Besicovitch-Musielak-Orlicz spaces of almost periodic functions equipped with the Orlicz norm

A. Daoui, M. Morsli, M. Smaali

On the strict convexity of

Besicovitch-Musielak-Orlicz spaces of almost periodic functions equipped with the Orlicz norm

In honour of professor Julian Musielak, on the occasion of his 85th birthday.

With esteem and consideration

Abstract. We characterize the strict convexity of Besicovitch-Musielak-Orlicz spa- ces of almost periodic functions, when it is equipped with the Orlicz norm. It is shown that this property is equivalent to the strict convexity of the Musielak-Orlicz function generating the space.

2000 Mathematics Subject Classification: 46B20, 42A75..

Key words and phrases: Orlicz norm, Amemiya norm, conjugate function, Besicovitch- Musielak-Orlicz spaces, almost periodic functions, strict convexity.

1. Introduction. The theory of almost periodic functions was the subject to various generalizations. It’s extension in the context of Lebesgue spaces was deve- lopped in [1] by A.S.Besicovitch to obtain the so called Besicovitch space of almost periodic functions.

In [5], T. R. Hillmann considered such class of functions in connection with the theory of Orlicz spaces, introducing the new class of Besicovitch-Orlicz almost periodic functions.

In [8], [9] and [10] the authors characterized some fundamental geometric pro- perties of this space.

Recently in [11] and [12] a natural extension of the class of almost periodic functions was considered in connection with the theory of Musielak-Orlicz spaces, where the strict and uniform convexity are characterized in case the space is endowed with the Luxemburg norm.

As these properties are of a metric nature, it is natural to consider them in case of the Orlicz norm. The later one was introduced for Besicovitch-Musielak-Orlicz space of almost periodic functions in [4], where some duality properties of this space are also stated. Here the strict convexity is characterized.

2. Preliminaries. Let ϕ be a Musielak-Orlicz function, i.e. ϕ: R × R⁺ −→

R⁺ is such that:

1. ∀t ∈ R, ϕ(t, .) is convex on R⁺.

2. ∀x ∈ R⁺, ϕ(., x) is measurable on R and ϕ(t, x) = 0 iff x = 0, ∀t ∈ R.

3. ∀t ∈ R, lim

x→+∞

ϕ(t,x)

x = +∞ and lim_x

→0 ϕ(t,x)

x = 0.

In the sequel we will moreover assume that ϕ verifies the following two condi- tions:

(4) ϕ(., .) is continuous on R × R⁺.

(5) ∀x ∈ R⁺, ϕ(., x) is periodic whith period T independent of x (we may assume that T = 1).

We denote by ψ the function complementary to ϕ, i.e.

ψ(t, x) = sup

y≥0{xy − ϕ(t, y)}, ∀t ∈ R, ∀x ∈ R⁺.

Recall that ψ is also a Musielak-Orlicz function (see [13]) and that the pair (ϕ, ψ) satisfies the Young inequality:

xy≤ ϕ(t, x) + ψ(t, y) for all t ∈ R and x, y ∈ R⁺.

Let M(R, C) = M be the set of all Lebesgue measurable functions on R with values in C.

The functional

ρϕ:M −→ [0, +∞]

f −→ ρ^ϕ(f ) = lim

T→+∞

1 2T

Z +T

−T

ϕ(t,|f(t)|)dt = M[ϕ(., |f(.)|)]

is a convex pseudo-modular on M (see [1]). The associated modular space B^ϕ = {f ∈ M : lim_α→0ρϕ(αf ) = 0},

= {f ∈ M : ρ^ϕ(λf ) < +∞, for some λ > 0}, is called Besicovitch-Musielak-Orlicz space.

This space is naturally endowed with the Luxemburg norm:

kfk^ϕ= inf



k > 0, ρϕ

f k



≤ 1

 .

Let A be the set of all generalized trigonometric polynomials, i.e.,

A = {Pⁿ(t) =

j=nX

j=1

aje^iλjt, aj∈ C, λ^j ∈ R, n ∈ N}.

The Besicovitch-Musielak-Orlicz space of almost periodic functions denoted by B^ϕa.p.is the closure of the set A with respect to the Luxemburg norm:

B^ϕa.p. ={f ∈ B^ϕ:∃{pⁿ} ∈ A s.t. lim

n→+∞kf − pⁿk^ϕ= 0}.

When ϕ(t, x) = |x| the space B^ϕa.p.is denoted by B¹a.p..

The closure of the set A with respect to the modular ρϕ is the subspace of B^ϕ denoted by eB^ϕa.p.:

Be^ϕa.p. ={f ∈ B^ϕ:∃{pⁿ} ∈ A s.t. lim

n→+∞ρϕ(α(f− pⁿ)) = 0for some α > 0}.

