Besov-Type and Triebel--Lizorkin-Type Spaces Associated with Heat Kernels

Let $(M,\rho ,\mu )$ be an RD-space satisfying the non-collapsing condition. In this paper, the authors introduce Besov-type spaces $B_{p,q}^{s,\tau }(M)$ and Triebel–Lizorkin-type spaces $F_{p,q}^{s,\tau }(M)$ associated to a non-negative self-adjoint operator $L$ whose heat kernels satisfy some Gaussian upper bound estimate, H"older continuity, and the stochastic completeness property. Characterizations of these spaces via Peetre maximal functions and heat kernels are established for full range of indices. Also, frame characterizations of these spaces are given. When $L$ is the Laplacian operator on ${\mathbb{R}}^n$, these spaces coincide with the Besov-type and Triebel-Lizorkin-type spaces on ${\mathbb{R}}^n$ studied in [Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010]. In the case $\tau =0$ and the smoothness index $s$ is around zero, comparisons of these spaces with the Besov and Triebel–Lizorkin spaces studied in [Abstr. Appl. Anal. 2008, Art. ID 893409, 250 pp] are also presented.