Besov-type and Triebel–Lizorkin-type spaces associated with heat kernels

Abstract

Let $$(M, \rho ,\mu )$$ be a space of homogeneous type satisfying the reverse doubling condition and 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 nonnegative self-adjoint operator $$L$$ whose heat kernel satisfies sub-Gaussian upper bound estimate, Hölder continuity, and stochastic completeness. The novelty in this article is that the indices $$p,q,s,\tau $$ here can be take full range of all possible values as in the Euclidean setting. Characterizations of these spaces via Peetre maximal functions and the heat semigroup are established for full range of possible 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 (Yuan et al. Lecture Notes in Mathematics, vol 2005, 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 (Han et al. Abstr Appl Anal 1–250, 2008, Art ID 893409) are also presented.