Let {u.a.p.} denote the classical Bohr’s algebra of almost periodic functions. It is known that {u.a.p.} is the closure of the set A with respect to the uniform norm and that {u.a.p.} ⊆ B^ϕa.p.⊆ B¹a.p.. Moreover, in view of Theorem 2.8 in [3] we have the following very useful property:

Let ϕ(., .) be continuous and such that ϕ(., x) is uniformly almost periodic with respect to x, then:

if f ∈ {u.a.p.} we have ϕ(., |f(.)|) ∈ {u.a.p.}.

Therefore a fortiori, this holds true for a Musielak-Orlicz function satisfying the conditions (4) and (5) cited above.

A fundamental and key result concerning the functions in B^ϕa.p.is the following:

(1) If f ∈ B^ϕa.p., then ϕ(., |f(.)|) ∈ B¹a.p.(see [12]).

This property ensures the existence of the limit in the expression of ρϕ(f ).

We can define on B^ϕa.p.the so called Orlicz norm:

kfk^oϕ= sup{M(|fg|), g ∈ B^ψa.p., ρψ(g)≤ 1}.

In [4], it was shown that this norm is also formulated by means of the Amemya formula:

kfk^oϕ = inf

1

k(ρϕ(kf ) + 1), k > 0

 (2) .

= 1

k0

(ρϕ(k0f ) + 1)for some k0> 0.

Moreover, the last norm is equivalent to the Luxumburg norm, more precisely:

(3) kfk^ϕ≤ kfk^oϕ≤ 2kfk^ϕ, for all f ∈ B^ϕa.p..

3. Auxiliary results. The fundamental convergence results of measure theory can’t be used directly in B^ϕa.p.. A key role in our computations is played by the set function µ defined on the σ-algebra Σ(R) = Σ of Lebesgue measurable sets as follows:

µ(A) = lim

T→+∞

1 2T

Z +T

−T

χA(t)dt = lim

T→+∞

1

2Tµ(A∩ [−T, +T ]), where µ denotes the Lebesgue measure on R.

We give here some definitions and convergence type results with respect to the set function µ.

Let {fn} be a sequence in B^ϕ. We say that:

• {fⁿ} is µ convergent to f (and denote by fⁿ−→ f) when, ∀η > 0, lim^µ _n

→+∞µ{t ∈ R : |fn− f| > η} = 0.

• {fn} is modular convergent to f when, ∃α > 0 such that:

n→+∞lim ρϕ(α(fn− f)) = 0.

The connection between these kinds of convergence are summarized as follows:

1. lim

n→+∞kfⁿ− fk^ϕ= 0 iff ∀α > 0, lim_n→+∞ρϕ(α(fn− f)) = 0 (see [13]).

2. If lim

n→+∞ρϕ(fn− f) = 0 then, fⁿ−→ f (see Lemma 2 in [12]).^µ

We now list some technical lemmas that we will use in the proof of the main result.

Lemma 3.1 [12] Let f ∈ B^ϕa.p., ρϕ(f ) > 0. Then there exist α, β, θ with 0 < α <

β; θ ∈]0, 1[ and G = {t ∈ R, α ≤ |f(t)| ≤ β} such that µ(G) ≥ θ .

Lemma 3.2 Let (an)_n≥1, an > 0 be a sequence of real numbers. With all n ∈ N, we associate measurable sets An s.t.

i. Ai∩ A^j= φ, for i 6= j and S

n≥1

An⊂ [0, 1] .

ii. P

n≥0

R1 0

ϕ (t, anχAn) dt < +∞, and if f = P

n≥1

anχAn defined on [0, 1] and ˜f is the periodic extension of f to the whole R ( with period τ = 1), then ˜f ∈ eB^ϕa.p.

Remark 3.3 Lemma 3.2 is proved in [12] with the assumption that S

n≥1

An ⊂ [0, α] , α < 1. The actual version is more general, it will be used in the next Lemma to prove the embedding of the usual Musielak-Orlicz class E^ϕ([0, 1]) in the Besicovitch-Musielak-Orlicz space of almost periodic functions B^ϕa.p..

Recall that E^ϕ([0, 1])is the subspace of the Musielak-Orlicz space L^ϕ([0, 1])defined as follows:



f measurable : ρEϕ(λf ) = Z 1

0

ϕ(t,|λf(t)|)dt < +∞, ∀λ > 0

 .

