Polynomial ergodicity and asymptotic behaviour of unbounded solutions of abstract evolution equations

In this paper we develop the notion of ergodicity to include functions dominated by a weight $w$. Such functions have polynomial means and include, amongst many others, the $w$-almost periodic functions. This enables us to describe the asymptotic behaviour of unbounded solutions of linear evolution, recurrence and convolution equations. To unify the treatment and allow for further applications, we consider solutions $\phi : G\rightarrow X$ of generalized evolution equations of the form $(*) (B\phi)(t)=A\phi (t)+\psi (t)$ for $t\in G$ where $G$\ is a locally compact abelian group with a closed subsemigroup $J$, $A$ is a closed linear operator on a Banach space $X$, $\psi :G\rightarrow X$ is continuous and $B$ is a linear operator with characteristic function $\theta_{B}:\hat{G}\rightarrow \mathbf{C}$. We introduce the resonance set $\theta_{B}^{-1}(\sigma (A))$ which contains the Beurling spectra of all solutions of the homogeneous equation $B\phi =A\circ \phi$. For certain classes ${\F}$ of functions from $J$ to $$ the spectrum $sp_{{\F}}(\phi)$ of $\phi$ relative to $$ is used to determine membership of ${\F}.$ Our main result gives general conditions under which $sp_{{\F}}(\phi)$\ is a subset of the resonance set. As a simple consequence we obtain conditions under which $\psi |_{J}\in \F$ implies $\phi |_{J}\in {\F}.$ An important tool is our generalization to unbounded functions of a theorem of Loomis. As applications we obtain generalizations or new proofs of many known results, including theorems of Gelfand, Hille, Katznelson-Tzafriri, Esterle et al., Ph\'{o}ng, Ruess and Arendt-Batty.