Variable Hardy spaces. (English) Zbl 1311.42053

E. Nakai and Y. Sawano [J. Funct. Anal. 262, No. 9, 3665–3748 (2012; Zbl 1244.42012)] developed a theory of variable Hardy spaces \(H^{p(\cdot)}\). The authors independently consider the same problem under a weaker condition for variable exponents \(p(x)\). Let \(0< p_{-} := \text{essinf}\, p(x) \) and \(0< p_{+} := \text{esssup}\, p(x) \). It is said that \(p(\cdot) \in M{\mathcal P}_0\) if there exists \(0 < p_0 < p_{-}\) such that the Hardy-Littlewood maximal operator \(M\) is bounded on \(L^{p(\cdot)/p_0}({\mathbb R}^n)\). Note that this condition is weaker than the \(\log\)-Hörder condition. When \(p(\cdot) \in M{\mathcal P}_0\), they define \(H^{p(\cdot)}\) by using the grand maximal function, and characterize \(H^{p(\cdot)}\) in terms of the radial maximal function and the tangential maximal function. They also prove an atomic decomposition by \((p(\cdot), \infty)\) atoms, and a finite atomic decomposition theorem by \((p(\cdot), q)\) atoms. The finite atomic decomposition is very useful for applications. They prove the boundedness of singular integral operators on \(H^{p(\cdot)}\). They also consider weighted Hardy spaces.


42B30 \(H^p\)-spaces
42B35 Function spaces arising in harmonic analysis
42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)


Zbl 1244.42012
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