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Upper Estimates for Gâteaux Differentiability of Bump Functions in Orlicz-Lorentz Spaces

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Upper Estimates for Gâteaux Differentiability of Bump Functions in Orlicz-Lorentz Spaces Volume 8 (2007), Issue 4, Article 113, 8 pp. UPPER ESTIMATES FOR GÂTEAUX DIFFERENTIABILITY OF BUMP FUNCTIONS IN ORLICZ-LORENTZ SPACES B. ZLATANOV DEPARTMENT OF MATHEMATICS AND INFORMATICS PLOVDIV UNIVERSITY, 24 “TZAR ASSEN” STR. PLOVDIV, 4000 BULGARIA [email protected] Received 29 January, 2007; accepted 17 November, 2007 Communicated by C.P. Niculescu ABSTRACT. Upper estimates for the order of Gâteaux smoothness of bump functions in Orlicz– Lorentz spaces d(w, M, Γ), Γ uncountable, are obtained. The best possible order of Gâteaux differentiability in the class of all equivalent norms in d(w, M, Γ) is found. Key words and phrases: Orlicz function, Orlicz–Lorentz space, Smooth bump functions. 2000 Mathematics Subject Classification. 46B25, 40E30. 1. INTRODUCTION The existence of higher order Fréchet smooth norms and bump functions and its impact on the geometrical properties of a Banach space have been subject to many investigations begin- ning with the classical result for Lp–spaces in [1] and [6]. An extensive study and bibliography may be found in [2]. As any negative result on the existence of Gâteaux smooth bump func- tions immediately applies to the problem of existence of Fréchet smooth bump functions and norms, the question arises of estimating the best possible order of Gâteaux smoothness of bump functions in a given Banach space. A variational technique (the Ekeland variational principle) was applied in [2] to show that in `1(Γ), Γ uncountable, there is no continuous Gâteaux dif- ferentiable bump function. Following the same idea and using Stegall’s variational principle, an extension of this result to Banach spaces with uncountable unconditional basis was given in [4] and to Banach spaces with uncountable symmetric basis in [9]. As an application in [4] it was shown that in `p(Γ), Γ uncountable, there is no continuous p–times Gâteaux differentiable bump function when p is odd and there is no continuous ([p] + 1)–times Gâteaux differentiable bump function in the case p 6∈ N. This is essentially different from the case `p(N), p-odd, where equivalent p–times Gâteaux differentiable and even uniformly Gâteaux differentiable norms are constructed (see [10] and [8] respectively). As examples of the main result in [9], Orlicz `M (Γ) Research is partially supported by National Fund for Scientific Research of the Bulgarian Ministry of Education and Science, Contract MM-1401/04. 042-07 2 B. ZLATANOV and Lorentz d(w, p, Γ), Γ uncountable are considered and estimates for the order of Gâteaux smoothness of bump functions are obtained. Recently a deep result on embedding of `p spaces in Orlicz–Lorentz sequence spaces d0(p, M) have been found in [5]. It is shown there that `p ,→ d0(w, M) iff `p ,→ hM iff p ∈ [αM , βM ]. From this result naturally arises the question of finding upper estimates for the order of Gâteaux smoothness of bump functions in Orlicz– Lorentz spaces. It is worthwhile to mention that results about differentiability of bump functions in `p(Γ) cannot be used directly for `M (Γ) and d(w, p, Γ). Indeed, in [3] it is proved that `p(A) is isomorphic to a subspace of d(w, p, Γ) iff A is countable. On the other hand `M (Γ) for M ≡ p q t (1 + | log t|) at zero, p ≥ 1, q 6= 0, contains an isomorphic copy of `p(A) iff A is countable. The problem of embedding `p(A) or `M (A) into d(w, M, Γ), Γ uncountable is open. In this note we give one new application of the main result of [9] in Orlicz–Lorentz spaces d(w, M, Γ), Γ uncountable for finding upper estimates for the order of Gâteaux smoothness of bump functions. Let U be an open set in a Banach space X and let f : U → R be continuous. Following [4] 0 + −k we shall say that f is Gω,k–smooth, k ∈ N in U for some ω : (0, 1] → R , limt→0 t ω(t) = 0 if for any x ∈ U, y ∈ X the representation holds k X ti f(x + ty) = f(x) + f (i)(x)(yi) + Rk(x, y, t), i! f i=1 |Rk(x,y,t)| where f (i), i = 1, 2, . , k are i–linear bounded symmetric forms on X and lim f = 0. t→0 ω(|t|) 0 0 If U = X we use the notation Gω,[p] instead of Gω,[p](X) and Gk, k ∈ N for the set of all con- k k tinuous k–times Gâteaux differentiable functions on X, for which limt→0 |Rf (x, y, t)|/|t| = 0. We say that the norm k · k in X is k–times Gâteaux differentiable if it is from the class Gk(X\{0}). 