Equivalent quasi-norms on generalized Orlicz spaces

Mathematical Inequalities &Applications Volume 26, Number 2 (2023), 401–413 doi:10.7153/mia-2023-26-26 EQUIVALENT QUASI–NORMS ON GENERALIZED ORLICZ SPACES ANTONIO FERN ´ ANDEZ∗,JOS´ EMANUEL S´ ANCHEZ-CUADRADO AND ENRIQUE ALFONSO S´ ANCHEZ-P´ EREZ (Communicated by P. T. Perez) Abstract. In this paper we show that the equivalence among the classical quasi-norms of the generalized Orlicz spaces XΦ— the Orlicz quasi-norm, the Luxemburg quasi-norm and the Amemiya quasi-norm — holds under some mild conditions on the underlying quasi-Banach function space X— mainly the weak Fatou property — improving previous results of [4]for which some lattice convexity requirements for the quasi-Banach function space Xwere needed. 1. Introduction The construction of the classical Orlicz spaces LΦ( μ )lies in the structure of the space of Lebesgue integrable functions L1( μ ). Indeed, given an N-function Φ,the finiteness of a given integral modular provides a criterion for determining when a given function belongs to the corresponding Orlicz space. In this case, it is well-known that the so-called Orlicz and Luxemburg norms are equivalent. The same ideas that allow the construction of the Orlicz spaces can be applied using as underlying space any other (quasi-) Banach function space Xof measurable functions instead of L1( μ ).In this way, for a Young function (or N-function) Φwe can construct the generalized Luxemburg XΦ L,Amemiya XΦ Aor Orlicz XΦ Ospace. As we will see later we will always have the equality XΦ L=XΦ A.The inclusion XΦ L⊆XΦ O always holds. However it is worth mentioning that generalized Orlicz spaces defined a la Orlicz and a la Luxemburg are in general different (see [4, Example 4.1]). In this case it is natural to ask under what conditions these spaces coincide and, when this occurs, if the equivalences among the different quasi-norms for the generalized Orlicz, Luxemburg or Amemiya spaces remain true. In recent years, some authors have devoted some attention to this question. Among them, and starting from the results that were published by Jain, Persson and Upreti in [6], some of the authors of the present paper, together with other mathematicians, have proved in [4, Theorem 5.5] for an N-function Φand a quasi-Banach function space Mathematics subject classification (2020): 46A16, 46E30. Keywords and phrases: Generalized Orlicz spaces, Amemiya quasi-norm, Luxemburg quasi-norm, Orlicz quasi-norm, quasi-Banach function space, weak Fatou property. The first and second authors were supported by La Junta de Andaluc´ıa (Spain) under the project FQM-133. The third author was supported by the grant PID2020-112759GB-I00 funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”. ∗Corresponding author. c    ,Zagreb Paper MIA-26-26 401 402 A. FERN ´ ANDEZ,J.M.S´ ANCHEZ-CUADRADO AND E. A. S´ ANCHEZ-P´ EREZ Xover a finite measure μ with the σ -Fatou property and having a strictly monotone quasi-renorming the equality XΦ L=XΦ Oholds, and the equivalence of the corresponding Luxemburg and Orlicz quasi-norms. On the other hand, also in [4] it is shown that every quasi-Banach function space that satisfies a minimal requirement of convexity (the so-called L-convexity used by Kalton in [7]) allows a quasi-renorming that is strictly monotone. This result entails obtaining the equality XΦ L=XΦ Oand the equivalence of the Luxemburg and Orlicz quasi-norms for a quasi-Banach function space Xover a finite measure μ with the σ - Fatou property and an N-function Φif Xis lattice r-convex for some 0 <r<∞(see [4, Corollary 6.6]). In this paper, continuing the work started in [4], we are interested in showing that these results can be strongly improved. In fact, we will show that we can remove the hypothesis on the lattice convexity of the space X,and we will also replace the σ - Fatou hypothesis by a weaker request, the so-called weak Fatou property, to obtain the equivalence of the Orlicz and Luxemburg quasi-norms on the generalized Orlicz space. For this purpose, we will use as a technical resource the Amemiya quasi-norm: we will prove that it is equivalent to both the Luxemburg and the Orlicz quasi-norms. 