Some Properties of Weighted Pseudo almost Periodic Functions

Abstract

Several interesting and new properties of weighted pseudo almost periodic functions are established. Firstly, we obtain an equivalent definition for weighted pseudo almost periodic functions, which shows a close relationship between asymptotically almost periodic functions and weighted pseudo almost periodic functions; secondly, we prove that the space of asymptotically almost periodic functions is always a proper subspace of the space of weighted pseudo almost periodic functions; thirdly, we show that under some cases, the space of weighted pseudo almost periodic functions equals the classical space of pseudo almost periodic functions.

1. Introduction

The notion of weighted pseudo almost periodic function is introduced by Diagana [1], which is an interesting generalization of the classical Bohr almost periodic function as well as a generalization of the classical pseudo almost periodic function introduced by Zhang [2].

Since the work of Diagana, it has been of great interest for many mathematicians to investigate weighted pseudo almost periodic functions and their applications to evolution equations. There is a large amount of literature on this topic. However, due to the influence of weighted term, the behaviors of weighted pseudo almost periodic functions are more tricky and changeable than those of the classical pseudo almost periodic functions.

It is needed to note that, recently, several authors have made interesting and important contributions on weighted pseudo almost periodic functions. For example, Blot et al. [3, 4] established many results including completeness, translation invariance, and composition theorem of weighted pseudo almost periodic functions in the framework of measure theory; Diagana [5] investigated the weighted mean for almost periodic functions; Liang et al. [6] showed that the decomposition of weighted pseudo almost periodic functions is not unique, in general, and that translation invariance implies unique decomposition; Ji and Zhang [7] obtained several results concerning translation invariance and ergodicity of an almost periodic function under a weight; Zhang et al. [8] proved that the space of weighted pseudo almost periodic functions is complete under a new norm, which is different from the supremum norm; Ding et al. [9] presented several basic properties about vector-valued weighted pseudo almost automorphic functions, including equivalence, completeness, translation invariance, composition theorem, and convolution theorem of these functions. In addition, we would like to refer the reader to [10–13] for some other recent development on weighted pseudo almost periodic functions and related topics.

Throughout the rest of this paper, we denote by ℕ the set of positive integers, by ℝ the set of real numbers, by 𝑋 a Banach space, by BC(ℝ,𝑋) the set of all bounded continuous functions 𝑓∶ℝ→𝑋, and by 𝕌 the set of functions (weights) 𝜌∶ℝ→[0,+∞), which are locally integrable over ℝ. In addition, for 𝜌∈𝕌 and 𝑟>0, we denote 𝜇𝑟,𝜌∶=^𝑟_−𝑟𝜌(𝑡)𝕌𝑑𝑡,_∞∶=𝜌∈𝕌∶lim_𝑟→+∞𝜇,𝕌𝑟,𝜌=+∞_𝐵∶=𝜌∈𝕌_∞∶𝜌isboundedwithinf_𝑡∈ℝ𝜌(𝑡)>0. (1) Obviously, 𝕌_𝐵⊂𝕌_∞⊂𝕌, with strict inclusions.

Next, let us recall some notions and basic results about almost periodic functions and pseudo almost periodic functions (for more details, see [14–16]).

Definition 1. A set 𝑃⊂ℝ is called relatively dense in ℝ if there exists a number 𝑙>0 such that, for all 𝑎∈ℝ, [𝑎,𝑎+𝑙]⋂𝑃≠Ø.

Definition 2. A continuous function 𝑓∶ℝ→𝑋 is called almost periodic if for every 𝜀>0 there exists a relatively dense set 𝑃(𝜀) such that sup_𝑡∈ℝ𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀 (2) for all 𝜏∈𝑃(𝜀). We denote the set of all such functions by AP(𝑋).

Definition 3. A continuous function 𝑓∶ℝ→𝑋 is called asymptotically almost periodic if it can be expressed as 𝑓=𝑔+ℎ, where 𝑔∈AP(𝑋) and ℎ∈𝐶₀(𝑋), where 𝐶₀(𝑋)=ℎ∈BC(ℝ,𝑋)∶lim_|𝑡|→∞ℎ(𝑡)=0. (3) The set of all such functions will be denoted by AAP(𝑋).

Lemma 4. The following holds true: (a)if 𝑓∈𝐴𝐴𝑃(𝑋), then 𝑓 is uniformly continuous on ℝ;(b)a necessary and sufficient condition for a continuous function 𝑓∶ℝ→𝑋 belonging to 𝐴𝐴𝑃(𝑋) is that for every 𝜀>0 there exist a constant 𝑇(𝜀)>0 and a relatively dense set 𝑃(𝜀) such that 𝑓(𝑡+𝜏)−𝑓(𝑡)<𝜀 (4)?for all 𝑡∈ℝ with |𝑡|≥𝑇(𝜀) and 𝜏∈𝑃(𝜀) with |𝑡+𝜏|≥𝑇(𝜀).

