Generalized vector valued almost periodic and ergodic distributions

2012/06/21 by Bolis Basit, Basit, Bolis, Hans Günzler +1
Mathematics · #37A30 #37A45 #43A07 #43A60 #44A10 #46E30 #46F05 (Primary) 34K25 #47A35 #47D03 (Secondary) #Advanced Banach Space Theory #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical and Theoretical Analysis #advanced mathematical theories #math.FA #msc:34K25 #msc:37A30 #msc:37A45 #msc:43A07 #msc:43A60 #msc:44A10 #msc:46E30 #msc:46F05 #msc:47A35 #msc:47D03

paper · pdf · doi:10.48550/arxiv.1206.4749

69 pages

arxiv created 2012/06/21 · openalex publication_date 2012/06/21 · arxiv updated 2012/06/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/04

Abstract

For \Cal A⊂ L¹_loc(\Bbb J,X) let \Cal M\Cal A consist of all f∈ L¹_loc with M_h f (⋅):=\frac 1h∫₀^hf(⋅ +s) ds ∈ \Cal A for all h>0. Here X is a Banach space, \Bbb J= (α,∞), [α,∞) or \Bbb R. Usually \Cal A⊂\Cal M\Cal A⊂ \Cal M²\Cal A⊂ .... The map \Cal A → \Cal D'_{\Cal A} is iteration complete, that is \Cal D'_\Cal D'_{\Cal A}= \Cal D'_{\Cal A}. Under suitable assumptions \widetilde \Cal Mⁿ \Cal A= \Cal A + \T⁽ⁿ⁾ : T ∈ \Cal A\, and similarly for \Cal Mⁿ \Cal A. Almost periodic X-valued distributions \h'_\A with \A = almost periodic (ap) functions are characterized in several ways. Various generalizations of the Bohl-Bohr-Kadets theorem on the almost periodicity of the indefinite integral of an ap or almost automorphic function are obtained. On \Cal D'_{\Cal E} , \Cal E the class of ergodic functions, a mean can be constructed which gives Fourier series. Special cases of \Cal A are the Bohr ap, Stepanoff ap, almost automorphic, asymptotically ap, Eberlein weakly ap, pseudo ap and (totally) ergodic functions (\T)\E. Then always \Cal Mⁿ \Cal A is strictly contained in \Cal Mⁿ⁺¹ \Cal A. The relations between \mⁿ \E, \mⁿ\T\E and subclasses are discussed. 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, we obtain new tauberian theorems for f∈ L¹_loc(\Bbb J,X) to belong to a class \A which are decisive in describing the asymptotic behavior of unbounded solutions of many abstract differential-integral equations. This generalizes various recent results

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