Nonlocal BV and nonlocal Sobolev spaces induced by nonfractional weight functions

In this paper, we expand upon the theory of the space of functions with nonlocal weighted bounded variation, first introduced by Kindermann et.al. in 2005 and later generalized by Wang et.al. in 2014. We consider nonfractional C^1 weights and, using an analogous formulation to the aforementioned works, we also introduce a (to our knowledge) new class of nonlocal weighted Sobolev spaces. After establishing some fundamental properties and results regarding the structure of these spaces, we study their relationship with the classical BV and Sobolev spaces, as well as with the space of test functions. We handle both the case of domains with finite measure and that of domains of infinite measure, and show that these two situations lead to quite different scenarios. As an application, we also show that these function spaces are suitable for establishing existence and uniqueness results of global minimizers for several classes of functionals. Some of these functionals were introduced in the above-mentioned references for the study of image deblurring problems.