Now, let us recall some notions and basic results about weighted pseudo almost periodic functions. Denote PAP₀()𝑋,𝜌∶=𝑓∈BCℝ,𝑋∶lim_𝑟→+∞1𝜇𝑟,𝜌^𝑟_−𝑟𝑓(𝑡)𝜌(𝑡)𝑑𝑡=0. (5)

Definition 5 (see [1]). Let 𝜌∈𝕌_∞. A function 𝑓∈BC(ℝ,𝑋) is called weighted pseudo almost periodic or 𝜌-pseudo almost periodic if it can be expressed as 𝑓=𝑔+ℎ, where 𝑔∈AP(𝑋) and ℎ∈PAP₀(𝑋,𝜌). The set of such functions will be denoted by PAP(𝑋,𝜌).

Remark 6. If 𝜌≡1, then a 𝜌-pseudo almost periodic function becomes a classical pseudo almost periodic function. So PAP(𝑋,1) is just the space of all pseudo almost periodic functions. In addition, it is easy to show that PAP(𝑋,𝜌)=PAP(𝑋,1) provided that 𝜌∈𝕌_𝐵.

Definition 7 (see [7]). Let 𝜌∈𝕌_∞. A set 𝐶⊂ℝ is said to be a 𝜌-ergodic zero set if lim_𝑟→+∞∫_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡𝜇𝑟,𝜌=0. (6)

The following lemma is due to [17, Lemma??3.2] (see also [7]).

Lemma 8. Let 𝑓∈𝐵𝐶(ℝ,𝑋) and 𝜌∈𝕌_∞. Then 𝑓∈𝑃𝐴𝑃₀(𝑋,𝜌) if and only if, for every 𝜀>0, 𝑀_𝜀(𝑓) is a 𝜌-ergodic zero set, where 𝑀_𝜀(𝑓)∶={𝑡∈ℝ∶‖𝑓(𝑡)‖≥𝜀}.

2. The Relationship between AAP(X) and PAP(X,?)

In this section, we discuss the relationship between AAP(𝑋) and PAP(𝑋,𝜌). Firstly, we will establish an equivalent definition for weighted pseudo almost periodic functions.

Theorem 9. Let 𝑓∈𝐵𝐶(ℝ,𝑋) and 𝜌∈𝕌_∞. Then a necessary and sufficient condition for 𝑓∈𝑃𝐴𝑃₀(𝑋,𝜌) is that there exists a 𝜌-ergodic zero set 𝐶⊂ℝ such that lim_{|𝑡|→∞𝑡∈ℝ⧵𝐶}𝑓(𝑡)=0. (7)