Proof For an arbitrary ε > 0, take n0∈ N such that P

n≥n0

R1

0 ϕ(t, anχAn)≤ ^ε3 and put M = max

n≤n0

sup

t ϕ(t, 2an), then choose δ > 0 with δ ≤ _3M^ε . Let f1 = Pn=n0

n=1 anχAn, and denote by f1^r the restriction of f1 to the interval [0, 1− δ].

By the Luzin theorem there exists gε^ra continuous function on [0, 1 − δ] s.t.:

µ{t ∈ [0, 1 − δ] : ϕ(t, |f1^r− gε^r|) > 0} ≤ ε 3M.

Moreover, since f1^ris bounded, g^rε is also bounded (with the same bound).

Let now gε be the linear extension of g^rε to [0, 1], precisely gε= g_ε^r on [0, 1 − δ], is linear on [1 − δ, 1] and satisfies g^ε(1) = gε(0). Then

Z 1 0

ϕ



t,|f(t) − g^ε(t)| 2

 dt ≤

Z 1 0

ϕ



t,|f(t) − f¹| + |f¹(t)− g^ε(t)| 2

 dt

≤ 1

2 Z 1

0

ϕ(t,|f − f¹|)dt +1 2

Z 1 0

ϕ(t,|f¹− g^ε|)dt

≤ 1

2 Z 1

0

ϕ



t, X

n≥n0

anχAn



 dt

+ 1

2 Z 1−δ

0

ϕ(t,|f1^r− gε^r|)dt

+ 1

2 Z 1

1−δ

ϕ(t,|f¹− g^ε|)dt

≤ 1

2 X

n≥n0

Z 1 0

ϕ(t, anχAn)dt +1 2M ε

3M +1 2M ε

3M

≤ ε

2.

Let ˜f and ˜gε be the respective periodic extensions of f and g to R. Since ˜gε ∈ {u.a.p.} ⊂ B^ϕa.p., then there exists pε∈ A s.t. ρ^ϕ

g˜ε− p^ε 2



≤2^ε.

Using the periodicity of ϕ we will have

ρϕ

f˜− ˜g^ε 2

!

= lim

T→+∞

1 2T

Z +T

−T

ϕ t,| ˜f− ˜gε| 2

! dt

= Z 1

0

ϕ



t,|f − g^ε| 2

 dt

≤ ε

2. Finally

ρϕ

f˜− p^ε 4

!

≤ 1

2ρϕ

f˜− ˜gε

2

! +1

2ρϕ

g˜ε− p^ε 2

 ,

≤ ε. 

Lemma 3.4 If f ∈ E^ϕ([0, 1]), then,

(1) If ˜f is the periodic extension of f to the whole R ( with period τ = 1), we have ˜f ∈ B^ϕa.p..

(2) The embedding map i : E^ϕ([0, 1]) ,→ B^ϕa.p., i (f ) = ˜f is an isometry with respect to the modulars and for the respective norms.

Proof (1) First suppose that f = Pi=n0

i=0 aiχAi with ∪ⁱ⁼ⁿi=1⁰Ai ⊂ [0, 1], and Aⁱ∩ Aj=∅ if i 6= j. By Lemma 3.2 we have ˜f ∈ eB^ϕa.p.. Moreover, since n ˜f ∈ eB^ϕa.p.

for each n ∈ N we have the following :

∀n > 0, ∃pⁿ∈ A s.t. ρϕ

n( ˜f− pⁿ) 4

!

≤ 1 n. Hence we deduce that lim

n→+∞k ˜f− pⁿk^ϕ= 0and then ˜f ∈ B^ϕa.p..

Let now f ∈ E^ϕ([0, 1]). Then there exists a sequence {fⁿ} of simple functions on [0, 1]such that lim

n→+∞kfⁿ− fk^Eϕ= 0.

Denote by ˜f and ˜fn the periodic extensions of f and fn respectively. Then using the periodicity of ϕ we deduce that lim

n→+∞k ˜fn− ˜fk^ϕ= lim

n→+∞kfⁿ− fk^Eϕ= 0.This means that ˜f ∈ B^ϕa.p..

(2) The embedding map i : E^ϕ([0, 1]) ,→ B^ϕa.p., i (f ) = ˜f is clearly an isometry with respect to the modulars and then also for the respective norms.

4. Main result. Here we will characterize the strict convexity of the B^ϕa.p.

space by means of it’s generating Musielak-Orlicz function.

Definition 4.1 The function ϕ(., .) is called strictly convex when, for almost all t∈ R the function ϕ(t, .) is strictly convex. More precisely:

There exists E ⊂ R with µ(E^c) = 0 such that ∀t ∈ E, ∀x, y ∈ R⁺ there holds:

(4) ϕ(t, αx + (1− α)y) < αϕ(t, x) + (1 − α)ϕ(t, y), ∀α ∈]0, 1[.

