Multi-dimensional Besicovitch almost periodic type functions and applications

Abstract

In this paper, we analyze multi-dimensional Besicovitch almost periodic type functions. We clarify the main structural properties for the introduced classes of Besicovitch almost periodic type functions, explore the notion of Besicovitch-Doss almost periodicity in the multi-dimensional setting, and provide some applications of our results to the abstract Volterra integrodifferential equations and the partial differential equations.

Metrical approximations of functions

“…provided that the function h ∈ L 1 (R n ) has a certain growth order. For instance, an extension of [24,Theorem 4.6] can be proved in this context (the use of general binary relations ρ is important, which can be seen from our recent applications to the Gaussain semigroup given in [21]; we will not reconsider such applications here). Furthermore, using a similar idea as above, we can consider the existence and uniqueness of metrical Besicovitch almost periodic solutions for some special classes of evolution equations of first order; for instance, we can analyze the evolution systems in the space Y := L r (R n ), where r ∈ [1, ∞), generated by the family of operators A(t) := ∆ + a(t)I, t ≥ 0, where ∆ is the Dirichlet Laplacian on L r (R n ) and a ∈ L ∞ ([0, ∞)).…”

“…Immediately from definition, it follows that, for every F ∈ e−(B, φ, F)−B P• (Λ× X : Y ) and λ ∈ R n , we have e i λ,• F ∈ e − (B, φ, F) − B P• (Λ × X : Y ); this also holds for the corresponding classes introduced in Definition 2.1, Definition 3.1 and Definition 3.6. The class e − (B, φ, F) − B p(•) (Λ × X : Y ) considered in [24] is nothing else but the class e − (B, φ, F) − B P• (Λ × X : Y ) with F(t, •) ≡ F(t) and P t = L p(•) (Λ t ) for all t > 0. The classes e − (B, φ, F) − B P• ω,ρ (Λ × X : Y ) and e − (B, φ, F) j∈Nn − B P• ωj ,ρj (Λ × X : Y ) have not been considered elsewhere even in the one-dimensional setting, with this choice of metric spaces.…”

“…Using the idea from the original proof of J. Marcinkiewicz (see also [26, pp. 249-252]), we have recently proved that the pseudometric space (e − (B, φ, [24,Theorem 2.5]). We will not consider here some sufficient conditions ensuring that the pseudometric space e − (B, φ, F)…”

“…In connection with Proposition 3.13, let us notice that the existence and uniqueness of Besicovitch-p-almost periodic solutions for certain classes of PDEs on some proper subdomains of R n have recently been analyzed in [24]. The interested reader may try to provide some applications of Proposition 3.13 in the analysis of the existence and uniqueness of metrical Besicovitch almost periodic solutions for certain classes of PDEs; see, e.g., the seventh application from [24,Section 4], where the Besicovitch-p-almost periodicity of solutions is clarified for the equations depending on two variables, on the sectors of form Λ = (−∞, 0] × R or Λ = [0, ∞) × R. Let us also emphasize that the equation (3.9) can be viewed as the abstract Hölder inequality in pseudometric spaces.…”

“…The first systematic study of metrical almost periodicity was conducted by the second named author in 2021 ( [21]). The Stepanov, Weyl and Besicovitch classes of metrical almost periodic functions were considered in [22], [23] and [24], respectively. As already mentioned in the abstract, the main aim of this research study is to consider the metrical approximations of functions F : Λ × X → Y by trigonometric polynomials and ρ-periodic type functions, where ∅ = Λ ⊆ R n , X and Y are complex Banach spaces and ρ is a general binary relation on Y (it would be very difficult to summarize here all relevant results concerning function spaces obtained by the closures of the set of trigonometric polynomials in certain norms; see, e.g., the investigations of A. Oliaro, L. Rodino, P. Wahlberg [29] and M. A. Shubin [31] for some non-trivial results established in this direction).…”