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Function spaces of variable smoothness and integrability. (English) Zbl 1179.46028

The spaces \(B^s_{p,q} (\mathbb R^n)\) and \(F^s_{p,q} (\mathbb R^n)\) with \(s \in \mathbb R\) \(0< p,q \leq \infty\) (\(p < \infty\) for the \(F\)-spaces), are well established nowadays, covering the Lebesgue spaces, (fractional) Sobolev spaces, Besov spaces, Hardy spaces, etc. Since the early 1990s, there have appeared several papers dealing with corresponding spaces with variable smoothness \(s(x)\) or variable integrability \(p(x)\). The paper under review studies spaces \(F^{s(\cdot)}_{p(\cdot), q(\cdot)} (\mathbb R^n)\), where simultaneously all three parameters may vary under some restrictions: \(0< p^- \leq p(x) \leq p^+ < \infty\), \(0< q^- \leq q(x) \leq q^+ < \infty\), \(0 \leq s(x) \leq s^+ < \infty\), \(s(x) \to s^\infty\) if \(|x| \to \infty\). Furthermore, \(p(\cdot)\), \(q(\cdot)\), \(s(\cdot)\) are assumed to be logarithmically continuous. The spaces \(F^{s(\cdot)}_{p(\cdot), q(\cdot)} (\mathbb R^n)\) are defined, as in the classical case, by Fourier-analytical decompositions. The authors show that this definition makes sense. They prove atomic and molecular representation theorems. There are some interesting assertions about traces on hyper-planes. Furthermore, it is shown that this approach covers many previously studied spaces with variable smoothness and/or variable integrability. The restriction \(s(x) \geq 0\) seems to be unavoidable so far.

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.)
42B35 Function spaces arising in harmonic analysis
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References:

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