Summary: Carleson measures for the Hardy space
1 Carleson measures for the Hardy spaces
Let 1 (1, ). A positive Borel measure µ on D is a Carleson measure for
(D) if the imbedding
i : Hp
is everywhere defined and continuous. If such is the case, we write µ CM(Hp
We let µ CM(Hp) to be the norm of i.
Theorem 1 A measure µ is Carleson for Hp
(D) if and only if there is C > 0
µ(S(z)) C|I(z)|, z D. (1)
Moreover, the least constant C for which (1) holds is comparable with µ CM(Hp).
Theorem 1 will follow almost immediately from the analogous statement for the
harmonic Hardy spaces. Recall that