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.Comment: Collect. Math. (to appear

Dyadic Sets, Maximal Functions and Applications on ax+bax+b --Groups

Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure $\rho$, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,\rho)$ admits a Calder\'on--Zygmund decomposition which involves a particular family of sets, called Calder\'on--Zygmund sets. In this paper, we first show the existence of a dyadic grid in the group $S$, which has {nice} properties similar to the classical Euclidean dyadic cubes. Using the properties of the dyadic grid we shall prove a Fefferman--Stein type inequality, involving the dyadic maximal Hardy--Littlewood function and the dyadic sharp dyadic function. As a consequence, we obtain a complex interpolation theorem involving the Hardy space $H^1$ and the $BMO$ space introduced in [Collect. Math. 60(2009), 277--295].Comment: Math. Z. (to appear