aa r X i v : . [ m a t h . F A ] O c t FUNCTIONAL ANALYTIC APPROACH TO CES `ARO MEAN
RYOICHI KUNISADA
Abstract.
We study a certain class P of positive linear functionals ϕ on L ∞ ([1 , ∞ ))for which ϕ ( f ) = α if lim x →∞ x R x f ( t ) dt = α . It turns out that translations f ( x ) f ( rx ) on L ∞ ([1 , ∞ )), where r ∈ [1 , ∞ ), which are induced by the action of themultiplicative semigroup [1 , ∞ ) on itself, plays an intrinsic role in the study of P .We also deal with an analogue K of P of positive linear functionals on L ∞ ([0 , ∞ ))partaining to the action of the additive semigroup [0 , ∞ ) on itself. In particular, wegive some expressions of maximal possible values of P and K for a given functionrespectively. Introduction
Let L ∞ ( R × + ) be the space of all essentially bounded functions on R × + = [1 , ∞ ). Let L ∞ ( R × + ) ∗ be the dual space of L ∞ ( R × + ). Given f ( x ) ∈ L ∞ ( R × + ), define M ( f ) = lim x x Z x f ( t ) dt if this limit exists. Obviously, it is an integral analogy of the notion of Ces`aro mean.Let E be the space of all functions in L ∞ ( R × + ) having the limit M ( f ) and E be itssubspace consisting of all f ∈ E for which M ( f ) = 0.Our main interest of this paper is the set C of normalized positive linear functionalson L ∞ ( R × + ) which vanish on E , that is, C = E ⊥ ∩ S + L ∞ ( R × + ) ∗ , where E ⊥ is the annihilatorof E and S + L ∞ ( R × + ) ∗ is the positive part of the unit sphere of L ∞ ( R × + ) ∗ . In other words, ϕ ∈ L ∞ ( R × + ) ∗ is an element of C if and only if it is an extension of the functional M : E → R .Of particular importance for our study is the following sublinear functional P ( f ) = lim θ → + lim sup x →∞ θx − x Z θxx f ( t ) dt = lim θ → + lim sup x →∞ θ − Z θ f ( xt ) dt, f ( x ) ∈ L ∞ ( R × + ) , which turns out to give the maximal value attained by elements of C for each f ∈ L ∞ ( R × + ). That is, sup ϕ ∈C ϕ ( f ) = P ( f ) olds. One of our main aim of this paper is to prove this assertion by functional analyticmethods and to give another expression of sup ϕ ∈C ϕ ( f ) which has a more simple form.It should be noted that these definitions derive from their summation counterparts,with which one is more familier. We give below a brief description of this notion. Let l ∞ be the set of all real-valued bounded functions on natural numbers N . Recall thatfor any f ∈ l ∞ , its Ces`aro mean M d ( f ) is define as M d ( f ) = lim n →∞ n n X i =1 f ( i )if this limit exists. Let C d be the set of normalized positive linear functionals on l ∞ which extend Ces`aro mean. Also we define a sublinear functional P d on l ∞ as follows. P d ( f ) = lim θ → − lim sup n →∞ P i ∈ [ θn,n ] f ( i ) n − θn , where f ∈ l ∞ . Then it holds that sup ϕ ∈C d ϕ ( f ) = P d ( f ) . Based on the results of P´olya in [5], this is proved in [3]. Remark that elements of C d and the functional P d are sometimes called density measures and P´olya densityrespectively when they are ristricted to the characteristic functions of subsets of N .Density measures have been studied by several authors, for example [1], [4], [6]. Inparticular, our main result mentioned above can be considered as an integral analogyof this result, which in fact, as we will see in Section 5, we can deduce from the