Variable Besov Spaces Associated with Heat Kernels

Abstract

Let $$({\mathcal {X}},\rho , \mu )$$ be a space of homogeneous type. Suppose that $$p(\cdot ),\ q(\cdot ):\ {\mathcal {X}}\rightarrow (0,\infty ]$$ are such that both $$1/p(\cdot )$$ and $$1/q(\cdot )$$ satisfy the globally log-Holder continuous condition, and $$s(\cdot ):\ {\mathcal {X}}\rightarrow \mathbb R$$ is a bounded function satisfying the locally log-Holder continuous condition. In this article, the authors introduce the variable Besov space $$B_{p(\cdot ),q(\cdot )}^{s(\cdot ),L}({\mathcal {X}})$$ , associated with a nonnegative self-adjoint operator L whose heat kernels satisfy small time Gaussian upper bound estimates, the Holder continuity, and the Markov property, which is new even on the sphere and the ball of $$\mathbb R^d$$ . Equivalent characterizations of this space, in terms of Peetre maximal functions and the heat semigroup, are established. Moreover, under the additional assumptions that $$\mu $$ satisfies the reverse doubling condition and the non-collapsing condition, its frame characterization is obtained. When L is the Laplacian operator on $$\mathbb R^d$$ , this space coincides with the existing variable Besov space.