Exploring simultaneous characterizations of the generalized local fractal dimension functions

For $ x\in \mathbb R^n $ and $ F\subseteq \mathbb R^n $, we introduce the generalized local Hausdorff dimension function of $ F $ at $ x $ and the generalized local packing dimension function of $ F $ at $ x $ as $ \mathscr D^{f,g}_{\mathscr H,\text{loc}}(x,F) = \lim\limits_{r\to 0}\mathscr D^{f,g}_{\mathscr H}\big(F\cap\mathsf{B}_r(x)\big) $ and$ \mathscr D^{f,g}_{\mathscr P,\text{loc}}(x,F) = \lim\limits_{r\to 0}\mathscr D^{f,g}_{\mathscr P}\big(F\cap\mathsf{B}_r(x)\big), $respectively, where $ \mathscr D^{f,g}_{\mathscr H} $ is the generalized Hausdorff dimension, $ \mathscr D^{f,g}_{\mathscr P} $ is the generalized packing dimension, and $ \mathsf{B}_r(x) $ is the ball of center $ x\in \mathbb R^n $ and radius $ r>0 $. Next, we provide a thorough description of the set of functions that serve as generalized local Hausdorff dimension functions and a complete characterization of functions that are generalized local Hausdorff dimension functions for generalized Hausdorff dimension and generalized local packing dimension functions for generalized packing dimension. Finally, we present a brief and straightforward demonstration indicating that any pair of continuous functions $ \varphi $, $ \psi $ defined on $ \mathbb R^n $ and bounded between 0 and $ \mathscr D^{f,g}_\mathscr H( \mathbb R^n) $ such that $ \varphi\leq\psi $ and under some assumptions, it is feasible to select a set $ F $ that concurrently serves as the generalized local Hausdorff dimension function for $ \varphi $ and the generalized local packing dimension function for $ \psi $.