arXiv subjects

Hafida Abbas

Publications and source records attributed to Hafida Abbas.

3 recordsLinked to original sources

arXiv2026

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

We propose a systematic Gagliardo-type formulation of fractional Sobolev spaces on arbitrary time scales, based on the Lebesgue Delta-measure and the off-diagonal interaction domain induced by the product measure. For fractional orders strictly between zero and one and for finite Lebesgue exponents, we define a nonlocal Gagliardo seminorm and the associated function space. This construction provides a notion of fractional regularity on time scales that is genuinely nonlocal and structurally distinct from the derivative-based approaches developed in the existing literature. We establish the basic functional properties of these spaces: they are Banach spaces in all admissible cases, reflexive in the strict range of exponents, and Hilbert in the quadratic case. On bounded time scales with finitely many connected components, we identify a sharp criterion for the construction to be nontrivial. We then compare the new framework with the derivative-based Riemann--Liouville fractional Sobolev spaces previously studied on time scales. On a continuous interval, in the supercritical regime, we obtain a norm equivalence with the bilateral Riemann--Liouville space on the subspace of functions with vanishing boundary trace. On hybrid time scales, we prove an explicit obstruction that rules out any analogous equivalence, due to the contribution of the mixed continuous--discrete interactions. On bounded hybrid time scales with finitely many connected components separated by a positive distance, we further establish a Poincar\'e-type inequality, a fractional Sobolev embedding, and fractional Hardy and Caffarelli--Kohn--Nirenberg-type inequalities for subcritical weights. Together, these results provide a complete functional and geometric framework, together with first geometric estimates, for the nonlocal Gagliardo-type approach to fractional Sobolev spaces on time scales.

math.AP↗

arXiv2026

A nonlocal transmission problem on a hybrid continuous-discrete domain

We study a quadratic nonlocal variational problem on a hybrid domain formed by a compact interval and finitely many discrete points. The associated energy splits into continuous, discrete, and interface contributions. Our main estimate shows that the interface term yields a coercive coupling between the two phases and provides an equivalent hybrid norm. As a consequence, we prove existence and uniqueness of a minimizer for the corresponding variational problem and characterize it as the unique weak solution of the associated hybrid Euler--Lagrange system. The latter combines a nonlocal integral equation on the continuous component with a finite nonlocal algebraic system on the discrete nodes.

math.AP↗

arXiv2026

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.

math.DS↗