Nonlocal gradients: Fundamental theorem of calculus, Poincar\'e inequalities and embeddings

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.