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Ball average characterizations of variable Besov-type spaces. (English) Zbl 1420.46034

The authors introduce and develop the Besov-type space with variable exponent \(B^{s(\cdot) \phi}_{p(\cdot)q(\cdot)}\), for any locally log-Hölder continuous functions \(s(\cdot)\in L^\infty(\mathbb{R}^n) \) and measurable function \(\phi \in \mathbb{R}^{n+1}_+\), under the assumption that \(p(\cdot),\, q(\cdot)\in P^{\log}(\mathbb{R}^n) \).
They characterize the spaces \(B^{s(\cdot) \phi}_{p(\cdot)q(\cdot)} \) by means of Peetre maximal functions and averages on balls. The latter one is new, even when \(\phi\) is equivalent to 1, and gives a way to introduce the variable Besov-type spaces on metric measure spaces.
Useful tools are the so-called doubling condition, the compatibility condition and a key pointwise estimate for some operators involving the decay function.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B35 Function spaces arising in harmonic analysis
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