11,228 research outputs found
New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators
We introduce a new class of Hardy spaces , called Hardy spaces of Musielak-Orlicz type, which generalize the
Hardy-Orlicz spaces of Janson and the weighted Hardy spaces of Garc\'ia-Cuerva,
Str\"omberg, and Torchinsky. Here, is a function such that is an Orlicz function
and is a Muckenhoupt weight. A function
belongs to if and only if its maximal
function is so that is integrable. Such a
space arises naturally for instance in the description of the product of
functions in and respectively (see
\cite{BGK}). We characterize these spaces via the grand maximal function and
establish their atomic decomposition. We characterize also their dual spaces.
The class of pointwise multipliers for characterized by
Nakai and Yabuta can be seen as the dual of where is the Hardy space of
Musielak-Orlicz type related to the Musielak-Orlicz function
. Furthermore, under
additional assumption on we prove that if is a
sublinear operator and maps all atoms into uniformly bounded elements of a
quasi-Banach space , then uniquely extends to a bounded
sublinear operator from to . These results are new even for the classical Hardy-Orlicz spaces on
.Comment: Integral Equations and Operator Theory (to appear
A note on -boundedness of Riesz transforms and -Calder\'on-Zygmund operators through molecular characterization
Let and in the Muckenhoupt class . Recently, by using
the weighted atomic decomposition and molecular characterization; Lee, Lin and
Yang \cite{LLY} (J. Math. Anal. Appl. 301 (2005), 394--400) established that
the Riesz transforms , are bounded on .
In this note we extend this to the general case of weight in the
Muckenhoupt class through molecular characterization. One
difficulty, which has not been taken care in \cite{LLY}, consists in passing
from atoms to all functions in . Furthermore, the
-boundedness of -Calder\'on-Zygmund operators are also given
through molecular characterization and atomic decomposition.Comment: to appear in Anal. Theory. Appl. 27 (2011), no. 3, 251-26
Endpoint estimates for commutators of singular integrals related to Schr\"odinger operators
Let be a Schr\"odinger operator on , ,
where is a nonnegative potential, , and belongs to the reverse
H\"older class . In this paper, we study the commutators for
in a class of sublinear operators containing the fundamental
operators in harmonic analysis related to . More precisely, when , we prove that there exists a bounded subbilinear operator
such that , where
is a bounded bilinear operator from
into which does not depend on . The subbilinear
decomposition (\ref{abstract 1}) explains why commutators with the fundamental
operators are of weak type , and when a commutator is of
strong type . Also, we discuss the -estimates for
commutators of the Riesz transforms associated with the Schr\"odinger operator
.Comment: Rev. Mat. Iberoam. (to appear
On the product of functions in and over spaces of homogeneous type
Let be an RD-space, which means that is a space of
homogeneous type in the sense of Coifman-Weiss with the additional property
that a reverse doubling property holds in . The aim of the present
paper is to study the product of functions in and in this setting.
Our results generalize some recent results in \cite{Feu} and \cite{LP}.Comment: J. Math. Anal. Appl. (to appear
Factorization of some Hardy type spaces of holomorphic functions
We prove that the pointwise product of two holomorphic functions of the upper
half-plane, one in the Hardy space , the other one in its dual,
belongs to a Hardy type space. Conversely, every holomorphic function in this
space can be written as such a product. This generalizes previous
characterization in the context of the unit disc.Comment: C. R. Math. Acad. Sci. Paris (to appear
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