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 $W^{\alpha(\cdot),p}_{\mathrm{rd}}(\mathcal I)$ 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 $W^{(\alpha,\beta),p}_{\mathrm{rd}}(\mathcal R)$ and establishing completeness, reflexivity, separability, and compact embeddings. To support boundary-value problems, we propose a boundary decomposition of $\partial\mathcal R$ into four sides and a corresponding trace framework (first on $C_{\mathrm{rd}}(\mathcal R)$ 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.