  We address the study of nonlocal gradients defined through general radial
kernels $\rho$. Our investigation focuses on the properties of the associated
function spaces, which depend on the characteristics of the kernel function.
Specifically, even with minimal assumptions on $\rho$, we establish Poincar\'e
inequalities and compact embeddings into Lebesgue spaces. Additionally, we
present a fundamental theorem of calculus that enables us to recover a function
from its nonlocal gradient through a convolution. This is used to demonstrate
embeddings into Orlicz spaces and spaces of continuous functions that mirror
the well-known Sobolev and Morrey inequalities for classical gradients.
Finally, we establish conditions for inclusions and equality of spaces
associated to different kernels.
