Generalizing algebraically defined norms

We extend the algebraic construction of finite dimensional varying exponent Lp(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p(\cdot )}$$\end{document} space norms, defined in terms of Cauchy polynomials to a more general setting, including varying exponent Lp(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p(\cdot )}$$\end{document} spaces. This boils down to reformulating the Musielak–Orlicz or Nakano space norm in an algebraic fashion where the infimum appearing in the definition of the norm should become a (uniquely attained) minimum. The latter may easily fail, as turns out, and in this connection we examine the Fatou type semicontinuity conditions on the modulars. Norms defined by ODEs are applied in studying such semicontinuity properties of Lp(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^{p(\cdot )}$$\end{document} space norms with p(·)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p(\cdot )$$\end{document} unbounded.


Introduction
The authors of Anatriello et al. [1] considered the variable (exponent) Lebesgue space L p(·) ( ) where ⊂ R n is a set of positive Lebesgue measure and the symbol p(·) is a simple function of the type Uniqueness of the positive root is due to the special form of the polynomial, which is a so-called Cauchy polynomial, i.e. a polynomial of the type a j x k− j (1.1) where the coefficients a j 's are non-negative. By the classical rule of signs, if (at least) one of the coefficients a j 's is nonzero, P(x) admits exactly a unique simple positive root. Historically, the importance of such root is linked to the problem, to which Cauchy gave an important contribution, to bound the modulus of the roots of polynomials. In recent years, in [1], it has been shown that-roughly speaking-when coefficients are norms, the unique positive root is again a norm; in view of old literature (see e.g. [11,Theorem 27.2,p. 123] and [3]), this root satisfies a number of norm inequalities. In this paper we start from the same observation, but we investigate the problem in order to understand when the norms of more general variable Lebesgue spaces (and even more general class of spaces) are solutions of equations and, in particular, the unique positive solutions of equations which are not necessarily algebraic but rather fixed point equations.
Intuitively, the norm of variable Lebesgue spaces whose exponent has k values should be the unique positive solution of an equation containing the sum of k monomials, and if the exponent has a countable number of values, the norm should be the unique positive solution of an equation containing some series; finally, it is natural to expect that for general exponents p(·) the series should be an integral. That is, analogously as in [1], this leads to a more general construction of norms, by using a variant of (1.1) as follows: or, more generally, where the fixed point condition R(λ) = λ, which in case of (1.2) leads to a nonlinear integral equation, defines the norm as f = λ. That is, up to suitable Fatou type semicontinuity properties of the modular which also depend on the structure of the space in question. In analyzing these Fatou type properties we apply some ideas from Talponen [14], where norms of variable Lebesgue spaces are considered, built as solutions of suitable ODEs.
the following hold: A typical example of norm generated by a modular is that one of Musielak-Orlicz spaces over a set ⊂ R n of positive Lebesgue measure: which gives, as a particular case, the variable Lebesgue space norm with exponent p(·) when (x, t) = t p(x) . An easy consequence of Theorem 1.1 is the following proposition, where a generalization of the classical Fatou lemma plays an essential role to establish that the infimum defining f ρ is actually a minimum.
Proof If f ∈ X ρ , then there exists μ > 0 such that ρ(μ f ) < 1. Therefore λ := 1/μ belongs to the set of the λ's defining f ρ , which is therefore nonempty. Let (λ h ) be a decreasing minimizing sequence, i.e. such that In spite of its simplicity, we obtained a popular property of the norm under a not usual assumption. We underline the weakeness of the condition of left lower semicontinuity. The Fatou property known from the theory of Banach function spaces (see e.g. [2,10]) is equivalent to the left continuity of the modular (see the details in Theorem 2.2.2, p. 19 in [8], stated in the framework of modular function spaces), and, most of all, the elements in X must be real valued or complex valued functions, in any case must be in some lattice. The notion of left-continuity in [12] is given in general vector spaces, but, as we will see, it is stronger with respect to our needs. In the case of the Musielak-Orlicz spaces, hence, in particular, of the variable Lebesgue spaces, the left lower semicontinuity is guaranteed by the classical Fatou property. For variable Lebesgue spaces, Proposition 1.2 reduces to the first part of Prop. 2.21 in [4], for Orlicz spaces reduces to Prop. 3, p. 60 in [13], for Musielak-Orlicz spaces reduces to Lemma 3.2.4 in [6].

