Towards a Gagliardo-Type Theory of Fractional Sobolev Spaces on Arbitrary Time Scales

Abstract

We introduce a Gagliardo-type approach to fractional Sobolev spaces on arbitrary time scales, based on the Lebesgue $Δ$-measure and on nonlocal interaction energies defined on the off-diagonal set $Ω_T \subset T \times T$. For constant fractional order $α\in (0,1)$ and $1 \le p < \infty$, we define the associated fractional seminorm and the corresponding space $W^{α,p}_Δ(T)$, and we show that this construction is well posed in the measure-theoretic setting of time scales. In particular, we establish that these spaces are Banach spaces for all $1 \le p < \infty$, reflexive for $1 < p < \infty$, and Hilbert spaces in the case $p=2$. We then prove a Poincaré-type inequality on a natural class of bounded hybrid time scales, showing that the geometry of the underlying time scale enters the theory already at the level of coercive estimates. The construction recovers, within a unified framework, the continuous, discrete, and hybrid settings. We conclude by briefly indicating a possible extension to variable-order nonlocal spaces, which will be studied separately.

Theorems & Definitions (37)

• Remark 1
• Lemma 1
• proof
• Definition 1
• Definition 2
• Remark 2: Non-triviality
• Proposition 1
• proof
• Definition 3
• Proposition 2