We study the reduced Beurling spectra $sp_{ \Cal { A},V} (F)$ of functions $F \in L^1_{ loc} (\jj,X)$ relative to certain function spaces $\Cal{ A}\st L^{ \infty}(\jj,X)$ and $V\st L^1 (\r)$ and compare them with other spectra including the weak Laplace spectrum. Here $\jj$ is $\r_+$ or $\r$ and $X$ is a Banach space. If $F$ belongs to the space $ \f'_{ ar}(\jj,X)$ of absolutely regular distributions and has uniformly continuous indefinite integral with $0\not\in sp_{ \A,\f(\r)} (F)$ (for example if F is slowly oscillating and $\A$ is $\{ 0\}$ or $C_0 (\jj,X)$), then $F$ is ergodic. If $F\in \f'_{ ar}(\r,X)$ and $M_h F (\cdot)= \int_0^h F(\cdot+s)\,ds$ is bounded for all $h > 0$ (for example if $F$ is ergodic) and if $sp_{ C_0(\r,X),\f} (F)=\emptyset$, then ${ F}*\psi \in C_0(\r,X)$ for all $\psi\in \f(\r)$. We show that tauberian theorems for Laplace transforms follow from results about reduced spectra. Our results are more general than previous ones and we demonstrate this through examples
Categories: math.FA