*Published Paper*

**Inserted:** 28 feb 2023

**Last Updated:** 1 jun 2024

**Journal:** Boll. Unione Mat. Ital.

**Volume:** 17

**Number:** 2

**Pages:** 259–281

**Year:** 2024

**Doi:** 10.1007/s40574-023-00370-y

**Abstract:**

Given $\alpha\in(0,1]$ and $p\in[1,+\infty]$, we define the space $\mathcal{DM}^{\alpha,p}(\mathbb R^n)$ of $L^p$ vector fields whose $\alpha$-divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the $\alpha$-divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss-Green formula. The sharpness of our results is discussed via some explicit examples.

**Keywords:**
Hausdorff measure, Gauss-Green formula, fractional calculus, fractional divergence-measure fields, Leibniz rule

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