International Monthly Seminar on Time Scales Analysis #9

We introduce a Gagliardo-type fractional Sobolev framework on arbitrary time scales, based on the Lebesgue $\Delta$-measure and the off-diagonal interaction domain
\begin{equation}
\Omega_{\mathbb T}={(t,s)\in \mathbb T\times\mathbb T:\ t\neq s}.
\end{equation}

For $\alpha\in(0,1)$ and $1\le p<\infty$, we define a nonlocal Gagliardo seminorm and the associated spaces $W_{\Delta}^{\alpha,p}(\mathbb T)$. This gives a nonlocal notion of fractional regularity on time scales, distinct from the existing derivative-based approaches.

We prove that $W_{\Delta}^{\alpha,p}(\mathbb T)$ is a Banach space for $1\le p<\infty$, reflexive for $1<p<\infty$, and Hilbert for $p=2$. On bounded time scales with finitely many connected components, we characterize when the construction is nontrivial. We also show that a direct norm equivalence with a single one-sided Riemann--Liouville fractional Sobolev norm cannot hold on the full space.

For bounded hybrid time scales with finitely many connected components separated by a positive distance, we establish a Poincaré-type inequality, a fractional Sobolev embedding, and fractional Hardy and Caffarelli--Kohn--Nirenberg-type inequalities. These results provide a functional and geometric framework for nonlocal fractional Sobolev spaces on time scales.