Proof. Consider the following.
Sufficiency. It follows from lim_{|𝑡|→∞𝑡∈ℝ⧵𝐶}𝑓(𝑡)=0 that, for every 𝜀>0, 𝑡∈ℝ⧵𝐶∶𝑓(𝑡)≥𝜀 (8) is a bounded set. Combining this with the fact that 𝐶 is a 𝜌-ergodic zero set, we conclude that 𝑀_𝜀(𝑓)={𝑡∈ℝ∶‖𝑓(𝑡)‖≥𝜀} is a 𝜌-ergodic zero set. Then, by Lemma 8, 𝑓∈𝑃𝐴𝑃₀(𝑋,𝜌).
Necessity. Let 𝑓∈PAP₀(𝑋,𝜌). We denote 𝐶_𝑛=𝑓(𝑡>1𝑡∈ℝ∶)𝑛,𝑛∈ℕ. (9) Then, by Lemma 8, every 𝐶_𝑛 is a 𝜌-ergodic zero set. Next, we divide the remaining proof into three steps.
Step??1. There exists an increasing sequence {𝑇_𝑛}^∞_𝑛=1⊂ℝ with lim_𝑛→∞𝑇_𝑛=∞, satisfying that, for every 𝑛∈ℕ and 𝑟≥𝑇_𝑛, there hold ∫_{[−𝑟,𝑟]∩𝐶𝑛+1}𝜌(𝑡)𝑑𝑡𝜇≤1𝑟,𝜌𝑛,∑^𝑛_𝑖=1∫_{[−𝑇𝑖,𝑇𝑖]∩𝐶𝑖}𝜌(𝑡)𝑑𝑡𝜇≤1𝑟,𝜌𝑛. (10)
In fact, for 𝑛=1, since 𝐶₁ and 𝐶₂ are both 𝜌-ergodic zero sets, there exists 𝑇₁>1 such that, for all 𝑟≥𝑇₁, there hold ∫_{[−𝑟,𝑟]∩𝐶2}𝜌(𝑡)𝑑𝑡𝜇∫𝑟,𝜌≤1,_{[−𝑇1,𝑇1]∩𝐶1}𝜌(𝑡)𝑑𝑡𝜇≤∫𝑟,𝜌^{[−𝑟,𝑟]∩𝐶1}𝜌(𝑡)𝑑𝑡𝜇𝑟,𝜌≤1. (11) For 𝑛=2, by using the fact that 𝐶₂ and 𝐶₃ are both 𝜌-ergodic zero sets, there exists 𝑇₂>max{𝑇₁,2} such that, for all 𝑟≥𝑇₂, there hold ∫_{[−𝑟,𝑟]∩𝐶3}𝜌(𝑡)𝑑𝑡𝜇≤1𝑟,𝜌2,∑²_𝑖=1∫_{[−𝑇𝑖,𝑇𝑖]∩𝐶𝑖}𝜌(𝑡)𝑑𝑡𝜇≤∫𝑟,𝜌_{[−𝑇1,𝑇1]∩𝐶1}𝜌(𝑡)𝑑𝑡+∫_{[−𝑟,𝑟]∩𝐶2}𝜌(𝑡)𝑑𝑡𝜇≤1𝑟,𝜌2. (12)
For 𝑛=𝑘, by using the fact that 𝐶_𝑘 and 𝐶_𝑘+1 are both 𝜌-ergodic zero, there exists 𝑇_𝑘>max{𝑇_𝑘−1,𝑘} such that, for all 𝑟≥𝑇_𝑘, there hold ∫_{[−𝑟,𝑟]∩𝐶𝑘+1}𝜌(𝑡)𝑑𝑡𝜇≤1𝑟,𝜌𝑘,∑^𝑘_𝑖=1∫_{[−𝑇𝑖,𝑇𝑖]∩𝐶𝑖}𝜌(𝑡)𝑑𝑡𝜇≤∑𝑟,𝜌_{𝑘−1𝑖=1}∫_{[−𝑇𝑖,𝑇𝑖]∩𝐶𝑖}𝜌(𝑡)𝑑𝑡+∫_{[−𝑟,𝑟]∩𝐶𝑘}𝜌(𝑡)𝑑𝑡𝜇≤1𝑟,𝜌𝑘. (13) Continuing by this way, we can get an increasing sequence {𝑇_𝑛}^∞_𝑛=1, which satisfies lim_𝑛→∞𝑇_𝑛=∞, (10).
Step??2.??𝐶∶=⋃^∞_𝑛=1𝐶_𝑛 is a 𝜌-ergodic zero set, where 𝐶₁=−𝑇₁,𝑇₁∩𝐶₁,𝐶_𝑛=−𝑇_𝑛,𝑇_𝑛⧵−𝑇_𝑛−1,𝑇_𝑛−1∩𝐶_𝑛,𝑛=2,3,…. (14)
In fact, for every 𝑟>0, there exists 𝑇_𝑛 such that 𝑟∈[𝑇_𝑛,𝑇_𝑛+1). Thus, we have [−𝑟,𝑟]⊂−𝑇₁,𝑇₁∪−𝑇₂,𝑇₂⧵−𝑇₁,𝑇₁∪⋯−𝑇_𝑛,𝑇_𝑛⧵−𝑇_𝑛−1,𝑇_𝑛−1∪[−𝑟,𝑟]⧵−𝑇_𝑛,𝑇_𝑛,([−𝑟,𝑟]∩𝐶)⊂𝐶₁∪⋯𝐶_𝑛∪[−𝑟,𝑟]∩𝐶_𝑛+1. (15) Combining this with (10), we conclude that ∫_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡𝜇≤∑𝑟,𝜌_𝑛𝑖=1∫_𝐶𝑖𝜌(𝑡)𝑑𝑡𝜇+∫𝑟,𝜌^{[−𝑟,𝑟]∩𝐶𝑛+1}𝜌(𝑡)𝑑𝑡𝜇≤∑𝑟,𝜌_𝑛𝑖=1∫_{[−𝑇𝑖,𝑇𝑖]∩𝐶𝑖}𝜌(𝑡)𝑑𝑡𝜇+∫𝑟,𝜌^{[−𝑟,𝑟]∩𝐶𝑛+1}𝜌(𝑡)𝑑𝑡𝜇≤2𝑟,𝜌𝑛, (16) which means that 𝐶 is a 𝜌-ergodic zero set.
Step??3. lim_{|𝑡|→∞𝑡∈ℝ⧵𝐶}𝑓(𝑡)=0.
In fact, for every 𝑛∈ℕ and 𝑡∈ℝ⧵𝐶 with |𝑡|>𝑇_𝑛, there exists 𝑘≥𝑛 such that 𝑡∈−𝑇_𝑘+1,𝑇_𝑘+1⧵−𝑇_𝑘,𝑇_𝑘, (17) which yields that 𝑡∉𝐶_𝑘+1, and thus 𝑓(𝑡<1)<1𝑘+1𝑛. (18) This completes the proof.

