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Besov Spaces Associated with Operators

Commun. Math. Anal.
Volume 16, Number 2 (2014), 89 - 104

Besov Spaces Associated with Operators

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Recent work of Bui, Duong and Yan in [2] defined Besov spaces associated with a certain operator $L$ under the weak assumption that $L$ generates an analytic semigroup $e^{-tL}$ with Poisson kernel bounds on $L^2({\mathcal X})$ where ${\mathcal X}$ is a (possibly non-doubling) quasi-metric space of polynomial upper bound on volume growth. This note aims to extend certain results in [2] to a more general setting when the underlying space can have different dimensions at $0$ and infinity. For example, we make some extensions to the Besov norm equivalence result in Proposition 4.4 of [2], such as to more general class of functions with suitable decay at $0$ and infinity, and to non-integer $k\geq1$.