Nonlocal gradient, the nonlocal Laplacian and maximum principles

We study the nonlocal $\rho$-Laplacian, defined as the composition of the nonlocal divergence and gradient operators associated with a general radial kernel $\rho$: $\Delta_\rho u=\mbox{div}_\rho\left(D_\rho u\right)$. Our first main contribution is to establish a precise connection between this operator and the class of integro-differential elliptic operators studied by Fern\'andez-Real and Ros-Oton (\cite{FernandezRos}), identifying explicit conditions on the kernel $\rho$ that guarantee membership in this class. Our second main contribution concerns maximum and comparison principles for the $\rho$-Laplacian. We establish both a strong and a weak maximum principle under conditions on $\rho$ that are strictly weaker than those required for membership in the integro-differential class, thereby covering a genuinely broader family of operators. The results require only minimal assumptions on the kernel, and in particular do not rely on any fractional-type comparability condition.