integralversion with ease.To this end, we introduce another class of continuous linear functionals on L ∞ ( R + ) ofthe space of all essentially bounded functions on R + = [0 , ∞ ). Given f ( x ) ∈ L ∞ ( R + ),define R ( f ) = lim x →∞ e x Z x f ( t ) e t dt if this limit exists. Similarly as preceding paragraphs, let F be the space of all functions f in L ∞ ( R + ) with the limit R ( f ) exists and F be its subspace consisting of all f ∈ F for which R ( f ) = 0. Then we consider the set D of normalized positive functionalson L ∞ ( R + ) which vanish on F , i.e., D = F ⊥ ∩ S + L ∞ ( R + ) ∗ . It may be said that theidea of introducing the class D in connection with C is very natural. We can formulatea similar question of giving an expression of the maximal value of D for any fixed f ∈ L ∞ ( R + ). Let us define a sublinear functional K on L ∞ ( R + ) by K ( f ) = lim θ → + lim sup x →∞ θ Z x + θx f ( t ) dt = lim θ → + lim sup x →∞ θ Z θ f ( t + x ) dt, f ∈ L ∞ ( R + ) . otice that the definition of K is analogous to that of P in the sense that we considerthe action of the additive group of R in place of the action of the multiplicative group R × = (0 , ∞ ) of R in the definition of P . Also, the additive counterpart of the precedingassertion about maximal values of C is given as follows:sup ϕ ∈D ϕ ( f ) = K ( f )holds for each f ∈ L ∞ ( R + ).The paper is organaized as follows. After Section 2, which contains necessary nota-tion and notions needed in the rest of the paper, we first consider the additive versionin Section 3. We deal with the multiplicative version in Section 4, where argumentsare similar to those in Section 3. Also we consider the relationship between C and D and show that there exists a natural affine homeomorphism between C and D . Lastsection deal with the descrete version of resutls in Section 4.2. Preliminaries
Let X be a set and Y be a compact space and f : X → Y be a mapping. Let U bean ultrafilter on X . Then there exists an element y of Y such that f − ( U ) ∈ U holdsfor every neighborhood U of y . In this case, we write U - lim x f ( x ) = y and say that y is the limit of f along U .Let N be the set of nonnegative integers and β N be its Stone- ˇCech compactification.Recall that β N is a compactification of N such that for any continuous mapping ι : N → X of N into a compact space X , there exists a continuous extension ι : β N → X of ι to β N .We denote by τ : β N → β N the continuous extension of the mapping N ∋ n n + 1 ∈ β N . Let us denote N ∗ = β N \ N and since the restriction of τ to N ∗ is ahomeomorphism of N ∗ onto itself, the pair ( N ∗ , τ ) is a topological dynamical system.Next we consider the compact space Ω of all maximal ideals of C ub ( R + ) of the spaceof all real-valued uniformly continuous bounded functions on R + , which can be viewedin a sense as a continuous version of β N . It is known that the maximal ideal spaceΩ of C ub ( R + ) is given as follows (see [2] for further details of the following assertions):Ω = ( β N × [0 , / ∼ , the quotient space of the product space β N × [0 , ∼ is an equivalence relation on β N × [0 ,
1] such that ω ∼ ω ′ if and only if ω = ( η, ω ′ = ( τ η,
