Abstract :
The Bohl–Bohr–Amerio–Kadets theorem states that the indefinite integral y = Pφ of an almost
periodic (ap) φ :R→X is again ap if y is bounded and the Banach space X does not contain a subspace
isomorphic to c0. This is here generalized in several directions: Instead of R it holds also for φ
defined only on a half-line J, instead of ap functions abstract classesAwith suitable properties are admissible,
φ ∈ A can be weakened to φ in some “mean” classMq+1A, then Pφ ∈MqA; hereMA
contains all f ∈ L1
loc with (1/h) h
0 f (· + s)ds in A for all h>0 (usually A ⊂MA⊂M2A⊂···
strictly); furthermore, instead of boundedness of y mean boundedness, y in some MkL∞, or in
MkE, E = ergodic functions, suffices. The Loomis–Doss result on the almost periodicity of a
bounded Ψ for which all differences Ψ(t +h)−Ψ(t) are ap for h>0 is extended analogously, also
to higher order differences. Studying “difference spaces” ΔA in this connection, we obtain decompositions
of the form: Any bounded measurable function is the sum of a bounded ergodic function
and the indefinite integral of a bounded ergodic function. The Bohr–Neugebauer result on the almost
periodicity of bounded solutions y of linear differential equations P(D)y = φ of degree m with ap φ
is extended similarly for φ ∈Mq+mA; then y ∈MqA provided, for example, y is in someMkU
with U = L∞ or is totally ergodic and, for the half-line, Re λ 0 for all eigenvalues P(λ) = 0.
Analogous results hold for systems of linear differential equations. Special case: φ bounded and Pφ
ergodic implies Pφ bounded. If all Reλ >0, there exists a unique solution y growing not too fast;
this y is inMqA if φ ∈Mq+mA, for quite general A.
2003 Elsevier Inc. All rights reserved.