Remark 10. In Theorem 9, we can choose 𝐶 to be an open set. In fact, letting 𝐶=𝐶^∞_𝑛=1𝑇_𝑛,𝑇_𝑛+𝛿_𝑛^∞_𝑛=1−𝑇_𝑛−𝛿_𝑛,−𝑇_𝑛, (19) where every 𝛿_𝑛 is a sufficiently small positive constant satisfying ^{𝑇𝑛+𝛿𝑛}_𝑇𝑛𝜌(𝑡)𝑑𝑡+^−𝑇𝑛_{−𝑇𝑛−𝛿𝑛}𝜌(𝑡)1𝑑𝑡<2^𝑛,𝑛∈ℕ, (20) it is easy to prove that 𝐶 is an open set and a 𝜌-ergodic zero set. Moreover, obviously, we have lim_{|𝑡|→∞𝑡∈ℝ⧵𝐶}𝑓(𝑡)=0.

Next, we present an equivalent definition of weighted pseudo almost periodic functions, which establishes a close relationship between asymptotically almost periodic functions and weighted pseudo almost periodic functions.

Theorem 11. Let 𝑓∈𝐵𝐶(ℝ,𝑋) and 𝜌∈𝕌_∞. Then a necessary and sufficient condition for 𝑓∈𝑃𝐴𝑃(𝑋,𝜌) is that there exist 𝑓∈𝐴𝐴𝑃(𝑋) and a 𝜌-ergodic zero set 𝐶⊂ℝ such that 𝑓(𝑡)=𝑓(𝑡),𝑡∈ℝ⧵𝐶. (21)

Proof. The sufficiency part is easy to prove. We only give the proof for the necessity part. Let 𝑓∈𝑃𝐴𝑃(𝑋,𝜌). Then, there exist 𝑔∈𝐴𝑃(𝑋) and ℎ∈𝑃𝐴𝑃₀(𝑋,𝜌) such that 𝑓=𝑔+ℎ. It follows from Theorem 9 that there exists a 𝜌-ergodic zero set 𝐶⊂ℝ such that lim_{|𝑡|→∞𝑡∈ℝ⧵𝐶}ℎ(𝑡)=0. (22) In addition, by Remark 10, without loss of generality, we can assume that 𝐶 is an open set. Then, we can conclude that there exists a function ℎ∈𝐶₀(𝑋) such that ℎ(𝑡)=ℎ(𝑡) for all 𝑡∈ℝ⧵𝐶. Letting 𝑓=𝑔+ℎ, we have 𝑓∈AAP(𝑋), and 𝑓(𝑡)=𝑓(𝑡) for all 𝑡∈ℝ⧵𝐶. This completes the proof.

Combining Lemma 4 and Theorem 11, we can get the following.

Corollary 12. Let 𝜌∈𝕌_∞ and 𝑓∈𝑃𝐴𝑃(𝑋,𝜌). Then, there exists a 𝜌-ergodic zero set 𝐶⊂ℝ satisfying that, for every 𝜀>0, there are two constants 𝛿(𝜀),𝑇(𝜀)>0 and a relatively dense set 𝑃(𝜀) such that𝑓()(𝑡(𝜀)[(𝜀)(𝜀,𝑓𝑡𝑡+𝜏−𝑓)<𝜀,𝜏∈𝑃,𝑡,𝑡+𝜏∉𝐶∪−𝑇,𝑇)]₁−𝑓𝑡₂<𝜀,𝑡₁,𝑡₂∉𝐶,𝑡₁−𝑡₂<𝛿(𝜀). (23)

Remark 13. For the case of 𝜌≡1, Zhang [18] established a similar result to Corollary 12. But here we use a different approach, and even for the case of 𝜌≡1, Corollary 12 improves the “if part” of [18, Theorem??11]. In fact, in [18], for every 𝜀>0, there exists a 1-ergodic zero set 𝐶(𝜀). Here, we find a common ergodic zero set 𝐶.

In the above, we show that there is a close relationship between asymptotically almost periodic functions and weighted pseudo almost periodic functions. Next, we will show that AAP(𝑋) is always a proper subspace of PAP(𝑋,𝜌).

Theorem 14. Let 𝜌∈𝕌_∞. Then 𝐴𝐴𝑃(𝑋)⫋𝑃𝐴𝑃𝑋,𝜌. (24)