0) for some η ∈ β N . Then there exists an algebraic isomorphism C ub ( R + ) ∋ f ( x ) → f ( ω ) ∈ C (Ω) of C ub ( R + ) onto C (Ω) of the set of all continuousfunctions on Ω. Notice that for each ω = ( η, t ) ∈ Ω, the class of sets { t + A : A ∈ η } isan ultrafilter on R + . In this way, we identify an element ω of Ω with an ultrafilter on R + from now on. Then for every f ∈ C ub ( R + ), it holds that f ( ω ) = ω - lim x f ( x ).Further, for each s ≥
0, let us define a continuous mapping τ s : Ω → Ω as follows: τ s ω = τ s ( η, t ) = ( τ [ t + s ] η, t + s − [ t + s ]) , here s ∈ R and [ x ] denotes the largest integer not exceeding a real number x . Let usdenote Ω ∗ = Ω \ R and remark that the restriction of each τ s to Ω ∗ is a homeomorphismof Ω ∗ onto itself. Thus the pair (Ω ∗ , { τ s } s ∈ R ) is a continuous flow.3. Extremal values of D We consider the set of continuous linear functionals ϕ on L ∞ ( R + ) for which ϕ ( f ) ≤ K ( f )holds for every f ∈ L ∞ ( R + ) and denote it by K .Given f ( x ) ∈ L ∞ ( R + ), we define a sequence { ˜ f n ( x ) } n ≥ of L ∞ ( T ), where T = [0 , f n ( x ) = f ( x + n ) , x ∈ [0 , , n = 0 , , , . . . . Notice that { ˜ f n ( x ) } n ≥ is a uniformly bounded sequence of L ∞ ( T ) and then it is a weak*relatively compact subset of L ∞ ( T ). Then the mapping N ∋ n ˜ f n ( x ) ∈ L ∞ ( T ) canbe extended continuously to β N . For each η ∈ β N , let us denote its image underthis extended mapping by ˜ f η ( x ) ∈ L ∞ ( T ). Notice that it can be expressed as the limitalong an ultrafilter η , i.e., ˜ f η ( x ) = η - lim n ˜ f n ( x ).We need the lemma below which asserts that we can interchange limit and integral.Let { f n ( x ) } n ≥ be a uniformly bounded sequence of L ∞ ( T ) and let η - lim n f n ( x ) = f η ( x )for η ∈ N ∗ = β N \ N , where the limit is taken with respect to the weak*-topology of L ∞ ( T ). Now we consider their indefinite integrals; let us define F n ( x ) = Z x f n ( t ) dt, x ∈ [0 , , n = 0 , , , . . . ,G η ( x ) = Z x f η ( t ) dt, x ∈ [0 , . Then it is obviously that { F n ( x ) } n ≥ is a uniformly bounded and equicontinuous se-quence of C ( T ), the set of all continuous functions on T . It means that { F n ( x ) } n ≥ isa relatively compact subset of C ( T ) in its uniform topology. Therefore we can considerfor each η ∈ N ∗ the limit F η ( x ) = η - lim n F n ( x ) in C ( T ). Further, since every F n ( x )is Lipschitz-continuous with Lipschitz-constant K = sup n k f n k ∞ , so is their uniformlimit F η ( x ). Thus F η ( x ) is differentiable a.e on T . Then we have the following result. Lemma 3.1. F η ( x ) = G η ( x ) holds for every η ∈ N ∗ . In other words, F ′ η ( x ) = f η ( x ) m-a.e on T . Proof .
Since the definition of weak*-convergence, for any x ∈ [0 ,
1] we have F η ( x ) = η - lim n F n ( x ) = η - lim n Z x f n ( t ) dt = Z x f η ( t ) dt = G η ( x ) . Hence F ′ η ( x ) = f η ( x ) also holds by the Lebesgue differentiation theorem. heorem 3.1. For each f ∈ L ∞ ( R + ) it holds that K ( f ) = sup η ∈ N ∗ ess sup x ∈ [0 , ˜ f η ( x ) . Proof .
