## 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

### Keywords:

Hardy spaces; variable exponents; Campanato spaces
Full Text:

### References:

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