In this work, we first investigate sufficient conditions of convolution invariance \(\mathcal {K}f=k*f\) in \((\mu , \nu )-\) Stepano pseudo-almost periodic and automorphic functions, where \(\mu , \nu \) are general real measures. Secondly, we established the fundamental fact that the convolution invariance is equivalent to the translation invariance of \((\mu , \nu )-\) Stepanov pseudo-almost periodic and automorphic functions with subsequent consequences. Thirdly, we investigate the existence and uniqueness of \((\mu , \nu )-\) Stepanov pseudo-almost periodic and automorphic solutions of some abstract differential equations. An example is provided to illustrate this work.
