Fractional Sobolev Spaces and Variational Problems with Variable-Order Operators on Time Scales

We construct fractional Sobolev spaces on arbitrary time scales, both in one dimension and on product time scales. In 1D, we define $W_{\mathrm{r}\mathrm{d}}^{\alpha (\cdot ),p}(\mathcal{I})$ through a variable-order Gagliardo-type seminorm and prove completeness and compact embedding properties under standard boundedness assumptions on the order. We then extend the framework to rectangles $\mathcal{R}={\mathcal{I}}_1\times {\mathcal{I}}_2\subset {\mathbb{T}}_1\times {\mathbb{T}}_2$, introducing the product spaces $W_{\mathrm{r}\mathrm{d}}^{(\alpha ,\beta ),p}(\mathcal{R})$ and establishing completeness, reflexivity, separability, and compact embeddings. To support boundary-value problems, we propose a boundary decomposition of $\mathrm{\partial }\mathcal{R}$ into four sides and a corresponding trace framework (first on $C_{\mathrm{r}\mathrm{d}}(\mathcal{R})$ and then by density). We also define variable-order Riemann–Liouville and Caputo fractional operators on time scales and derive an Euler–Lagrange equation for variational functionals depending on these operators. The resulting toolkit provides a functional-analytic basis for fractional dynamic equations on mixed time scales and for anisotropic nonlocal models on product time scales.