Let f ( x ) ∈ L ∞ ( R ) and let us define˜ F n ( x ) = Z x ˜ f n ( t ) dt, x ∈ [0 , , ˜ F η ( x ) = η - lim n ˜ F n ( x ) , x ∈ [0 , . Remark that by the Lemma 3.1, it holds that˜ F η ( x ) = Z x ˜ f η ( t ) dt, i.e., ˜ F ′ η ( x ) = ˜ f η ( x ) , x ∈ [0 , . Then we have K ( f ) = lim θ → + lim sup x →∞ θ Z x + θx f ( t ) dt = lim θ → + lim sup n →∞ sup x ∈ [0 , − θ ] θ Z x + θx ˜ f n ( t ) dt = lim θ → + lim sup n →∞ sup x ∈ [0 , − θ ] ˜ F n ( x + θ ) − ˜ F n ( x ) θ = lim θ → + sup η ∈ N ∗ sup x ∈ [0 , − θ ] η - lim n ˜ F n ( x + θ ) − ˜ F n ( x ) θ = lim θ → + sup η ∈ N ∗ sup x ∈ [0 , − θ ] ˜ F η ( x + θ ) − ˜ F η ( x ) θ ≥ sup η ∈ N ∗ sup x ∈ [0 , lim sup θ → + ˜ F η ( x + θ ) − ˜ F η ( x ) θ ≥ sup η ∈ N ∗ ess sup x ∈ [0 , lim sup θ → + ˜ F η ( x + θ ) − ˜ F η ( x ) θ = sup η ∈ N ∗ ess sup x ∈ [0 , ˜ f η ( x ) . On the other hand, for any η ∈ N ∗ , θ > x ∈ [0 , − θ ], we have˜ F η ( x + θ ) − ˜ F η ( x ) θ = 1 θ Z x + θx ˜ f η ( t ) dt ≤ θ · θ · ess sup x ∈ [0 , ˜ f η ( x )= ess sup x ∈ [0 , ˜ f η ( x ) . hus we have lim θ → + sup η ∈ N ∗ sup x ∈ [0 , − θ ] ˜ F η ( x + θ ) − ˜ F η ( x ) θ ≤ sup η ∈ N ∗ ess sup x ∈ [0 , ˜ f η ( x ) . ∴ K ( f ) = sup η ∈ N ∗ ess sup x ∈ [0 , ˜ f η ( x ) . Now we define for each ω = ( η, t ) ∈ Ω ∗ a linear operator T ω : L ∞ ( R + ) → L ∞ ( R ) asfollows; let us consider a set of functions { f ( x + s ) } s ≥ in L ∞ ( R + ). This is boundedand weak* relatively compact set of L ∞ ( R + ). Thus for each ω ∈ Ω ∗ we can defineits limit along ω , i.e., ω - lim s f ( x + s ) in L ∞ ( R + ) with respect to the weak* topology.Moreover, we can extend it to the negative direction by( T ω f )( x ) = f τ − N ω ( N + x ) , x ∈ [ − N, , for every N >
0. Then it is easy to see that this definition of ( T ω f )( x ) is equal to thefollowing. ( T ω f )( x ) = ˜ f τ [ x + t ] η ( x + t − [ x + t ]) , x ∈ R . Notice that for f ( x ) ∈ C ub ( R + ), it holds that ( T f ) ω ( x ) = f ( τ x ω ), i.e., the restrictionof f ( ω ) to the orbit o ( ω ) of ω .Thus Theorem 3.1 can be expressed as follows. Corollary 3.1. K ( f ) = sup ω ∈ Ω ∗ ess sup x ∈ R ( T ω f )( x ) . For a fucntion f ∈ L ∞ ( R + ), let us denote K ( f ) = α if ϕ ( f ) = α for every ϕ ∈ K .Then the following corollary follows immediately by the above corollary. Corollary 3.2. K ( f ) = α if and only if w ∗ - lim s f ( x + s ) = α , where the symbol w ∗ - lim represents weak*-convergence in L ∞ ( R + ) . Now the following result is important for our intention. For the sake of simplicity,for any f ∈ L ∞ ( R + ), let us denote ( T ω f )( x ) = f ω ( x ) and( Sf )( x ) = 1 e x Z x f ( t ) e t dt, x ≥ . Theorem 3.2. R ( f ) = α if and only if K ( f ) = α . Proof .
First we observe that R ( f ) = α ⇐⇒ lim x →∞ ( Sf )( x ) = α ⇐⇒ ( Sf )( ω ) = α on Ω ∗ ⇐⇒ ( Sf ) ω ( x ) = α f or every ω ∈ Ω ∗ . For each f ( x ) ∈ L ∞ ( R + ), notice that f ( x ) = ( Sf )( x ) + ( Sf ) ′ ( x ) , x ≥ . perating T ω both sides and recalling that (( Sf ) ′ ) ω ( x ) = (( Sf ) ω ) ′ ( x ) by Lemma 3.1,we have f ω ( x ) = ( Sf ) ω ( x ) + (( Sf ) ω ) ′ ( x ) , x ∈ R . Hence ( e x · ( Sf ) ω ( x )) ′ = e x · f ω ( x ) , x ∈ R . Since ( Sf ) ω ( x ) is bounded on R , we obtain immediately that for each ω ∈ Ω ∗ , ( Sf ) ω ( x ) = α if and only if f ω ( x ) = α . This completes the proof. Corollary 3.3. R ( f ) = α if and only if w ∗ - lim s f ( x + s ) = α . Hnece we have F = { f ( x ) ∈ L ∞ ( R + ) : w ∗ - lim s f ( x + s ) = 0 } . Then by Corollary3.1 it is immediate that K ⊆ D . In fact, we can show the reverse inclusion:
Theorem 3.3. K = D holds. Proof .
