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Session 4. Banach Spaces and Operator Theory with Applications
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Carleson measures and compostion operators on abstract Hardy spaces |
Luis Rodríguez-Piazza, Universidad de Sevilla, Spain
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In this talk I will report on some recent results about abstract
Hardy spaces obtained in collaboration with Mieczysław Mastyło
(Adam Mickiewicz University Poznań, Poland).
Let \(\mathbb{D}\) be the open unit disk of the complex plane and
the torus \(\mathbb{T}=\partial\mathbb{D}\) its border equipped with
the normalized length measure \(m\). Essentially in the same way
that Hardy spaces \(H^p(\mathbb{D})\) are defined out of Lebesgue
spaces \(L^p(\mathbb{T})\), one can define an abstract Hardy space
\(H\!X\) out of any (quasi-)Banach function space \(X\) on the torus
\(\mathbb{T}\):
\[
H\!X:= \big\{ f\in H(\mathbb{D});\; \|f\|_{H\!X}:=\sup_{0\leq r<1}
\|f_r\|_X<+\infty \big\}.
\]
We will consider mainly the case where \(X\) is a symmetric (that
is, rearrangement invariant) Banach space on \((\mathbb{T},m)\).
Particular classes of abstract Hardy spaces are Hardy-Orlicz,
Hardy-Lorentz and Hardy-Marcinkie\-wicz spaces, which correspond
respectively to the cases where \(X\) is an Orlicz, a Lorentz or a
Marcinkiewicz space on \((\mathbb{T},m)\). In these cases we have
\(H\!X=\{ f\in H^1(\mathbb{D}) : \widetilde{f}\in X\}\), where
\(\widetilde{f}\) is the boundary value of \(f\) (the radial
limits).
Motivated by the study of composition operators on \(H\!X\) we study
the inclusion of \(H\!X\) into a Banach symmetric space \(Y(\mu)\),
for \(\mu\) a finite measure on \(\mathbb{D}\). We investigate the
relationships between boundedness or compactness of the inclusion of
\(H\!X\) in \(Y(\mu)\) and some conditions on \(\mu\) which are
variants of the requirement for \(\mu\) to be a Carleson measure. An
important role in the definition of these conditions is played by
the fundamental functions of \(X\) and \(Y\). In particular we will
give a characterization of the compactness of composition operators
on Hardy-Lorentz and Hardy-Marcinkiewicz spaces. This
characterization is similar to the one given for Hardy-Orlicz spaces
by Lef\`evre, Li, Queffélec and Rodríguez-Piazza
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References- P. Lefévre, D. Li, H. Queffélec and
L.\, Rodríguez-Piazza, Composition operators on Hardy
Orlicz spaces , Memoirs of the American Mathematical Society,
Vol. 207, American Mathematical Society, Providence, RI, 2010.
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