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Inhomogeneous Besov spaces associated to operators with off-diagonal semigroup estimates. (English) Zbl 1373.46023

The classical Besov spaces \(B^\alpha_{p,q} (\mathbb R^n)\) with \(0<p,q \leq \infty\) and \(\alpha \in \mathbb R\) on the Euclidean \(n\)-space \(\mathbb R^n\) can be introduced in the context of the spectral theory of the Laplacian \(L = -\Delta\) and related to the Gauss-Weierstrass semigroup \(e^{t \Delta}\), \(t>0\), extended to \(t^a (-\Delta)^b e^{t \Delta}\). The paper deals with a generalization of this approach with homogeneous spaces \((X, d, \mu)\) as a substitute of \(\mathbb R^n\). Here, \(X\) is a set, \(d\) is a distance and \(\mu\) is a doubling measure. The substitute of \(-\Delta\) is a suitable class of operators \(L\) admitting a functional calculus and generating a counterpart of the Gauss-Weierstrass semigroup defined by related kernel estimates. The authors study the corresponding spaces \(B^{\alpha, L}_{p,q} (X)\), concentrating on molecular decompositions and lifting properties. They compare these spaces with previously introduced spaces on \((X,d,\mu)\).

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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Full Text: Euclid