dt 1/p and the classical Lorentz spacesL p (v) and G q (w). Moreover, we solve the only missing open case of the embedding L p (v) ,! G q (w), where 0 < q < p 6 1.