New Hardy spaces of Musielak-Orlicz type and boundedness of sublinear operators

Autor: Ky, Luong Dang
Přispěvatelé: Mathématiques - Analyse, Probabilités, Modélisation - Orléans (MAPMO), Centre National de la Recherche Scientifique (CNRS)-Université d'Orléans (UO), Ky, Luong Dang
Jazyk: angličtina
Rok vydání: 2013
Předmět:
Mathematics::Functional Analysis
Hardy spaces
Hardy-Orlicz spaces
[MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA]
Mathematics::Classical Analysis and ODEs
BMO-multipliers
[MATH.MATH-GM] Mathematics [math]/General Mathematics [math.GM]
[MATH.MATH-CA]Mathematics [math]/Classical Analysis and ODEs [math.CA]
[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA]
[MATH.MATH-CA] Mathematics [math]/Classical Analysis and ODEs [math.CA]
Musielak-Orlicz functions
quasi-Banach spaces
Functional Analysis (math.FA)
Mathematics - Functional Analysis
atomic decompositions
[MATH.MATH-GM]Mathematics [math]/General Mathematics [math.GM]
Muckenhoupt weights
Mathematics - Classical Analysis and ODEs
Classical Analysis and ODEs (math.CA)
FOS: Mathematics
sublinear operators
42B35 (46E30
42B15
42B30)
Zdroj: Integral Equations and Operator Theory
Integral Equations and Operator Theory, Springer Verlag, 2014, 36 p
ISSN: 0378-620X
1420-8989
Popis: We introduce a new class of Hardy spaces $H^{\varphi(\cdot,\cdot)}(\mathbb R^n)$, 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, $\varphi: \mathbb R^n\times [0,\infty)\to [0,\infty)$ is a function such that $\varphi(x,\cdot)$ is an Orlicz function and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight. A function $f$ belongs to $H^{\varphi(\cdot,\cdot)}(\mathbb R^n)$ if and only if its maximal function $f^*$ is so that $x\mapsto \varphi(x,|f^*(x)|)$ is integrable. Such a space arises naturally for instance in the description of the product of functions in $H^1(\mathbb R^n)$ and $BMO(\mathbb R^n)$ 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 $BMO(\mathbb R^n)$ characterized by Nakai and Yabuta can be seen as the dual of $L^1(\mathbb R^n)+ H^{\rm log}(\mathbb R^n)$ where $ H^{\rm log}(\mathbb R^n)$ is the Hardy space of Musielak-Orlicz type related to the Musielak-Orlicz function $\theta(x,t)=\displaystyle\frac{t}{\log(e+|x|)+ \log(e+t)}$. Furthermore, under additional assumption on $\varphi(\cdot,\cdot)$ we prove that if $T$ is a sublinear operator and maps all atoms into uniformly bounded elements of a quasi-Banach space $\mathcal B$, then $T$ uniquely extends to a bounded sublinear operator from $H^{\varphi(\cdot,\cdot)}(\mathbb R^n)$ to $\mathcal B$. These results are new even for the classical Hardy-Orlicz spaces on $\mathbb R^n$.
Comment: Integral Equations and Operator Theory (to appear)
Databáze: OpenAIRE
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