Proof. It is easy to show that AAP(𝑋)⊂PAP(𝑋,𝜌) since 𝐶₀(𝑋)⊂PAP₀(𝑋,𝜌). So we only need to show that AAP(𝑋)≠PAP(𝑋,𝜌). Without loss for generality, we only give the proof for the case of 𝑋=ℝ. We divide the remaining proof into three steps.
Step??1. Let 𝐶=⋃^∞_𝑛=1𝐶_𝑛, where 𝐶_𝑛()()<1∶=𝑡∈𝑛−1,𝑛∶𝐹𝑡,𝑛𝑛𝐹(𝑛−1,𝑛),𝑛∈ℕ, (25) where 𝐹𝑥,𝑦∶=_{𝐴(𝑥,𝑦)}𝜌(𝑠)𝑑𝑠,𝐴𝑥,𝑦∶=𝑠∈ℝ∶𝑥≤|𝑠|≤𝑦𝑦≥𝑥≥0. (26) It is easy to see that every 𝐶_𝑛 is open. Moreover, there are infinitely many nonempty 𝐶_𝑛, and thus we can assume that every 𝐶_𝑛 is nonempty without loss for generality. In addition, we claim that 𝐶 is a 𝜌-ergodic zero set. In fact, we have the following two cases.
Case??I. If 𝑟∈𝐶, then 𝑟∈𝐶_[𝑟]+1, which yields that ∫_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡𝜇≤∫𝑟,𝜌_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡+∫_{𝐴(𝑟,[𝑟]+1)}𝜌(𝑡)𝑑𝑡∫_{[−𝑟,𝑟]}𝜌(𝑡)𝑑𝑡+∫_{𝐴(𝑟,[𝑟]+1)}𝜌(𝑡)=∫𝑑𝑡_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡+𝐹(𝑟,[𝑟]+1)𝜇𝑟]≤∫[+1,𝜌_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡+(1/([𝑟]+1))𝐹([𝑟],[𝑟]+1)𝜇[𝑟]≤∑+1,𝜌_{[𝑟]+1𝑘=1}∫_{𝐴(𝑘−1,𝑘)∩𝐶}𝜌(𝑡)𝑑𝑡+(1/([𝑟]+1))𝐹([𝑟],[𝑟]+1)𝜇[𝑟]≤∑+1,𝜌_{[𝑟]+1𝑘=1}∫_𝐶𝑘𝜌(𝑡)𝑑𝑡+(1/([𝑟]+1))𝐹([𝑟],[𝑟]+1)𝜇[𝑟]≤∑+1,𝜌_{[𝑟]+1𝑘=1}(1/𝑘)𝐹(𝑘−1,𝑘)+(1/([𝑟]+1))𝐹([𝑟],[𝑟]+1)𝜇[𝑟]=∑+1,𝜌_[𝑟]𝑘=1(1/𝑘)𝐹(𝑘−1,𝑘)+(2/([𝑟]+1))𝐹([𝑟],[𝑟]+1)𝜇𝑟]=[+1,𝜌_𝑚𝑘=11𝑘𝐹(𝑘−1,𝑘)+^[𝑟]_𝑘=𝑚+11𝑘𝐹()+2𝑘−1,𝑘[𝑟]𝐹𝑟],[𝑟])×𝜇[𝑟]+1([+1+1,𝜌⁻¹≤𝜇𝑚,𝜌+(2/(𝑚+1))𝜇[𝑟]+1,𝜌𝜇[𝑟]≤𝜇+1,𝜌𝑚,𝜌𝜇[𝑟]+2+1,𝜌,𝑚+1 (27) where 𝑚<[𝑟]−1 is a positive integer. Then, we get lim_𝑟→+∞∫_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡𝜇𝑟,𝜌=0. (28)
Case??II. If 𝑟∉𝐶, then ([𝑟],𝑟]∩𝐶=∅, which yields that ∫_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡𝜇≤∑𝑟,𝜌_[𝑟]𝑘=1∫_{𝐴(𝑘−1,𝑘)∩𝐶}𝜌(𝑡)𝑑𝑡+∫_{𝐴([𝑟],𝑟)∩𝐶}𝜌(𝑡)𝑑𝑡𝜇=∑𝑟,𝜌_[𝑟]𝑘=1∫_{𝐴(𝑘−1,𝑘)∩𝐶}𝜌(𝑡)𝑑𝑡𝜇.𝑟,𝜌 (29) Then, similar to Case??I, one can also obtain that lim_𝑟→+∞∫_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡𝜇𝑟,𝜌=0. (30) Thus, 𝐶 is a 𝜌-ergodic zero set.
Step??2. For every 𝑛∈ℕ, noting that 𝐶_𝑛 is open, there exist 𝑡_𝑛∈𝐶_𝑛 and 𝛿_𝑛>0 such that (𝑡_𝑛−𝛿_𝑛,𝑡_𝑛+𝛿_𝑛)⊂𝐶_𝑛. Now, we construct a bounded and continuous function on ℝ by 𝜑(𝑡)=𝑡−𝑡_{𝑛(𝑛+1)/2}+𝛿_{𝑛(𝑛+1)/2}𝛿_{𝑛(𝑛+1)/2},𝑡∈𝑡_{𝑛(𝑛+1)/2}−𝛿_{𝑛(𝑛+1)/2},𝑡_{𝑛(𝑛+1)/2}𝑡,𝑛∈ℕ,_{𝑛(𝑛+1)/2}+𝛿_{𝑛(𝑛+1)/2}−𝑡𝛿_{𝑛(𝑛+1)/2},𝑡∈𝑡_{𝑛(𝑛+1)/2},𝑡_{𝑛(𝑛+1)/2}+𝛿_{𝑛(𝑛+1)/2},𝑛∈ℕ,0,otherwise. (31)
Step??3. 𝜑∈PAP(𝑋,𝜌)⧵AAP(𝑋).
Since {𝑡∈ℝ∶𝜑(𝑡)≠0}⊂𝐶 and 𝐶 is a 𝜌-ergodic zero set, we have 𝜑∈PAP₀(𝑋,𝜌)⊂PAP(𝑋,𝜌). It remains to show that 𝜑∉AAP(𝑋). We prove it by contradiction, assuming that there exist 𝜑₁∈AP(𝑋) and 𝜑₂∈𝐶₀(𝑋) such that 𝜑=𝜑₁+𝜑₂. For sufficiently large 𝑛, since 𝜑₁∈AP(𝑋), we can choose 𝜏_𝑛∈𝑛(𝑛+1)2−𝑡_{𝑛(𝑛+1)/2},(𝑛+1)(𝑛+2)2−1−𝑡_{𝑛(𝑛+1)/2} (32) such that 𝜑₁𝑡_{𝑛(𝑛+1)/2}+𝜏_𝑛−𝜑₁𝑡_{𝑛(𝑛+1)/2}<12. (33) Moreover, since 𝜑₂∈𝐶₀(𝑋), for sufficiently large 𝑛, we also have 𝜑₂𝑡_{𝑛(𝑛+1)/2}+𝜏_𝑛−𝜑₂𝑡_{𝑛(𝑛+1)/2}<12. (34) So, we get 𝜑𝑡_{𝑛(𝑛+1)/2}+𝜏_𝑛−𝜑𝑡_{𝑛(𝑛+1)/2}<1, (35) which contradicts the fact that 𝜑𝑡_{𝑛(𝑛+1)/2}=1,𝜑𝑡_{𝑛(𝑛+1)/2}+𝜏_𝑛=0. (36) This completes the proof.

