Molecular Characterizations and Dualities of Variable Exponent Hardy Spaces Associated with Operators

Let $L$ be a linear operator on $L^2(\mathbb R^n)$ generating an analytic semigroup $\{e^{-tL}\}_{t\ge0}$ with kernels having pointwise upper bounds and $p(\cdot):\ \mathbb R^n\to(0,1]$ be a variable exponent function satisfying the globally log-H\"older continuous condition. In this article, the authors introduce the variable exponent Hardy space associated with the operator $L$, denoted by $H_L^{p(\cdot)}(\mathbb R^n)$, and the BMO-type space ${\mathrm{BMO}}_{p(\cdot),L}(\mathbb R^n)$. By means of tent spaces with variable exponents, the authors then establish the molecular characterization of $H_L^{p(\cdot)}(\mathbb R^n)$ and a duality theorem between such a Hardy space and a BMO-type space. As applications, the authors study the boundedness of the fractional integral on these Hardy spaces and the coincidence between $H_L^{p(\cdot)}(\mathbb R^n)$ and the variable exponent Hardy spaces $H^{p(\cdot)}(\mathbb R^n)$.Comment: 47 pages, Ann. Acad. Sci. Fenn. Math. (to appear

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

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