It is sufficient to show that
D ⊆ K and by the Krein-Milman theorem it isequivalent to the assertion that K ( f ) ≤ K ( f ) for every f ∈ L ∞ ( R + ), where K ( f ) = sup ϕ ∈ S + L ∞ ( R +) ∗ ∩ F ⊥ ϕ ( f ) . First, remark that K ( f ) = inf h ∈ F ess sup x ∈ [0 , ∞ ) ( f ( x ) − h ( x ))= inf h ∈ F sup n ∈ N ess sup x ∈ [0 , ( ˜ f n ( x ) − ˜ h n ( x )) . Now for each h ∈ F we put H n ( x ) = Z x ( ˜ f n ( t ) − ˜ h n ( t )) dt, x ∈ [0 , , n = 0 , , , . . . . Then by the assumption that h ( x ) ∈ F we have H n ( x ) → F n ( x ) unif ormly in x ∈ [0 , as n → ∞ . Also we note˜ f n ( x ) − ˜ h n ( x ) = lim θ → H n ( x + θ ) − H n ( x ) θ m - a.e. on [0 , , n = 0 , , , . . . . ence we get that K ( f ) = inf h ∈ F sup n ∈ N sup x ∈ [0 , lim sup θ → H n ( x + θ ) − H n ( x ) θ ≤ inf h ∈ F lim sup θ → sup n ∈ N sup x ∈ [0 , − θ ] H n ( x + θ ) − H n ( x ) θ = inf h ∈ F lim sup θ → lim sup n ∈ N sup x ∈ [0 , − θ ] H n ( x + θ ) − H n ( x ) θ = inf h ∈ F lim sup θ → lim sup n ∈ N sup x ∈ [0 , − θ ] F n ( x + θ ) − F n ( x ) θ = lim sup θ → lim sup n ∈ N sup x ∈ [0 , − θ ] θ Z n + x + θn + x f ( t ) dt = lim sup θ → lim sup x →∞ θ Z x + θx f ( t ) dt = K ( f ) Theorem 3.4. sup ϕ ∈D ϕ ( f ) = K ( f ) holds for each f ∈ L ∞ ( R + ) . Extremal values of C We consider the set of continuous linear functionals ψ on L ∞ ( R × + ) for which ψ ( f ) ≤ P ( f )holds for every f ∈ L ∞ ( R × + ) and denote it by P . In this section we deal with themultiplicative version of the preceding section. First, for given f ( x ) ∈ L ∞ ( R × + ), wedefine in a similar way as the former section the sequence { ˆ f n ( x ) } n ≥ of elements of L ∞ ([1 , e ]) by ˆ f n ( x ) = f ( e n x ) , x ∈ [1 , e ] , n = 0 , , , · · · . Similarly we consider its limit along η ∈ β N , denoted by ˆ f η ( x ) = η - lim n ˆ f n ( x ), in L ∞ ([1 , e ]) with respect to its weak*-topology. Also let us defineˆ F n ( x ) = Z x ˆ f n ( t ) dt ∈ C ([1 , e ]) , n = 0 , , , · · · . ˆ F η ( x ) = η - lim n ˆ F n ( x ) , η ∈ N ∗ . Then we have an analogous result to Lemma 3.1:
Lemma 4.1. ˆ F ′ η ( x ) = ˆ f η ( x ) m-a.e. on [1 , e ] . Then we can show an analogue of Theorem 3.1 by similar arguments:
Theorem 4.1.
For every f ∈ L ∞ ( R × + ) it holds that P ( f ) = sup η ∈ N ∗ ess sup x ∈ [1 ,e ] ˆ f η ( x ) . roof . For given f ( x ) ∈ L ∞ ( R × + ), we have P ( f ) = lim θ → + lim sup x →∞ θx − x Z θxx f ( t ) dt = lim θ → + lim sup n →∞ sup x ∈ [1 ,e/θ ] θx − Z θxx ˆ f n ( t ) dt = lim θ → + lim sup n →∞ sup x ∈ [1 ,e/θ ] ˆ F n ( θx ) − ˆ F n ( x ) θx − x = lim θ → + sup η ∈ N ∗ sup x ∈ [1 ,e/θ ] η - lim n ˆ F n ( θx ) − ˆ F n ( x ) θx − x = lim θ → + sup η ∈ N ∗ sup x ∈ [1 ,e/θ ] ˆ F η ( θx ) − ˆ F η ( x ) θx − x ≥ sup η ∈ N ∗ sup x ∈ [1 ,e ] lim sup θ → + ˆ F η ( θx ) − ˆ F η ( x ) θx − x ≥ sup η ∈ N ∗ ess sup x ∈ [1 ,e ] lim sup θ → + ˆ F η ( θx ) − ˆ F η ( x ) θx − x = sup η ∈ N ∗ ess