2. Preliminaries and notation In this article we will consider function spaces over a measure space (Ω,Σ, μ ), where Ωis a nonempty set, Σis a σ -algebra of subsets of Ωand μ is a finite positive measure defined on Σ.We will denote by L0( μ )the space of ( μ -a.e. equivalence classes of) measurable functions f:Ω−→ Requipped with the topology of convergence in measure. Aquasi-normed function space over μ is any order ideal X⊆L0( μ )which is a quasi-normed lattice with respect to the μ -a.e. order, that is, if f∈L0( μ ),g∈X and |f|⩽|g| μ -a.e., then f∈Xand fX⩽gX,where ·Xis the quasi-norm of X.We usually denote by C⩾1a quasi-triangle constant of X,that is, f+gX⩽ C(fX+gX)for all f,g∈X.We will always assume that characteristic function of Ω, χ Ω,belongs to X.Note that any quasi-normed function space over μ is continuously embedded into L0( μ ),as it is proved in [10, Proposition 2.2]. A complete quasi-normed function space is called a quasi-Banach function space. If, in addition, the quasi-norm happens to be a norm, then Xis called a Banach function space. We say that a quasi-normed function space Xhas the σ -Fatou property if for any positive increasing sequence (fn)nin Xwith sup n⩾1 fnX<∞,we have that sup n⩾1 fn∈X and     sup n⩾1 fn   X =sup n⩾1 fnX.We say that a quasi-normed function space Xhas the weak Fatou property if for any positive increasing sequence (fn)nin X,with sup n⩾1 fnX<∞, we have that sup n⩾1 fn∈X.It is known that if a quasi-normed function space has the σ - Fatou property (weak Fatou property), then it is complete and hence a quasi-Banach function space (see [10, Proposition 2.35]). EQUIVALENT QUASI-NORMS ON GENERALIZED ORLICZ SPACES 403 Note that the only difference among the σ -Fatou and the weak Fatou properties is that the first one requires in addition that     sup n⩾1 fn   X =sup n⩾1 fnX.Thus, each quasinormed function space Xwith the σ -Fatou property obviously satisfies also the weak Fatou property. However, in general the σ -Fatou property is not quasi-renorming invariant, while the weak Fatou property clearly is. This fact will be relevant for us, since we will use quasi-renormings in several steps of our work. The following result is the adaptation (regarding the proof; the statement is exactly the same) for quasi-Banach function spaces of a well-known result of Amemiya for Banach function spaces. It states that although the equality     sup n⩾1 fn   X =sup n⩾1 fnXis not guaranteed by the weak Fatou property, it implies a weaker inequality that properly relates the norm     sup n⩾1 fn   X with sup n⩾1 fnX.The proof of next result is adapted from Theorem 2 in [12, Ch. 15 §65], that refers to an original proof of Amemiya [2]. THEOREM 1. (Amemiya) Let X be a quasi-Banach function space with the weak Fatou property. Then there is a constant G ⩾1(only depending on ·X) such that fX⩽Gsup n⩾1 fnXwhenever 0⩽fn↑f∈X. Proof. Let C⩾1 be a quasi-triangular constant for ·X.Suppose by contradiction that there is no such a constant G⩾1.Then for each natural number p⩾1there is a sequence (fpn)n⊆Xand a function fp∈Xsuch that 0 ⩽fpn ↑fpand  fp X>p3Cpsup n⩾1 fpn X,p=1,2,... (1) By multiplying by appropriate constants we can further assume that sup n⩾1 