# Continuous characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators

## TL;DR

This paper provides new continuous characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces associated with certain self-adjoint operators on metric measure spaces, extending classical Euclidean results to more general settings.

## Contribution

It extends classical characterizations of Besov and Triebel-Lizorkin spaces to inhomogeneous spaces linked to self-adjoint operators on metric measure spaces, covering the full range of indices.

## Key findings

- Characterizations via Littlewood-Paley functions
- Characterizations via Lusin area functions
- Extension to general metric measure spaces

## Abstract

Let $(M,\rho,\mu)$ be a metric measure space satisfying the doubling, reverse doubling and non-collapsing conditions, and $\mathscr{L}$ be a self-adjoint operator on $L^2 (M, d\mu)$ whose heat kernel $p_t (x,y)$ satisfy the small-time Gaussian upper bound, H\"{o}lder continuity and Markov property. In this paper, we give characterizations of inhomogeneous "classical" and "non-classical" Besov and Triebel-Lizorkin spaces associated to $\mathscr{L}$ in terms of continuous Littlewood-Paley and Lusin area functions defined by the heat semigroup, for complete range of indices. This extends related classical results for Besov and Triebel-Lizorkin spaces on $\mathbb{R}^n$ to more general setting, and extends corresponding results in [Trans. Amer. Math Soc. 367 (2015), 121-189] to complete range of indices.