sup x ∈ [1 ,e ] ˆ f η ( x ) . On the other hand, for any η ∈ N ∗ , θ > x ∈ [1 , e/θ ], we haveˆ F η ( θx ) − ˆ F η ( x ) θx − x = 1 θx − x Z θxx ˆ f η ( t ) dt ≤ θx − x · ( θx − x ) · ess sup x ∈ [1 ,e ] ˆ f η ( x )= ess sup x ∈ [1 ,e ] ˆ f η ( x ) . Thus we get lim θ → + sup η ∈ N ∗ sup x ∈ [1 ,e/θ ] ˆ F η ( θx ) − ˆ F η ( x ) θx − x ≤ sup η ∈ N ∗ ess sup x ∈ [1 ,e ] ˆ f η ( x ) . ∴ P ( f ) = sup η ∈ N ∗ ess sup x ∈ [1 ,e ] ˆ f η ( x ) . Similarly as K ( f ), we define for each ω ∈ Ω a linear operator P ω : L ∞ ( R × ) −→ L ∞ ( R × ) as follows; let us consider the set of functions { f ( rx ) } r ≥ in L ∞ ( R × + ). Thisis bounded and weak* relatively compact set of L ∞ ( R × + ). Thus for each ω ∈ Ω ∗ wecan define its limit along the ultrafilter e ω = { e A : A ∈ ω } , i.e., e ω - lim r f ( rx ) = ω - lim r f ( e r x ). Remark that in this way one obtains just a function in L ∞ ( R × + ), but an extend it to the whole space R × in the same way as the definiton of T ω . Then wehave ( P ω f )( x ) = ˆ f τ [log xθ ] η ( xθ/e [log xθ ] ) , x ∈ R × , where ω = ( η, t ) and θ = e t . Hence Theorem 3.1 can be expressed as follows. Corollary 4.1. P ( f ) = sup ω ∈ Ω ∗ ess sup x ∈ R ( P ω f )( x ) . Now we take up the relation between ˜ f η ( x ) ∈ L ∞ ([0 , f η ( x ) ∈ L ∞ ([1 , e ]). We define a linear operator W by W : L ∞ ([1 , e ]) −→ L ∞ ([0 , , ( W f )( x ) = f ( e x ) . Then notice that ( W ˆ f n )( x ) = ( g W f ) n ( x ) , n = 0 , , , . . . , holds for each f ∈ L ∞ ( R × + ). Further, we have the following result. Lemma 4.2. ( W ˆ f η )( x ) = ( g W f ) η ( x ) for each η ∈ N ∗ and f ∈ L ∞ ( R × + ) . Proof .
For any f ∈ L ∞ ( R × + ) and η ∈ N ∗ , by the definition of the weak* topology, η - lim n ˆ f n ( x ) = ˆ f η ( x ) ⇐⇒ η - lim n Z e ˆ f n ( t ) φ ( t ) dt = Z e ˆ f η ( t ) φ ( t ) dt for every φ ∈ L ([1 , e ]). Notice that by integration by substitution for any g ( x ) ∈ L ([1 , e ]) we have Z e g ( t ) dt = Z g ( e t ) · e t dx. Hence for every φ ∈ L ([1 , e ]), we have η - lim n Z ˆ f n ( e t ) φ ( e t ) e t dt = Z ˆ f η ( e t ) φ ( e t ) e t dt. Namely, it holds that η - lim n Z ( g W f ) n ( t ) ψ ( t ) dt = Z ( W ˆ f η )( t ) ψ ( t ) dt where ψ ( x ) = φ ( e x ) e x . Notice that for every ψ ( x ) ∈ L ([0 , φ ( y ) = ψ (log y ) y ∈ L ([1 , e ]) and φ ( e x ) e x = ψ ( x ) holds. This means that η - lim n Z ( g W f ) n ( t ) ψ ( t ) dt = Z ( W ˆ f η )( t ) ψ ( t ) dt for every ψ ( x ) ∈ L ([0 , g W f ) η ( x ) = η - lim n ( g W f ) n ( x ) = ( W ˆ f η )( x ) in L ∞ ([0 , η ∈ N ∗ . We complete the proof.It is possible to extend this result to the relation between ( T ω f )( x ) and ( P ω f )( x ) asfollows; let us define a linear operator W by W : L ∞ ( R × + ) −→ L ∞ ( R ) , ( W f )( x ) = f ( e x ) . orollary 4.2. For every f ∈ L ∞ ( R × + ) and ω ∈ Ω , T ω W f = W P ω f holds. Namely, P ω = W − T ω W holds. Proof .