fpn X=1 p2Cp,p=1,2,... (2) From (1)and(2)wehavethat fp X⩾pfor all p=1,2,... Take now the functions gn:=f1n+f2n+···+fnn,n=1,2,... Then 0 ⩽g1⩽g2⩽···,and for every n⩾1 we have gnX=     n ∑ p=1 fpn    X ⩽ n ∑ p=1 Cp fpn X ⩽ n ∑ p=1 Cp1 p2Cp ⩽ ∞ ∑ p=1 1 p2<∞. Thus, sup n⩾1 gnX<∞and then g:=sup n⩾1 gn∈X,as a consequence of the weak Fatou property of X.But this gives a contradiction, since for each natural number p⩾1,if 404 A. FERN ´ ANDEZ,J.M.S´ ANCHEZ-CUADRADO AND E. A. S´ ANCHEZ-P´ EREZ we take n⩾pwe get g⩾gn=f1n+f2n+···+fnn ⩾fpn,n⩾p and so g⩾fpfor all p=1,2,... We conclude that gX⩾ fp X⩾pfor all p= 1,2,..., what gives a contradiction.  Finally, recall that a quasi-normed function space Xis said to be σ -order continuous if for any positive increasing sequence (fn)nin Xconverging μ -a.e. to a function f∈X,we have that f−fnX→0. 3. Young functions and N-functions We collect here some results, all known, on Young’s functions that we will need next. AYoung function is any strictly increasing convex function (and so continuous) Φ:[0,∞)−→ [0,∞)such that Φ(0)=0 and lim x→∞Φ(x)=∞.From the convexity of Φ we have the following useful inequality Φ( α x)⩽ α Φ(x)if 0 ⩽ α ⩽1,x⩾0.(3) A Young function Φis called an N-function if Φsatisfies the two limit conditions lim x→0 Φ(x) x=0 and lim x→∞ Φ(x) x=∞.The complementary function of the Young function Φis defined as ˆ Φ(y):=sup x⩾0 {xy −Φ(x)},for all y⩾0.From the definition of ˆ Φit is clear that Φand ˆ Φsatisfy the Young inequality xy⩽Φ(x)+ ˆ Φ(y),x,y⩾0.(4) Every Young function Φhas a right derivative, that is, a non-decreasing, right continuous function ϕ :[0,∞)−→ [0,∞),with ϕ (0)=0,such that Φ(x)=x 0 ϕ (t)dt for all x∈[0,∞)(see [9, Theorem 1.1] or [11, Theorem 1.3.1]). This function ϕ also satisfies the following equality (see [9, (2.7)] or [11, Theorem 1.3.3]) that we will use later x ϕ (x)=Φ(x)+ ˆ Φ( ϕ (x)),x⩾0.(5) A Young function Φhas the Δ2-property, written Φ∈Δ2,if there exists a constant K>1 such that Φ(2x)⩽KΦ(x)for all x⩾0. Next lemma is well-known and will be used later. The proof can be seen in [9, Ch. I §1p.9] LEMMA 1. Let Φbe an N-function and take its right derivative ϕ .Then we have that lim x→∞ ϕ (x)=∞and lim x→0 ϕ (x)=0. EQUIVALENT QUASI-NORMS ON GENERALIZED ORLICZ SPACES 405 4. Quasi-norms on generalized Orlicz spaces In this section we introduce the Luxemburg, Amemiya and Orlicz quasi-Banach function spaces whose relations, mainly the equivalence of their quasi-norms, will be the aim of our work. The reader can find information about in [3]and[4], in which there is an analysis of the relations among two of them, and in [6], where the same topic is studied for the Banach space case under the assumption of the σ -Fatou property. Let Φbe a Young function and let Xbe a quasi-normed function space over a finite measure μ . The (generalized) Luxemburg space XΦ Lis defined as the following set: XΦ L:=f∈L0( μ ):∃c>0:Φ|f| c∈X. Given f∈XΦ L,we define the Luxemburg lattice quasi-norm of fby fXΦ L:=infc>0:Φ|f| c∈X,with     Φ|f| c   X ⩽1.(6) The (generalized) Amemiya space XΦ Ais defined as the following set: XΦ A:=f∈L0( μ ):∃k>0:Φ(k|f|)∈X. Given f∈XΦ A,we define the Amemiya lattice quasi-norm of fby fXΦ A:=inf1 k(1+Φ(k|f|)X),k>0.(7) The Luxemburg and Amemiya spaces defined above, equipped with their corresponding quasi-norms, are really quasi-normed function spaces over the finite measure μ with the same quasi-triangle constant as the one of the quasi-norm of the space X. PROPOSITION 1. Let (X,·X)be a quasi-normed function space over the measure μ with the weak Fatou property. Let Φbe Young function. Then the Amemiya space XΦ Ahas also the weak Fatou property. This is a consequence of the equality XΦ A=XΦ L,the equivalence of the quasinorms · XΦ Aand · XΦ Lwhich will be shown in Theorem 2, and of the fact that the Luxemburg space XΦ Lhas the weak Fatou property (see [3, Theorem 4]). Now, let Φbe an N-function. The corresponding (generalized) Orlicz space XΦ O is defined as the following set: XΦ O:=f∈L0( μ ):fXΦ O<∞, where ·XΦ Ois the Orlicz quasi-norm defined by fXΦ O:=supfgX:ˆ Φ(|g|)∈X, ˆ Φ(|g|) X⩽1.