3. Equivalence

Just as noted in Remark 6, we know that PAP(𝑋,𝜌)=PAP(𝑋,1) provided that 𝜌∈𝕌_𝐵. Then, there is a natural question:

Does PAP(𝑋,𝜌)=PAP(𝑋,1) imply that 𝜌∈𝕌_𝐵?

In fact, the above question has a negative answer. For example, recently, it is proved in [7] and [9] (by a different method) that PAP𝑋,|⋅|^𝑛=PAP(𝑋,1),𝑛∈ℕ. (37)

In this section, we will make further study on this question. We will prove that for some other 𝜌∉𝕌_𝐵, there still holds PAP(𝑋,𝜌)=PAP(𝑋,1). Firstly, we recall a theorem, which is due to [7, Theorem??4.3].

Theorem 15. Let 𝜌₁,𝜌₂∈𝕌_∞ and 𝐶⊂ℝ∶𝐶𝑖𝑠𝑎𝜌₁=-𝑒𝑟𝑔𝑜𝑑𝑖𝑐𝑧𝑒𝑟𝑜𝑠𝑒𝑡𝐶⊂ℝ∶𝐶𝑖𝑠𝑎𝜌₂-𝑒𝑟𝑔𝑜𝑑𝑖𝑐𝑧𝑒𝑟𝑜𝑠𝑒𝑡. (38) Then 𝑃𝐴𝑃₀(𝑋,𝜌₁)=𝑃𝐴𝑃₀(𝑋,𝜌₂).

Theorem 16. Let 𝜌∈𝕌_∞ be a periodic function with 𝜌(𝑡)>0 almost everywhere on ℝ. Then 𝑃𝐴𝑃₀(𝑋,𝜌)=𝑃𝐴𝑃₀(𝑋,1), and thus 𝑃𝐴𝑃(𝑋,𝜌)=𝑃𝐴𝑃(𝑋,1).