Let x ∈ R , ω = ( η, t ) ∈ Ω ∗ , θ = e t and f ∈ L ∞ ( R × + ). Then we have( W P ω f )( x ) = ˆ f τ [log ex · θ ] η ( e x · θ/e [log e x · θ ] )= ˆ f τ [ x +log θ ] η ( e x · θ/e [ x +log θ ] )= ˆ f τ [ x + t ] η ( e x + t /e [ x + t ] )= ˆ f τ [ x + t ] η ( e x + t − [ x + t ] )= ( W ˆ f τ [ x + t ] η )( x + t − [ x + t ])= ( g W f ) τ [ x + t ] η ( x + t − [ x + t ])= ( T ω W f )( x ) . Remark that for each ω ∈ Ω ∗ , ess sup x ∈ R × ( P ω f )( x ) = ess sup x ∈ R ( P ω f )( e x ) = ess sup x ∈ R ( T ω W f )( x )holds and then we have the following result. Theorem 4.2. P ( f ) = K ( W f ) for every f ∈ L ∞ ( R × + ) . Let us consider the adjoint operator W ∗ of W , which is a linear isomeory from L ∞ ( R + ) ∗ onto L ∞ ( R × + ) ∗ . Theorem 4.2 shows that K and P are affinely homeomorphicvia W ∗ .For a function f ∈ L ∞ ( R × + ), we denote P ( f ) = α if ψ ( f ) = α for every ψ ∈ P . Theorem 4.3.
Let f ∈ L ∞ ( R × + ) . Then M ( f ) = α if and only if P ( f ) = α . Proof .
By Theorem 4.2, P ( f ) = α is equivalent to K ( W f ) = α . Also, notice thatlim x →∞ x Z x f ( t ) dt = lim x →∞ x Z log x ( W f )( t ) e t dt = lim x →∞ e x Z x ( W f )( t ) e t dt, which shows that M ( f ) = α is equivalent to R ( W f ) = α . Thus, by Theorem 3.2, wehave M ( f ) = α ⇐⇒ R ( W f ) = α ⇐⇒ K ( W f ) = α ⇐⇒ P ( f ) = α. Corollary 4.3. M ( f ) = α if and only if w ∗ - lim r f ( rx ) = α . As we have seen in the above proof, f ( x ) ∈ E is equivalent to ( W f )( x ) ∈ F , whichmenas that W E = F , and thus W ∗ D = C . Therefore since we have already shownthat W ∗ K = P and K = D , we get P = C . Now the following result of the extremalvalue attined by C follows immediately. Theorem 4.4. sup ψ ∈C ψ ( f ) = P ( f ) for each f ∈ L ∞ ( R × + ) . Finally, we will give another expression of sup ψ ∈C ψ ( f ). We define a sublinear func-tional Q on L ∞ ( R × + ) as follows. Q ( f ) = lim θ → + lim sup x →∞ θ Z θxx f ( t ) dtt , here f ( x ) ∈ L ∞ ( R × + ). Theorem 4.5. Q ( f ) = K ( W f ) holds for every f ∈ L ∞ ( R × + ) . Proof .
For any f ∈ L ∞ ( R × + ) , x ≥ θ >
0, we have that1 θ Z x + θx ( W f )( t ) dt = 1 θ Z x + θx f ( e t ) dt = 1 θ Z e x · e θ e x f ( s ) 1 ds Now we put r = e x , then r tends to ∞ as x tends to ∞ . And then put y = e θ andthen y tends to 1 + as θ tends to 0 + . Hence we have K ( W f ) = lim y → + lim sup r →∞ y Z ryr f ( s ) dss = Q ( f ) . Hence by Theorem 4.2 and Theorem 4.4, we get the following result.
Corollary 4.4. sup ψ ∈C ψ ( f ) = Q ( f ) for each f ∈ L ∞ ( R × + ) . Applications to Ces`aro mean
In this section, we deal with the discrete version of the preceding results, that is,normalized positive linear functionals on l ∞ which extend Ces`aro mean. First, weconsider the linear operators V and V defined as follows. V : l ∞ −→ L ∞ ( R × + ) , ( V f )( x ) = f ([ x ]) . and V : L ∞ ( R × + ) −→ l ∞ , ( V f )( n ) = Z n +1 n f ( t ) dt. Let V ∗ and V ∗ be their adjoint operators respectively. Then we have the followingresult. Theorem 5.1. C and C d are affinely homeomorphic via V ∗ . Proof .
