Motivated by the recent works by the first and the second named authors, in this paper we introduce the notion of doubly-weighted pseudo-almost periodicity (respectively, doubly-weighted pseudo-almost automorphy) using theoretical measure theory. Basic properties of these new spaces are studied. To illustrate our work, we study, under Acquistapace-Terreni conditions and exponential dichotomy, the existence of (µ, ν)pseudo almost periodic (respectively, (µ, ν)-pseudo almost automorphic) solutions to some nonautonomous partial evolution equations in Banach spaces. A few illustrative examples will be discussed at the end of the paper. RESUMEN Motivado por los trabajos recientes del primer y segundo autor, en este artículo introducimos la noción de seudo-casi periodicidad con doble peso (seudo-casi automorfía con doble peso respectivamente) usando Teoría de la Medida. Se estudian las propiedades básicas de estos espacios nuevos. Para ilustrar nuestro trabajo, bajo las condiciones de Acquistapace-Terreni y dicotomía exponencial estudiamos la existencia de soluciones (respectivamente, (µ, ν) seudo-casi periódicas (µ, ν) seudo-casi automórficas) para algunas ecuaciones parciales de evolución autónomas en espacios de Banach. Algunos ejemplos ilustrativos se discutirán al final del artículo.

The aim of this work is to study the new concept of the (µ, ν)-pseudo almost automorphic functions for some non-autonomous differential equations. We suppose that the linear part has an exponential dichotomy. The nonlinear part is assumed to be (µ, ν)-pseudo almost automorphic. We show some results regarding the completness and the invariance of the space consisting in (µ, ν)-pseudo almost automorphic functions. Then we propose to study the existence of (µ, ν)-pseudo almost automorphic solutions for some differential equations involving reflection of the argument.

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