2. PRELIMINARIES We use the standard Banach space terminology from [7]. Let us recall that an Orlicz func- tion M is an even, continuous, non-decreasing convex function such that M(0) = 0 and limt→∞ M(t) = ∞. We say that M is a non–degenerate Orlicz function if M(t) > 0 for every t > 0. ∞ A weight sequence w = {wn}n=1 is a positive decreasing sequence such that w1 = 1 and Pn limn→∞ W (n) = ∞, where W (n) = j=1 wj, for any n ∈ N. The Orlicz–Lorentz space d(w, M, Γ) is the space of all real functions x = x(α) defined on the set Γ, for which ( ∞ ) X I(λx) = sup wiM(λx(αi)) < ∞ i=1 ∞ for some λ > 0, where the supremum is taken over all sequences {αi}i=1 of different elements ∗ ∞ ∗ ∗ ∗ on Γ. There exists a sequence {αi }i=1, such that |x(α1)| ≥ |x(α2)| ≥ · · · ≥ |x(αi )| ≥ · · · , ∗ ∗ ∗ P∞ ∗ limi→∞ x(αi ) = 0, |x(α )| = 0 if α 6= αi for i ∈ N and I(λx) = i=1 wiM(λx(αi )). The space d(w, M, Γ), equipped with the Luxemburg norm: n x o kxk = inf λ > 0 : I ≤ 1 λ is a Banach space. By supp x we denote the set {α ∈ Γ : x(α) 6= 0}. The symbol eγ, γ ∈ Γ will stand for the unit vectors. J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 113, 8 pp. http://jipam.vu.edu.au/ GÂTEAUX DIFFERENTIABILITY 3 p If M(u) = u , 1 ≤ p < ∞ then d(w, M, Γ) is the Lorentz space d(w, p, Γ). If wi = 1 for every i ∈ N then d(w, M, Γ) is the Orlicz space `M (Γ). In this case we use the notation I(x) = Mf(x) To every Orlicz function M the following numbers are associated: M(uv) αM = sup p > 0 : sup p < ∞ , 0<u,v≤1 u M(v) M(uv) βM = inf q > 0 : inf > 0 . 0<u,v≤1 uqM(v) We consider only spaces generated by an Orlicz function M satisfying the ∆2–condition at zero, i.e., βM < ∞, which implies of course that (2.1) M(uv) ≥ uqM(v), u, v ∈ [0, 1] for some q > βM (see [7]). Finally we mention that the unit vectors {eγ}γ∈Γ form a symmetric basis of d(w, M, Γ) with symmetric constant 1, which is boundedly complete [5], [7]. For a function g : (0, 1] → R+ denote: M(uv) d (g) = sup : u, v ∈ (0, 1] . M g(u)M(v) Let us recall a well known definition. Let X have symmetric basis {eγ}γ∈Γ with a symmetric P∞ ∞ constant 1 and let z ∈ X, z 6= 0, z = i=1 uieγi , γi 6= γj for i 6= j. A sequence {zk}k=1, P∞ zk = i=1 uieαi,k , αi,k 6= αj,l for (i, k) 6= (j, l), αi,k ∈ Γ is called a block basis generated by the vector z. We will apply a general result for upper estimates for the order of Gâteaux smoothness of bump functions in a Banach space with a symmetric, boundedly complete basis with a symmet- ric constant 1, obtained in [9]. Theorem 2.1. [9] Let X be a Banach space, let {eγ}γ∈Γ, ]Γ > ℵ0 be a symmetric, boundedly complete basis in X with a symmetric constant 1 and let: n X − 1 lim zj n k = 0 n→∞ j=1 for every z ∈ X. Let ω : [0, 1] → R+ be such that for every x ∈ X there exist y ∈ X, suppy ∩ suppx = ∅ and a sequence tn & 0, which satisfy the inequality kx + tnyk − kxk ≥ ω(tn), n ∈ N. Then in X there is no continuous: 0 k (i) Gω,k–smooth bump when ω(t) = o(t ); 0 k+1 (ii) Gω,k+1–smooth bump when ω(t) = o(t ), k–even; (iii) k–times Gâteaux differentiable bump if ω(t) = tk; (iv) (k + 1)–times Gâteaux differentiable bump if ω(t) = tk+1, k–even. J. Inequal. Pure and Appl. Math., 8(4) (2007), Art. 113, 8 pp. http://jipam.vu.edu.au/ 4 B. ZLATANOV 3. MAIN RESULT Theorem 3.1. Let M be an Orlicz function. If f is a continuous k–times Gâteaux differentiable bump function in d(w, M, Γ), then ( [αM ] [αM ], dM (t ) < ∞ k ≤ EM = αM αM − 1, αM ∈ N, dM (t ) = ∞. 4. AUXILIARY LEMMAS To apply Theorem 2.1 for d(w, M, Γ) we need the following lemmas. M(t) Lemma 4.1. Let p ≥ 1 and let M be an Orlicz function satisfying the conditions limt→0 tp = p ∞ 0, dM (t ) = c < ∞. Then every block basis {zj}j=1 of the unit vector basis {eγ}γ∈Γ in d(w, M, Γ), generated by one vector, satisfies n X − 1 lim zj n p = 0.
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