Theorem 4.2 B^ϕa.p.,k.k^oϕ

is strictly convex if and only if ϕ is strictly convex.

Proof Sufficiency: Let K(f) = {k > 0, s.t. kfk^oϕ = ¹_k(ρϕ(kf ) + 1)}, then in view of (2), K(f) 6= ∅.

Let f1 and f2 ∈ B^ϕa.p. s.t. kf¹k^oϕ = kf²k^oϕ = 1 and kf¹− f²k^oϕ > 0. Then, for s ∈ K(f¹) and m ∈ K(f²) it is clair that ksf¹− mf²k^oϕ > 0. Indeed in the opposite case we will have ksf¹k^oϕ=kmf²k^oϕ and consequently s = m which leads to kf¹− f²k^oϕ= 0, contradiction.

Now by Lemma3.1 there exists σ > 0, θ ∈]0, 1[ s.t. : µ(G) > θ, where G ={t ∈ R : |sf¹(t)− mf²(t)| ≥ σ}.

For a given k > 0, consider the following sets:

A1={t ∈ R : |f¹(t)| ≥ k}, A2={t ∈ R : |f²(t)| ≥ k}.

We have clearly

1 =kfⁱk^oϕ≥ kfⁱk^ϕ≥ kfχ^Aik^ϕ≥ kkχ^Aik^ϕ, i = 1, 2, i.e.

kχ^Aik^ϕ≤ 1

k, i = 1, 2.

Now, we can choose k > 1 such that:

∀A ∈ Σ, µ(A) ≥ θ

4 =⇒ kχ^Ak^ϕ> 1 k.

Indeed from Lemma 1 in ([12]) we know that ∀α > 0, ∃δ > 0, such that ρϕ(χA)≤ δ ⇒ µ(A) < α, ∀A ∈ Σ.

The result follows easily since ρϕ(χA)≤ kχ^Ak^ϕwhenever kχ^Ak^ϕ< 1. Consequently we have

µ(Ai) < θ

4, i = 1, 2.

Now, for k, σ as above and b = max(s, m) let us define the set:

Q ={(x, y) ∈ R⁺ s.t. : |x| ≤ bk, |y| ≤ bk, |x − y| ≥ σ}.

For all α ∈]0, 1[, consider on R × Q the following function:

F (t, x, y) = ϕ(t, αx + (1− α)y) αϕ(t, x) + (1− α)ϕ(t, y). We have F (t, x, y) < 1, ∀(t, x, y) ∈ E × Q.

By (4) and the periodicity of ϕ(., x), there exists E1 ⊂ [0, 1] with µ(E1^c) = 0 such that for all t ∈ E1, ϕ(t, .) is strictly convex.

Let K^θ=_n∪

i∈Z{K1^θ+ni} where, K1^θ⊂ E¹is a compact subset such that µ(E1\K1^θ) <

θ

4. We have also µ(E \ K^θ) < ^θ₄.

Using the periodicity and the continuity of F on E × Q, we can find δ > 0 s.t.

sup

K^θ×Q

F (t, x, y) = 1− δ.

Put H = (G ∩ K^θ)\ (A¹∪ A²)and note that µ(H) ≥ ^θ4. Now for all t ∈ H, we have

ϕ(t, m

s + m(s|f¹(t)|) + s

s + m(m|f²(t)|)) = ϕ(t, sm

s + m(|f¹(t)| + |f²(t)|))

≤ (1 − δ)( m

s + mϕ(t, s|f¹(t)|) + s

s + mϕ(t, m|f²(t)|)).

Next,

2− kf1+ f2k^oϕ≥ 1

s(1 + ρϕ(sf1)) + 1

m(1 + ρϕ(mf2))−s + m

sm (1 + ρϕ( sm

s + m(f1+ f2))

≥1

sρϕ(sf1) + 1

mρϕ(mf2)−s + m sm ρϕ( sm

s + m(f1+ f2))

≥ lim

T→+∞

1 2T

Z +T

−T

1

sϕ(t, s|f1|) + 1

mϕ(t, m|f2|) −s + m

sm ϕ(t, sm

s + m(|f1+ f2|))

 dt

≥ lim

T→+∞

1 2T

Z +T

−T

1

sϕ(t, s|f1|) + 1

mϕ(t, m|f2|) −s + m

sm ϕ(t, sm

s + m(|f1| + |f2|))

 χHdt

≥ lim

T→+∞

1 2T

Z +T

−T

1

sϕ(t, s|f1|) + 1

mϕ(t, m|f2|) −s + m

sm (1− δ)( m

s + mϕ(t, s|f1|)

+ s

s + mϕ(t, m|f2|)

 χHdt

≥2δ b lim

T→+∞

1 2T

Z +T

−T

ϕ(t,|sf1| + |mf2| 2 )χHdt

≥2δ b lim

T→+∞

1 2T

Z +T

−T

ϕ(t,|sf1− mf2| 2 )χHdt

≥2δ b

σ 2µ(H)

≥2δ b

σ 2 θ 4

> 0.

