Variable Hajłasz–Besov–Triebel–Lizorkin spaces and capacities in metric measure spaces

Abstract

Let \((\mathcal {X},d,\mu )\) be a metric measure space with \(\mu \) being doubling. In this article, via Hajłasz gradients, we define the variable Besov–Triebel–Lizorkin spaces on \(\mathcal {X}\) and then establish an approximation property of those spaces in terms of discrete \(\gamma \)-median. By a new concept of the variable power of variable mixed Lebesgue-sequence (quasi-)norms, we introduce the variable Besov–Triebel–Lizorkin \(p(\cdot )\)-capacity on \(\mathcal {X}\) and also obtain an equivalent expression of the \(p(\cdot )\)-capacity. Moreover, the lower and upper bound estimates for these \(p(\cdot )\)-capacity in terms of a modified version of the generalized Netrusov–Hausdorff content are established