Pointwise characterizations of variable Besov and Triebel-Lizorkin spaces via Hajłasz gradients

Let \(p(\cdot ),\ q(\cdot )\) $p(·),q(·)$ and \(\alpha (\cdot )\) $\alpha (·)$ be variable exponents satisfying some Hölder continuous conditions. In this paper, the authors characterize the variable inhomogeneous Besov space \(B_{p(\cdot ),q(\cdot )}^{\alpha (\cdot )}(\mathbb {R}^n)\) $B_{p(·),q(·)}^{\alpha (·)}({\mathbb{R}}^n)$ and the variable inhomogeneous Triebel-Lizorkin space \(F_{p(\cdot ),q(\cdot )}^{\alpha (\cdot )}(\mathbb {R}^n)\) $F_{p(·),q(·)}^{\alpha (·)}({\mathbb{R}}^n)$ in terms of Hajłasz gradient, via establishing the boundedness of a kind of almost diagonal operator A on the corresponding sequence spaces.