In the arXive paper Control Functionals for Monte Carlo Integration, $\S$ 2.3.1 presents theorem $1$ on p. 7:
Assume $\phi\in\mathcal{H}^{d}$ and $(A1,3)$. Then $\psi$ belongs to $\mathcal{H}_{0}$, the reproducing kernel Hilbert space with kernel $$ k_{0}(\mathbf{x},\mathbf{x}'):= \nabla_{x} \cdot \nabla_{x'} k(\mathbf{x},\mathbf{x}') + \mathbf{u}(\mathbf{x})\cdot\nabla_{x'} k(\mathbf{x},\mathbf{x}') + \mathbf{u}(\mathbf{x}') \cdot \nabla_{x} k(\mathbf{x},\mathbf{x}') + \mathbf{u}(\mathbf{x}) \cdot \mathbf{u}(\mathbf{x}') k(\mathbf{x},\mathbf{x}') $$
There is a term
$$ \nabla_{x} \cdot \nabla_{x'} k(\mathbf{x},\mathbf{x}') $$
where $\mathbf{x}$ and $\mathbf{x}'$ are two equal sized vectors.
Does this $\nabla_{x} \cdot \nabla_{x'} f(x,x')$ operator have a name?