(8) 406 A. FERN ´ ANDEZ,J.M.S´ ANCHEZ-CUADRADO AND E. A. S´ ANCHEZ-P´ EREZ As we have pointed out above XΦ A=XΦ Las sets. Nevertheless, the spaces XΦ Land XΦ Oare in general different (see [4, Example 4.1]), but we proved in [4, Proposition 3.3]) that the inclusion XΦ L⊆XΦ Oholds and fXΦ O ⩽2CfXΦ L,f∈XΦ L,(9) where C⩾1 is a quasi-triangular constant for X. 5. Equivalence of the Amemiya and Luxemburg quasi-norms Let us now check the inequalities that relate the Amemiya and Luxemburg quasinorms. The next result provides the main tool in order to do it. PROPOSITION 2. Let X be a quasi-normed function space over a measure μ ,and let Φbe a Young function. Given a function f ∈XΦ L,we have that fXΦ L=infmax1 k,1 kΦ(k|f|)X,k>0 =inf1 kmax{1,Φ(k|f|)X},k>0.(10) Proof. Let us show first the inequality fXΦ L ⩽infmax1 k,1 kΦ(k|f|)X,k>0. Let k>0.If Φ(k|f|)X⩽1,then (by the definition of fXΦ L) we can conclude that fXΦ L ⩽1 k.On the other hand, if we have Φ(k|f|)X>1,using (3)weget      Φ|f| 1 kΦ(k|f|)X    X =    Φk|f| Φ(k|f|)X   X ⩽Φ(k|f|)X Φ(k|f|)X =1. In this second case we have that fXΦ L ⩽1 kΦ(k|f|)X.And in any case, we have that fXΦ L ⩽max1 k,1 kΦ(k|f|)X for all k>0.Computing the infimum with respect to k>0,we conclude that fXΦ L ⩽infmax1 k,1 kΦ(k|f|)X,k>0. Let us show now the converse inequality, that is, infmax1 k,1 kΦ(k|f|)X,k>0⩽fXΦ L. EQUIVALENT QUASI-NORMS ON GENERALIZED ORLICZ SPACES 407 Let c>0 such that     Φ|f| c   X ⩽1.Then infmax1 k,1 kΦ(k|f|)X,k>0⩽maxc,c    Φ|f| c   X⩽c. Computing the infimum with respect to c>0,we get that infmax1 k,1 kΦ(k|f|)X,k>0⩽fXΦ L. The next result establishes the equivalence of the Amemiya and Luxemburg quasinorms. THEOREM 2. Let X be a quasi-normed function space over a measure μ and Φ a Young function. Then the quasi-norms ·XΦ Aand ·XΦ Lare equivalent in XΦ A=XΦ L. In particular, we have that fXΦ L ⩽fXΦ A ⩽2fXΦ L,(11) for all f ∈XΦ A=XΦ L. Proof. It is enough to consider Proposition 2and the inequality max{1,x}⩽1+x⩽2max{1,x}, for all x⩾0.So we get 1 kmax{1,Φ(k|f|)X}⩽1 k(1+Φ(k|f|)X)⩽2 kmax{1,Φ(k|f|)X} and then we obtain the equivalence among the quasi-norms just by computing the infimum on k>0. 6. Stability of generalized Orlicz spaces by equivalence of Young functions and quasi-renorming of X Later we will need to renormalize the underlying space Xand change the function Φfor another one with better properties. In this section we will check that these changes will not affect the corresponding generalized Orlicz space. Something similar will be shown for the Luxemburg space and the Amemiya space. Most of these results are probably known, and, in many cases, the proofs are straightforward. DEFINITION 1. Let Φand ΨYoung functions. We say that Ψis stronger than Φif there is a constant a>0 such that Φ(x)⩽Ψ(ax),for all x⩾0.In this case we write Φ≺Ψ.We say that Φand Ψare equivalent if Φ≺Ψand Ψ≺Φ.In this case we write Φ≡Ψ. 408 A. FERN ´ ANDEZ,J.M.S´ ANCHEZ-CUADRADO AND E. A. S´ ANCHEZ-P´ EREZ REMARK 1. It is proved in [11, Theorem 2, p. 16] that if Φand Ψare N-functions such that Φ≺Ψ,then ˆ Ψ≺ˆ Φ.In particular, if Φ≡Ψ,then ˆ Φ≡ˆ Ψ. PROPOSITION 3. Let Φand Ψbe Young functions such that Φ≺Ψwith constant a >0.Then XΨ L⊆XΦ L, and we also have that fXΦ L ⩽afXΨ Lfor all f ∈XΨ L. In particular, if Φ≡Ψthen XΦ L=XΨ Land the quasi-norms · XΦ Land · XΨ Lare equivalent. We have similar results for the Amemiya and Orlicz quasi-norms. PROPOSITION 4. Let Φand Ψbe Young functions such that Φ≺Ψwith constant a >0.Then XΨ A⊆XΦ Aand fXΦ A ⩽afXΨ Afor all f ∈XΨ A,and if Φ≡Ψthen XΦ A=XΨ Aand the quasi-norms ·XΦ Aand ·XΨ Aare equivalent. PROPOSITION 5. Let Φand Ψbe N-functions such that Φ≺Ψ.Let b >0a constant associated to ˆ Ψ≺ˆ Φ.Then fXΦ O ⩽bfXΨ Ofor all f ∈XΨ O.Thus XΨ O⊆XΦ O. In particular, if Φ≡Ψthen XΦ O=XΨ Oand the quasi-norms · XΦ Oand · XΨ Oare equivalent. To conclude the first part of this section we present the following result that will allow us to assume that the Young function (N-function) Φused for the construction of the spaces XΦ O,XΦ Aand XΦ Lcan be chosen with a continuous derivative. PROPOSITION 6. Let Φbe a Young function (an N-function). Then there is a Young function (an N-function) Ψwith continuous derivative such that Φ≡Ψ. Proof. Given the Young function (N-function) Φ,let us consider its right derivative ϕ :[0,∞)−→ [0,∞).We know that it is non-decreasing, right continuous and Φ(x)=x 0 ϕ (t)dt for all x∈[0,∞).Consider now the function Ψ:[0,∞)−→ [0,∞), given by Ψ(x):=x 0 Φ(u) udu =x 01 uu 0 ϕ (t)dtdu =x 0 ϕ (t)log x tdt.(12) Let us see that it satisfies the requirements of the statement of the proposition. i) There is a>0 such that Ψ(x)⩽Φ(x)⩽Ψ(ax),for all x⩾0.Indeed, let us prove that the above inequalities are satisfied for a=2.Since the function Φ(u) uis increasing (due to the fact that Φis convex), we have that Ψ(x)=x 0 Φ(u) udu ⩽Φ(x) xx=Φ(x) for all x⩾0.On the other hand, Ψ(2x)=2x 0 Φ(u) udu ⩾2x x Φ(u) udu ⩾Φ(x) xx=Φ(x) EQUIVALENT QUASI-NORMS ON GENERALIZED ORLICZ SPACES 409 for all x⩾0.In fact, it can be proved that Φ(x)⩽Ψ(ax)for all x⩾0 and each a⩾1. ii) We claim that Ψhas continuous derivative. Indeed, note that the function x→1 xx 0 ϕ (u)du is continuous, and by the Fundamental Theorem of Calculus, we have that Ψ(x)=1 xx 0 ϕ (u)du for all x⩾0. iii) It is also clear that Ψis increasing, since Ψ(x)⩾0forallx⩾0. iv) We also have that Ψis convex. Indeed, at the points where ϕ is continuous (except countable many points), the function Ψhas a derivative and it is given by Ψ(x)=1 x ϕ (x)−1 x2x 0 ϕ (u)du,for all x⩾0.Now it is clear that Ψ(x)⩾0ifand only if x ϕ (x)⩾x 0 ϕ (u)du and the last inequality holds since ϕ is non-decreasing. In fact, any function Ψgiven by formula (12) is convex. Now, assume that Φis an N-function. Then we have as a direct consequence of i) that lim x→0 Ψ(x) x=0 and lim x→∞ Ψ(x) x=∞. Finally, we will show that, if we renorm the space Xwe get equivalent (quasi-) norms for the associated generalized Orlicz spaces. For the proof, see [4, Proposition 5.10]. PROPOSITION 7. Let Φbe a Young function and let X an ideal of L0( μ ).Consider two equivalent lattice quasi-norms ·1and ·2on X,and denote by X1:= (X,·1)and X2:=(X,·2)the corresponding quasi-normed function spaces. Then 1) the Luxemburg quasi-norms ·XΦ 1Land ·XΦ 2Lare also equivalent, and 2) the Amemiya quasi-norms ·XΦ 1Aand ·XΦ 2Aare equivalent too. If moreover, Φis an N-function, then 3) the Orlicz quasi-norms ·XΦ 1Oand ·XΦ 2Oare equivalent. Recall that a quasi-norm · on Xis not necessarily a continuous function ·: x∈X−→  x∈[0,+∞); which of course it is if · is a norm. DEFINITION 2. Let 0 <p⩽1.Ap-norm on a linear space Xis a function |·|: X−→ [0,+∞)satisfying a) |x| =0 if and only if x=0. b) | α x| =| α ||x|, α ∈R,x∈X. c) |x+y| p⩽|x|p+|y| p,x,y∈X. Ap-norm on Xis a quasi-norm with quasi-triangular constant C=21 p−1,which is always continuous in X.Let us recall that the Aoki-Rolewicz Theorem guarantees that we can always find an equivalent continuous quasi-norm for the quasi-normed space X.