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Atomic decompositions of Hardy spaces with variable exponents and its application to bounded linear operators. (English) Zbl 1293.42025

Let \(p(\cdot): \mathbb{R}^n \to (0,\infty)\), and assume a certain log-Hölder continuity and a decay condition on \(p\). The Hardy space \(H^{p(\cdot)}(\mathbb{R}^n)\) with variable exponent \(p(\cdot)\) is defined as the set of all distributions \(f \in \mathcal{S'}(\mathbb{R}^n)\) for which \[ \|f\|_{H^{p(\cdot)}} :=\|\sup_{t>0} |e^{t\Delta}f|\|_{L^{p(\cdot)}}< \infty. \] Here, \(L^{p(\cdot)}\) denotes the variable exponent Lebesgue space as a special case of generalised Orlicz spaces. The theory of such spaces goes back to the work of H. Nakano [Modulared semi-ordered linear spaces. Tokyo Math. Book Series, Vol. 1. Tokyo: Maruzen (1950; Zbl 0041.23401)] and J. Musielak and W. Orlicz [Stud. Math. 18, 49–65 (1959; Zbl 0086.08901)] and was motivated by the study of differential equations with \(p(\cdot)\)-growth conditions and related energy estimates.
In continuation of [E. Nakai and Y. Sawano, J. Funct. Anal. 262, No. 9, 3665–3748 (2012; Zbl 1244.42012)], the paper under review establishes a refined atomic decomposition of Hardy spaces with variable exponents. More precisely, the result allows for an atomic decomposition with \(q\)-atoms, with \(q>p_+\) when \(p_+=\sup_{x \in \mathbb{R}^n} p(x) \geq 1\), and \(q\geq 1\) when \(p_+<1\), whereas the earlier result required \(q \geq 1\) to be sufficiently large. As an application, the author shows boundedness of fractional integral operators \(I_\alpha\) from \(H^{p(\cdot)}\) to \(H^{q(\cdot)}\), with \(\frac{1}{q(\cdot)} = \frac{1}{p(\cdot)} -\frac{\alpha}{n}\), boundedness of commutators of Calderón-Zygmund operators with \(\mathrm{BMO}(\mathbb{R}^n)\) functions from \(H^{p(\cdot)}\) to \(L^{p(\cdot)}\), and boundedness of the Hardy operator from \(H^{p(\cdot)}\) to \(L^{p(\cdot)}\). In a last paragraph, it is shown that the Hausdorff-Young inequality cannot be extended to variable exponent Lebesgue spaces in general.

MSC:

42B30 \(H^p\)-spaces
42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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