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Hardy spaces with variable exponents and generalized Campanato spaces. (English) Zbl 1244.42012

The authors define Hardy spaces with variable exponents and prove the atomic decomposition. As a corollary they prove the boundedness of singular integral operators. They also investigate the Littlewood-Paley characterization, and prove that the dual space of Hardy spaces are Campanato spaces with variable growth. Their theorem admits exponential functions \(p(x)\) such that \(p(x_0) <1\) and \(p(x_1) >0\) for some \(x_0\) and \(x_1\). The properties of the ordinary Hardy spaces \(H^p({\mathbb R}^n)\) are quite different according as \(p<1, p=1\) and \(p>1\). Therefore they develop methods which are valid for all cases \(0<p<\infty\) simultaneously.
Let \(\varphi \in S({\mathbb R}^n)\) be a function such that \(\int \varphi (x) dx \neq 0\). We define \[ H^{p(\cdot)}_{\varphi}({\mathbb R}^n) = \Big\{ f \in S'({\mathbb R}^n) : \| f \|_{H^{p(\cdot)}_{\varphi}} := \Big\| \sup_{t>0} | t^{-n} \varphi(t^{-1} \cdot)*f | \Big\|_{L^{p(\cdot)}} < \infty \Big\}. \] Let \(0 < p_{-} := \inf_{x \in {\mathbb R}^n} p(x) \leq p_{+} := \sup_{x \in {\mathbb R}^n} p(x) < q \leq \infty\) and \(q \geq 1\). Fix an integer \(d \geq \min \{ d \in {\mathbb N} \cup \{ 0 \} : p_{-} (n+d+1) >n \}\). A function \(a\) on \({\mathbb R}^n\) is called a \((p(\cdot), q)\)-atom if there exists a cube \(Q\) such that \[ \begin{aligned} &\text{supp} (a) \subset Q, \\ & \| a \|_{L^q} \leq \frac{| Q |^{1/q}}{\| \chi_Q \|_{L^{p(\cdot)}}}, \\ &\int a(x) x^{\alpha} dx = 0 \quad \text{for} \quad | \alpha | \leq d. \end{aligned} \] For sequences of nonnegative numbers \(\{ c_j \}\) and cubes \(\{ Q_j \}\), define \[ A( \{ c_j \}, \{ Q_j \}) := \inf \Big\{ \lambda >0 : \int_{{\mathbb R}^n} \Big( \sum_{j=1}^{\infty} \big( \frac{c_j \chi_{Q_j} (x)}{ \lambda \| \chi_{Q_j} \|_{L^{p(\cdot)}}} \big)^{\underline{p}} \Big)^{p(x)/{\underline{p}}} dx \leq 1 \Big\}, \] where \(\underline{p} := \min (p_{-},1)\).
They prove the following: Under suitable conditions on \(p(x)\), every \(f \in H^{p(\cdot)}_{\varphi}({\mathbb R}^n)\) can be written as \[ f = \sum_{j=1}^{\infty}c_j a_j \] where \(a_j\) are \((p(\cdot), q)\)-atoms supported in \(Q_j\) and \(A( \{ c_j \}, \{ Q_j \}) \) is finite.

MSC:

42B30 \(H^p\)-spaces
46E15 Banach spaces of continuous, differentiable or analytic functions
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