First, notice that for each f ∈ l ∞ , we have1 n n X i =1 f ( i ) = 1 n Z n +11 ( V f )( t ) dt. Hence if ϕ ∈ C and f ∈ l ∞ with M d ( f ) exists, then( V ∗ ϕ )( f ) = ϕ ( V f ) = M ( V f ) = M d ( f ) . Thus V ∗ ϕ ∈ C d holds for every ϕ ∈ C . Next we show that V ∗ is one to one. Suppose that ϕ , ϕ ∈ C with ϕ = ϕ . Then there is some f ∈ L ∞ ( R × + ) such that ϕ ( f ) = ϕ ( f ).On the other hand, for every ϕ ∈ C , it holds that ϕ ( f ) = ϕ ( V V f ) sincelim x →∞ x Z x ( f ( t ) − ( V V f )( t )) dt = 0 . hen we have ϕ ( f ) = ϕ ( V V f ) = ( V ∗ ϕ )( V f ) , ϕ ( f ) = ϕ ( V V f ) = ( V ∗ ϕ )( V f )Thus, by the assumption that ϕ ( f ) = ϕ ( f ), we have ( V ∗ ϕ )( V f ) = ( V ∗ ϕ )( V f ).Hence V ∗ ϕ = V ∗ ϕ . This shows that V ∗ : C → C d is one to one.Next we show that V ∗ : C → C d is onto. Given any ψ ∈ C d , let us take ϕ = V ∗ ψ ∈ C .Remark that V V f = f for every f ∈ l ∞ and then we have V ∗ ϕ = V ∗ V ∗ ψ = ψ . Hence V ∗ : C → C d is onto. We obtain the result. Lemma 5.1.
For each f ∈ l ∞ , it holds that lim θ → + lim sup n →∞ θn − n Z θnn ( V f )( t ) dt = lim θ → + lim sup n →∞ θn − n X i ∈ [ n,θn ] f ( i ) . Proof .
Let us n ∈ N , θ > f ∈ l ∞ . The assertion follows from the followingequation. Z [ θn ]+1 n ( V f )( t ) dt = [ θn ] X i = n f ( i ) . Theorem 5.2. sup ψ ∈C d ψ ( f ) = P d ( f ) holds for each f ∈ l ∞ . Proof .
We have thatsup ψ ∈C d ψ ( f ) = sup ϕ ∈C ( V ∗ ϕ )( f ) = sup ϕ ∈C ϕ ( V f ) = P ( V f )= lim θ → + lim sup x →∞ θx − x Z θxx ( V f )( t ) dt = lim θ → + lim sup n →∞ θn − n Z θnn ( V f )( t ) dt = lim θ → + lim sup n →∞ θn − n X i ∈ [ n,θn ] f ( i ) . = P d ( f ) . We define a sublinear functional Q d on l ∞ , which is a discrete version of the sublinearfunctional Q as follows: Q d ( f ) = lim θ → + lim sup n →∞ θ X i ∈ [ n,θn ] f ( i ) i , f ∈ l ∞ . Lemma 5.2.
For each f ∈ l ∞ , it holds that lim θ → + lim sup n →∞ θ Z θnn ( V f )( t ) dtt = lim θ → + lim sup n →∞ θ X i ∈ [ n,θn ] f ( i ) i . roof . Let us n ∈ N , θ > f ∈ l ∞ . The lemma follows immediately from thefollowing computation. (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Z [ θn ]+1 n ( V f )( t ) dtt − X i ∈ [ n,θn ] f ( i ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f ( n ) · (cid:18) n − log (cid:18) n (cid:19)(cid:19) + f ( n + 1) · (cid:18) n + 1 − log (cid:18) n + 1 (cid:19)(cid:19) + . . . + f ([ θn ]) · (cid:18) θn ] − log (cid:18) θn ] (cid:19)(cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | f ( n ) | · · n + | f ( n + 1) | · · n + 1) + . . . | f ([ θn ]) | · · θn ] ≤ k f k ∞ · π . Then we can show the following result in a similar way as Theorem 5.2.
Theorem 5.3. sup ψ ∈C d ψ ( f ) = Q d ( f ) holds for each f ∈ l ∞ . References [1] M. Bl¨umlinger,
Levy group action and invariant measures on β N , Trans. Amer. Math. Soc. (12) (1996) 5087-5111.[2] R. Kunisada,
Density measures and additive property , J. Number Theory, (2017), 184-203.[3] P. Letavaj, L. Miˇs´ık, M. Sleziak,
Extreme points of the set of density measures,
J. Math. Anal.Appl. (2015), 1150-1165.[4] D. Maharam,
Finitely additive measures on the integers,
Sankhya Ser. A (1976) 44-59.[5] G. P´olya, Untersuchungen ¨uber L¨ucken und Singularit¨aten von Potenzreihen , Math. Z. (1929),549-640.[6] E. K. van Douwen, Finitely additive measures on N , Topology Appl. (1992) 223-268. Faculty of Education and Integrated Arts and Science, Waseda University, Shinjuku-ku, Tokyo 169-8050, Japan
E-mail address : [email protected]@aoni.waseda.jp