Intermediate Assouad-like dimensions for measures

The upper and lower Assouad dimensions of a metric space are local variants of the box dimensions of the space and provide quantitative information about the thickest' and thinnest’ parts of the set. Less extreme versions of these dimensions for sets have been introduced, including the upper and lower quasi-Assouad dimensions, $\theta$-Assouad spectrum, and $\mathrm{\Phi }$-dimensions. In this paper, we study the analogue of the upper and lower $\mathrm{\Phi }$-dimensions for measures. We give general properties of such dimensions, as well as more specific results for self-similar measures satisfying various separation properties and discrete measures.