Necessity: Suppose that ϕ(t, .) is not strictly convex. Then (see [2] page 182.), we can find a set E ⊂ R with µ(E) > 0 and an interval [a, b] ∈ R⁺ such that ∀t ∈ E,

ϕ(t, .) is affine on [a, b]. Now since ϕ(., x) is periodic with the period τ = 1 there exists a set A ∈ [0, 1] with µ(A) > 0 and an interval [a, b] ∈ R⁺ such that ∀t ∈ A, ϕ(t, .)is affine on [a, b] i.e.

ϕ(t, x) = α(t)x + β(t),∀x ∈ [a, b], ∀t ∈ A.

Pick now x^∗ ∈]a, b[ and ε > 0 such that x^∗+ ε∈ [a, b] and x^∗− ε ∈ [a, b]. We define on [0, 1] the function f by

f = x^∗χA+ y^∗χB,

where B ⊂ [0, 1] \ A and y^∗∈ R⁺ are such that

Z 1 0

ψ(t, ϕ⁰(t, f (t)))dt = 1,

which leads to the fact that kfk^oEϕ = 1 + ρEϕ(f ),see [2].

Now, let {C¹, C2} be a partition of A so that:

Z

C1

α(t)dt = Z

C2

α(t)dt.

Define g1 and g2as follows:

g1(t) =









1

kfk^o_Eϕ(x^∗+ ε) if t∈ C¹,

1

kfk^o_Eϕ(x^∗− ε) if t∈ C²,

f

kfk^oϕ elsewhere.

g2(t) =









1

kfk^oϕ(x^∗− ε) if t∈ C¹,

1

kfk^o_Eϕ(x^∗+ ε) if t∈ C²,

f

kfk^o_Eϕ elsewhere.

Then, clearly g16= g² and g1+ g2= 2_kfk^fo

Eϕ. Moreover kg¹k^oEϕ ≤ 1

kfk^oEϕ

{1 + ρ^ϕ(kfk^oEϕg1)}

≤ 1

kfk^oEϕ

{1 + ρ^ϕ(kfk^oEϕ

f kfk^oEϕ

χ[0,1]/A)

+ Z

C1

(α(t)(kfk^oEϕ

x^∗ kfk^oEϕ

+ ε) + β(t))dt

+ Z

C2

(α(t)(kfk^oEϕ

x^∗ kfk^oEϕ

− ε) + β(t))dt}

≤ 1

kfk^oEϕ

{1 + ρ^ϕ(kfk^oEϕ

f kfk^oEϕ

χ[0,1]/A)

+ Z

C1

(α(t)(kfk^oEϕ

x^∗ kfk^oEϕ

) + β(t))dt

+ Z

C2

(α(t)(kfk^oEϕ

x^∗ kfk^oEϕ

) + β(t))dt}

≤ 1

kfk^oEϕ

{1 + ρ^ϕ(kfk^oEϕ

f kfk^oEϕ

)}

= 1. _

In the same way we may get that kg²k^oEϕ≤ 1.

Now, consider the 1− periodic extensions ˜g1, ˜g2 and ˜f of g1, g2 and f, respectively.

Then we have the following: k ˜g1k^oϕ = kg¹k^oEϕ ≤ 1, k ˜g2k^oϕ = kg²k^oEϕ ≤ 1 and k_{k ˜f}^f˜_k^o_ϕk^oϕ=k_kfk^fo_Eϕk^oEϕ= 1with _{k ˜f}^f˜_ko

ϕ = ¹₂( ˜g1+ ˜g2), which means that B^ϕa.p.is not strictly convex.

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A. Daoui

Department of Mathematics, Faculty of Sciences University of Tizi-Ouzou Algeria

E-mail: daoui_aminadz@yahoo.fr M. Morsli

Department of Mathematics, Faculty of Sciences University of Tizi-Ouzou Algeria

E-mail: mdmorsli@yahoo.fr M. Smaali

Department of Mathematics, Faculty of Sciences University of Tizi-Ouzou Algeria

E-mail: mannal-smaali@mail.ummto.dz

(Received: 10.06.2013)