Generalized vector valued almost periodic and ergodic distributions

Schwartz’s almost periodic distributions are generalized to the case of Banach space valued distributions D AP(R,X), and furthermore for a given arbitrary class A to D A(R,X) := {T ∈ D (R,X): T ∗ ϕ ∈ A for ϕ ∈ test functions D(R,C)}. It is shown that this extension process A→D A is iteration complete, i.e. D D A = D A . Moreover the T from D A are characterized in various ways, also tempered distributions S (R,X) = D P (R,X) with P = {X-valued functions of polynomial growth} are shown. Under suitable assumptionsD A (R,X) = ∞n=0 ˜MnA,D A(R,X)∩ L1 loc(R,X) = ∞n=0MnA, where MA= {f ∈ L1 loc: Mhf (·) := (1/h) h 0 f (· + s)ds ∈ A for all h>0}, M˜ A is defined with the corresponding extension of Mh. With an extension of the indefinite integral from L1 loc to D (R,X) a distribution analogue to the Bohl–Bohr–Amerio–Kadets theorem on the almost periodicity of bounded indefinite integrals of almost periodic functions is obtained, also for almost automorphic, Levitan almost periodic and recurrent functions, similar for a result of Levitan concerning ergodic indefinite integrals. For many of the above results a new (Δ)-condition is needed, we show that it holds for most of the A needed in applications. Also an application to the study of asymptotic behavior of distribution solutions of neutral integro-differential–difference systems is given.  2005 Elsevier Inc. All rights reserved.