Fractional Sobolev spaces with kernel function on compact Riemannian manifolds

In this paper, a new class of Sobolev spaces with kernel function satisfying a Lévy-integrability type condition on compact Riemannian manifolds is presented. We establish the properties of separability, reflexivity, and completeness. An embedding result is also proved. As an application, we prove the existence of solutions for a nonlocal elliptic problem involving the fractional $p(\cdot ,\cdot )$-Laplacian operator. As one of the main tools, topological degree theory is applied.