Ball average characterizations of variable Besov-type spaces (Q2631765)

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) $. \par 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. \par Useful tools are the so-called doubling condition, the compatibility condition and a key pointwise estimate for some operators involving the decay function.