Localization of spherical and radial nonlocal anisotropic gradients

We establish nonlocal difference quotient representations for the $L^p$ and total variation norms of distributions $\mathcal{A}u$, where $u$ is a distribution, and $\mathcal{A}$ is a constant coefficient homogeneous first-order linear differential operator. Our results provide a criterion for the boundedness of $\mathcal{A}$-gradients that does not require the use of distributional derivatives.