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In this paper we develop a new set of results based on a nonlocal gradientjointly inspired by the Riesz s-fractional gradient and Peridynamics, in thesense that its integration domain depends on a ball of radius delta > 0(horizon of interaction among particles, in the terminology of Peridynamics),while keeping at the same time the singularity of the Riesz potential in itsintegration kernel. Accordingly, we define a functional space suitable fornonlocal models in Calculus of Variations and partial differential equations.Our motivation is to develop the proper functional analysis framework in orderto tackle nonlocal models in Continuum Mechanics, which requires working withbounded domains, while retaining the good mathematical properties of Rieszs-fractional gradients. This functional space is defined consistently withSobolev and Bessel fractional ones: we consider the closure of smooth functionsunder the natural norm obtained as the sum of the Lp norms of the function andits nonlocal gradient. Among the results showed in this investigation wehighlight a nonlocal version of the Fundamental Theorem of Calculus (namely, arepresentation formula where a function can be recovered from its nonlocalgradient), which allows us to prove inequalities in the spirit of Poincar\'e,Morrey, Trudinger and Hardy as well as the corresponding compact embeddings.These results are enough to show the existence of minimizers of general energyfunctionals under the assumption of convexity. Equilibrium conditions in thisnonlocal situation are also established, and those can be viewed as a new classof nonlocal partial differential equations in bounded domains.

Nonlocal gradients in bounded domains motivated by Continuum Mechanics: Fundamental Theorem of Calculus and embeddings