Weighted Ergodic Components in R^n

In this paper, we analyze various classes of weighted Stepanov ergodic spaces, weighted Weyl ergodic spaces and weighted pseudo-ergodic spaces in ${\mathbb R}^{n}$ with the help of results from the theory of Lebesgue spaces with variable exponents $ L^{p(x)}.$ Several structural results of ours seem to be new even in the case of consideration of the constant exponents $p(x)\equiv p\in [1,\infty)$. We especially examine the Stepanov asymptotical almost periodicity at minus infinity and the Weyl asymptotical almost periodicity at minus infinity, providing also some interesting applications to the abstract Volterra integro-differential equations in Banach spaces.