Homogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators
The theory of the homogeneous spaces \(\dot{B}^s_{p,q}\) and \(\dot{F}^s_{p,q}\) on \(\mathbb R^n\) with \(s\in\mathbb R\), \(0<p,q \leq \infty\) (\(p<\infty\) for \(F\)-spaces) can be developed on the basis of the (essentially) self-adjoint Laplacian \(L = -\Delta\) in \(L_2 (\mathbb R^n)\), its powers \((-\Delta)^m\) and, in particular, the related Gauss-Weierstrass semi-group \(e^{-tL} = e^{t\Delta}\), \(t>0\), and their kernels with the well-known miraculous properties. Spectral theory, that is \(f(L)\) for suitable real bounded measurable functions \(f\) on \(\mathbb R_+\) (preferably cut-off functions) can be taken as a starting point to construct suitable frame representations for \(\dot{B}^s_{p,q}\) and \(\dot{F}^s_{p,q}\) on \(\mathbb R^n\) in the context of the dual pairing \(\big( \dot{S} (\mathbb R^n), \dot{S}' (\mathbb R^n) \big)\) (modulo polynomials). The paper extends this approach to metric measure spaces \((M,\rho, \mu)\) where \(\rho\) is a distance and \(\mu\) a doubling Radon measure. It is assumed that there is a substitute \(L\) of the above Laplacian, a self-adjoint non-negative operator in \(L_2 (M, d\mu)\), imitating crucial properties of the Gauss-Weierstrass semi-group and the related kernels. On this basis the authors develop a theory of corresponding homogeneous spaces \(\dot{B}^s_{p,q} (L)\) and \(\dot{F}^s_{p,q} (L)\) (in two versions), including frame representations.
scientific article; zbMATH DE number 6682102
scientific article