The map $(-\Delta + 1)^{-\frac{1}{2}}: L^p (\Omega) \to W^{1, p} (\Omega)$ is a linear bijection when $\Omega$ is smooth and $1 < p < +\infty$.
When $\Omega=\mathbb{R}^d$, this operator correspond to the convolution with the Bessel kernel of order $1$.