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Carleson measures for the Hardy space 1 Carleson measures for the Hardy spaces
 

Summary: Carleson measures for the Hardy space
N.A.
07
1 Carleson measures for the Hardy spaces
Let 1 (1, ). A positive Borel measure µ on D is a Carleson measure for
Hp
(D) if the imbedding
i : Hp
(D) Lp
(µ)
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
such that
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

  

Source: Arcozzi, Nicola - Dipartimento di Matematica, Universitą di Bologna

 

Collections: Mathematics