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 \((X,d,\mu)\).


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
Full Text: Euclid