Weighted pseudo almost periodic functions, convolutions and abstract integral equations

This paper deals with a systematic study of the convolution operator Kf=f⁎kKf=f⁎k defined on weighted pseudo almost periodic functions space PAP(X,ρ)PAP(X,ρ) and with k∈L1(R)k∈L1(R). Upon making several different assumptions on k,fk,f and ρ, we get five main results. The first two main results establish sufficient conditions on k and ρ   such that the weighted ergodic space PAP0(X,ρ)PAP0(X,ρ) is invariant under the operator KK. The third result specifies a sufficient condition on all functions (k,fk,f and ρ  ) such that the Kf∈PAP0(X,ρ)Kf∈PAP0(X,ρ). The fourth result is a sufficient condition on the weight function ρ   such that PAP0(X,ρ)PAP0(X,ρ) is invariant under KK. The hypothesis of the convolution invariance results allows to establish a fifth result related to the translation invariance of PAP0(X,ρ)PAP0(X,ρ). As a consequence of the fifth result, we obtain a new sufficient condition such that the unique decomposition of a weighted pseudo almost periodic function on its periodic and ergodic components is valid and also for the completeness of PAP(X,ρ)PAP(X,ρ) with the supremum norm. In addition, the results on convolution are applied to general abstract integral and differential equations.