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Variable Besov and Triebel–Lizorkin spaces. (English) Zbl 1160.46025

In recent times, the scales of the spaces \(B^s_{pq} (\mathbb R^n)\) and \(F^s_{pq} (\mathbb R^n)\) have been generalised admitting that one or several of the parmeters \(s,p,q\) are no longer constant but may vary on \(\mathbb R^n\). In the paper under review, \(s,q\) are constant, but \(p\) is replaced by \(p(x)\), \(x \in \mathbb R^n\), such that maximal inequalities for the underlying variable Lebesgue spaces \(L_{p(\cdot)} (\mathbb R^n)\) remain valid. Under this hypothesis, the author proves maximal inequalities for the corresponding spaces \(B^s_{p(\cdot),q} (\mathbb R^n)\) and \(F^s_{p(\cdot),q} (\mathbb R^n)\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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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