Xists a finite true quantity l0 0 such that, for every single l l0 and x R, we havex l xf (t ) – f (t)pdt lp.Plugging x = 0 and x = -l here, we very easily get that, for each and every real quantity l l0 , we have:l-lf (s ) – f (s)pds 2lp,which just implies that f ( is Doss-p-almost periodic. In [7] (Theorem eight.3.eight), we’ve got specifically proved the following: Suppose that (0, 1), p [1,), (1 -) p 1 and a 1 – (1 -) p. Define f ( x) := | x | , x R. Then the function f ( is Weyl-p-almost periodic, Besicovitch GNF6702 Protocol p-unbounded and has no mean worth (see [7] for the notion). As a consequence, we have that a Weyl-p-almost periodic function (Doss-p-almost periodic function) just isn’t necessarily Besicovitch-p-almost periodic; also, a Doss-p-almost periodic function f : R R has no imply value and may be Besicovitch p-unbounded in general (1 p). The above consideration can be merely extended towards the multi-dimensional setting. In an effort to do that, we will 1st recall the following definition from [16]: Definition 2. Assume that the following situation holds: (WM): = Rn , = Rn and = Rn is really a Lebesgue measurable set such that m 0, p P , l and l for all l 0, : [0,) [0,) and F : (0,) (0,). By W, ,B ( X : Y), we denote the set consisting of all functions F : X Y such that, for each 0 and B B , there exists a finite actual quantity L 0 such that for each and every t0 there exists B(t0 , L) such that, for each x B, the mapping u ( F (u; x)), u is well defined, and lim sup sup sup F(l, t)l x B t( p(u),F),F ( u; x) – ( F (u; x))Y L p(u) (tl) .The usual concept of multi-dimensional Weyl-p-almost periodicity is obtained by plugging p( p [1,), ( x) x, x 0, F(l, t) l -n/p , l 0, t , = [0, 1]n , = = [0,)n or Rn and = I. The proof of following proposition is fairly easy and hence omitted (we employ pretty much all the above-mentioned circumstances but we allow the predicament in which = and ( x) is not identically equal to x for all x 0): Proposition 8. Suppose that (WM) holds with = [0,)n or Rn , p( p [1,), F(l, t) F(l), l 0, t , = [0, 1]n , and is single-valued on R( F). Suppose that for each l0 0 there exists a finite real quantity t0 l0 such that t/ln/pF l0)/F(t),t t0 .(17)Mathematics 2021, 9,19 ofIf F W, ,B ( X : Y), then F ( is Doss-( p, , F, B , ,)-almost periodic. It is worth noting that situation (17) holds within the classical scenario F l, t) l -n/p and F(t) t-n/p (l, t 0; t). We continue with the following instructive instance, which has not been published in any study article by now and which will be published quickly as [7] (Example 3.2.14): Instance 7. Let 1 and 0 := 0. Define the complex-valued function: f (t) := 1 t sin l , l 2 l =( p,F),t R.Then the function f ( is Lipschitz continuous and uniformly recurrent. To prove the Lipschitz continuity of function f (, it suffices to observe that the function t sin (t), t R is continuous and sin x – sin y | x – y|, x, y R. (18)To find out that the function f ( is uniformly AZD1208 Purity recurrent (cf. [7] for the notion), it suffices to view that for each integer k N \ 1 we have f t 2k – f ( t) =k -t 2k t 1 sin – sin l l 2l two l == =l =1 t 2k t sin – sin l l l 21 t 2k t sin – sin l l l 2 two l =k1 t t 2k – sin l sin l l 2 2 l =kl =k1 t 2k t sin – sin l l l 2l =kk-l 2 2 = , l kt R,where we’ve got applied (18) in the final line of computation. Inside the case that = 2v for some integer v N, we’ve got that the function f ( is Besicovitch unbounded. This can be inspected as in the proof of [30] (Theorem 1.1), using the added observation that:2k – lsin2v2 (2v -.