Proof. It suffices to prove that PAP₀(𝑋,𝜌)=PAP₀(𝑋,1). We divide the remaining proof into two steps.
Step??1. Every 𝜌-ergodic zero set is a 1-ergodic zero set.
Let 𝐶 be a 𝜌-ergodic zero set. Then, we have lim_𝑟→+∞∫_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡𝜇𝑟,𝜌=0. (39) Assuming that lim_𝑟→+∞mes([−𝑟,𝑟]∩𝐶)2𝑟≠0, (40) then there exist 𝜀₀>0 and a sequence of positive numbers {𝑇_𝑛}^∞_𝑛=1, which satisfies lim_𝑛→∞𝑇_𝑛=∞, and mes−𝑇_𝑛,𝑇_𝑛∩𝐶>2𝑇_𝑛𝜔⋅𝜀₀, (41) where 𝜔 is a positive periodic of 𝜌. On the other hand, since 𝜌(𝑡)>0 almost everywhere on ℝ, by Lusin's Theorem, there exists a closed set 𝐹⊂[0,𝜔] with mes([0,𝜔]−𝐹)<𝜀₀/2, such that 𝜌 is continuous on 𝐹 and 𝜌(𝑡)>0 for all 𝑡∈𝐹. Let 𝑚₀∶=inf_𝑡∈𝐹𝜌(𝑡). (42) Then 𝑚₀>0. Then, we have mes𝑡∈[0,𝜔]∶𝜌(𝑡)<𝑚₀<𝜀₀2, (43) which yields that mes𝑡∈−𝑇_𝑛,𝑇_𝑛∶𝜌(𝑡)<𝑚₀<2𝑇_𝑛𝜔⋅𝜀+1⁰2. (44)
Combining this with (41), we get ∫_{[−𝑇𝑛,𝑇𝑛]∩𝐶}𝜌(𝑡)𝑑𝑡𝜇𝑇_𝑛≥𝑚,𝜌₀⋅2𝑇_𝑛/𝜔⋅𝜀₀−2𝑇_𝑛/𝜔+1⋅𝜀₀/2∫_{[−𝑇𝑛,𝑇𝑛]}𝜌(𝑡)≥𝑚𝑑𝑡₀⋅2𝑇_𝑛/𝜔⋅𝜀₀−2𝑇_𝑛/𝜔+1⋅𝜀₀/22𝑇_𝑛/𝜔+1⋅∫^𝜔₀𝜌(𝑡)⟶𝑚𝑑𝑡₀⋅𝜀₀/2∫^𝜔₀𝜌(𝑡)𝑑𝑡,𝑛⟶∞, (45) which contradicts (39). Therefore, 𝐶 is a 1-ergodic zero set.
Step??2. Every 1-ergodic zero set is a 𝜌-ergodic zero set.
Let 𝐶 be a 1-ergodic zero set. For every 𝜀>0, there exists 𝛿>0 such that, for all set 𝐸⊂[0,𝜔] with mes𝐸<𝛿, there holds _𝐸𝜌(𝑡)𝑑𝑡<𝜀. (46) In addition, again by Lusin's Theorem, for the above 𝛿>0, there exists a closed set 𝐹_𝛿⊂[0,𝜔] with mes([0,𝜔]−𝐹_𝛿)<𝛿, such that 𝜌 is continuous on 𝐹_𝛿. Let 𝑀_𝛿=sup_{𝑡∈𝐹𝛿}𝜌(𝑡). (47) Then 𝑀_𝛿∈(0,+∞). Recalling that 𝐶 is a 1-ergodic zero set, for the above 𝜀 and 𝑀_𝛿, there exists 𝑟_𝜀>0 such that lim_𝑟→+∞mes([−𝑟,𝑟]∩𝐶)+2𝜔<𝜀2𝑟𝑀_𝛿,𝑟≥𝑟_𝜀. (48) Combining (46) and (48), by using the periodicity of 𝜌, we conclude that, for all 𝑟≥𝑟_𝜀, there holds _{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡≤^[𝑟/𝜔]_{𝑘=0[𝑘𝜔,𝑘𝜔+𝜔]∩𝐶}𝜌(𝑡)+𝑑𝑡_{[𝑟/𝜔]𝑘=0[−𝑘𝜔−𝜔,−𝑘𝜔]∩𝐶}𝜌(𝑡)≤𝑟𝑑𝑡𝜔+1⋅2𝜀+𝑀_𝛿⋅(]))≤𝑟mes([−𝑟,𝑟∩𝐶+2𝜔𝜔+1⋅2𝜀+2𝑟𝜀, (49) where ^[𝑟/𝜔]_{𝑘=0[𝑘𝜔,𝑘𝜔+𝜔]∩𝐶}𝜌(𝑡)≤𝑑𝑡_{[𝑟/𝜔]𝑘=0[[𝑘𝜔,𝑘𝜔+𝜔]−(𝐹𝛿+𝑘𝜔)]∩𝐶}𝜌(𝑡)+𝑑𝑡_{[𝑟/𝜔]𝑘=0[[𝑘𝜔,𝑘𝜔+𝜔]∩(𝐹𝛿+𝑘𝜔)]∩𝐶}𝜌(𝑡)≤𝑑𝑡_{[𝑟/𝜔]𝑘=0[𝑘𝜔,𝑘𝜔+𝜔]−(𝐹𝛿+𝑘𝜔)}𝜌(𝑡)+𝑑𝑡_{[𝑟/𝜔]𝑘=0}𝑀_𝛿])≤⋅mes([𝑘𝜔,𝑘𝜔+𝜔∩𝐶_{[𝑟/𝜔]𝑘=0[0,𝜔]−𝐹𝛿}𝜌(𝑡)𝑑𝑡+𝑀_𝛿⋅(]))≤𝑟mes([0,𝑟∩𝐶+𝜔𝜔+1⋅𝜀+𝑀_𝛿⋅(mes([0,𝑟]∩𝐶)+𝜔), (50) and, similarly, ^[𝑟/𝜔]_{𝑘=0[−𝑘𝜔−𝜔,−𝑘𝜔]∩𝐶}𝜌(𝑡)≤𝑟𝑑𝑡𝜔+1⋅𝜀+𝑀_𝛿⋅(mes([−𝑟,0]∩𝐶)+𝜔). (51) Noting that ^𝑟_−𝑟𝜌(𝑡)𝑟𝑑𝑡≥2𝜔⋅^𝜔₀𝜌(𝑡), (52) we get ∫_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡∫^𝑟_−𝑟𝜌(𝑡)≤])𝑑𝑡([𝑟/𝜔+1⋅2𝜀+2𝑟𝜀2[𝑟/𝜔]⋅∫^𝜔₀𝜌(𝑡)⟶𝑑𝑡𝜔+1∫^𝜔₀𝜌(𝑡)𝑑𝑡⋅𝜀,𝑟⟶+∞. (53) Then, by the arbitrariness of 𝜀, we conclude that lim_𝑟→+∞∫_{[−𝑟,𝑟]∩𝐶}𝜌(𝑡)𝑑𝑡∫^𝑟_−𝑟𝜌(𝑡)𝑑𝑡=0; (54) that is, 𝐶 is a 𝜌-ergodic zero set.

Example 17. Let 𝜌(𝑡)=cos^2𝑛(𝑡), where 𝑛∈ℕ. Then by Theorem 16, PAP(𝑋,𝜌)=PAP(𝑋,1).

Except for the case in Theorem 16, there is some other 𝜌, which satisfies that PAP₀(𝑋,𝜌)=PAP₀(𝑋,1). For example, we have the following.

Theorem 18. Let 𝜌(𝑡)=|𝑡|,|𝑡||𝑡|ln>1,0,≤1. (55) Then 𝑃𝐴𝑃₀(𝑋,𝜌)=𝑃𝐴𝑃₀(𝑋,1).

Proof. We first show that every 1-ergodic zero set is a 𝜌-ergodic zero set. Let 𝐶 be a 1-ergodic zero set. For every 𝜀>0, there exists 𝑅>0 such that, for all 𝑟>𝑅, mes([−𝑟,𝑟]∩𝐶)<2𝑟⋅𝜀. (56) Combining this with the fact that 𝜌 is even and increasing on [0,+∞), we get 𝜇[−𝑟,𝑟]∩𝐶,𝜌𝜇[]⩽−𝑟,𝑟,𝜌2𝑟ln𝑟⋅𝜀2(𝑟ln𝑟−𝑟+1),𝑟>𝑅, (57) which yields that limsup_𝑟→+∞𝜇[−𝑟,𝑟]∩𝐶,𝜌𝜇[−𝑟,𝑟],𝜌≤𝜀. (58) Then, by the arbitrariness of 𝜀, we know that lim_𝑟→+∞𝜇[−𝑟,𝑟]∩𝐶,𝜌𝜇[−𝑟,𝑟],𝜌=0; (59) that is, 𝐶 is a 𝜌-ergodic zero set.
Next, let us show that every 𝜌-ergodic zero set is a 1-ergodic zero set. Let 𝐶 be a 𝜌-ergodic zero set. Assuming that lim_𝑟→+∞mes([−𝑟,𝑟]∩𝐶)2𝑟≠0 (60) then there exist 𝜀₀>0 and a sequence of positive numbers {𝑇_𝑛}^∞_𝑛=1, which satisfies lim_𝑛→∞𝑇_𝑛=∞, and mes−𝑇_𝑛,𝑇_𝑛∩𝐶>2𝑇_𝑛𝜀₀. (61) By (61) and the fact that 𝜌 is even and increasing on [0,+∞), we conclude that 𝜇−𝑇_𝑛,𝑇_𝑛∩𝐶,𝜌𝜇−𝑇_𝑛,𝑇_𝑛⩾𝜇,𝜌−𝑇_𝑛𝜀₀,𝑇_𝑛𝜀₀,𝜌𝜇−𝑇_𝑛,𝑇_𝑛=𝜀,𝜌₀𝑇_𝑛ln𝜀₀𝑇_𝑛−𝑇_𝑛+1𝑇_𝑛ln𝑇_𝑛−𝑇_𝑛=𝜀+1₀𝑇_𝑛ln𝑇_𝑛−𝑇_𝑛+𝑇_𝑛𝜀₀ln𝜀₀+1𝑇_𝑛ln𝑇_𝑛−𝑇_𝑛.+1 (62) Letting 𝑛→∞, we get liminf_𝑛→∞𝜇−𝑇_𝑛,𝑇_𝑛∩𝐶,𝜌𝜇−𝑇_𝑛,𝑇_𝑛,𝜌≥𝜀₀, (63) which contradicts the fact that 𝐶 is a 𝜌-ergodic zero set. Thus, 𝐶 is a 1-ergodic zero set. This completes the proof.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Ding acknowledges support from the NSF of China (11101192), the Program for Cultivating Young Scientist of Jiangxi Province (20133BCB23009), and the NSF of Jiangxi Province; Zheng acknowledges support from the Graduate Innovation Fund of Jiangxi Province.