Functions whose norms are fixed points
The question of the continuity of λ ∈ [0, ∞[→ ρ( f /λ) has been recently considered by the second named author for certain particular Musielak-Orlicz spaces, namely, certain weighted variable Lebesgue spaces. The following statement is contained in Prop. 5.3 in [15].
Let us record the following consequence. First of all, by Proposition 1.2 and Theorem 1.1 (see also, e.g., Theorem 3, p. 54 in [13]) Now let p(·) be (finite and) unbounded in [0, 1]. Since the necessary condition on w established in Proposition 2.1 is not satisfied by the constant weight w ≡ 1, then there We can go further and make a more general assertion: by Prop. 2.21 in [4], there exists a function f 0 having norm ) < 1, and this suggests (2.1) by virtue of the following proposition, which is a complement to Proposition 1.2: Proof By assumption F( f ρ ) < 1 and by definition of f ρ for any ε > 0 it must be F( f ρ − ε) > 1.
The discontinuity of F considered in Proposition 2.2 occurs also in Orlicz spaces theory. In fact, in [13] (see Proposition 6, p. 77 therein for details) it is proved that continuity of F is equivalent to the so-called 2 condition for the Young function generating the Orlicz space, hence discontinuity may occur for not 2 Young functions.

Norms in terms of algebraic equations
Let us consider the variable Lebesgue space L p(·) ( ) where ⊂ R n is a set of positive Lebesgue measure and p(·) is a bounded function (so that (2.3) holds) having at most a countable number of values: In this case the norm f MO,L p(·) ( ) is a solution of the equation where the function R is in (2.2). In this case the equation reads i.e. (next equality should be compared with that one in [15], p. 752) last equality being exactly the algebraic equation of degree k ∈ N in the case that p(·) has the values 1, . . . , k.

Norms in terms of integral equations
In the case of a general, bounded exponent p(·), the above chain of equalities cannot be performed, because of possibly infinity values of the exponent. However, the equality between the extreme left and right sides is still true. In fact, we have

Proof of Proposition 3.1 We already observed that since p(·) is a bounded function, then (2.3) holds and we may apply Theorem 2.3. Hence by (2.2) the norm
Uniqueness of the solution is easy to see and the proof is complete.
We remark that (3.1), with the substitution λ = f MO,L p(·) ( ) , is known from the general theory of variable Lebesgue spaces (see the second part of Prop. 2.21 in [4]), and that it holds also for exponents which may assume value +∞, but which must be bounded in the set where they are finite (if such set exists). In particular, in the case of the space L ∞ , we remark that its norm could be seen as generated by the pseudomodular ρ( f ) = f ∞ which satisfies all the assumptions of Theorem 2.3. This shows that the conclusion of Theorem 2.3 holds also for spaces which do not have an absolutely continuous norm (see e.g. [2,9] for details about spaces having absolutely continuous norm).

Norms in terms of minima
Fan [5] observed that the Amemiya norm and the Luxemburg norm coincide for a given Orlicz functional. This result includes the variable exponent Lebesgue spaces as a special case.
Recall that the Amemiya norm is defined as follows in the latter case: It is easy to see that the term inside the infimum tends to infinity as K 0 or K ∞ for an essentially non-zero funtion f . On the other hand, by convexity considerations the infimum is actually a minimum occuring for a unique value of K . Thus, a natural question occurs: What is the value of K , depending on f and the space L p(·) (μ)? It turns out that there is a neat expression for the minimum. We may rewrite the above definition in a form more compatible with our approach here: Here the function λ → λ is convex and λ → dx is strictly convex, provided that f = 0 and p(·) > 1. The latter is due to the fact that ]0, ∞[→]0, ∞[, x → ax − p is strictly convex for every a > 0 and p > 0.

Proposition 3.2
Consider an exponent p(·) > 1 (uniformly) bounded from above, f ∈ L p(·) ( ), f = 0, and let K > 0 be the value where the minimum in the Amemiya norm is attained. Then where ν is the special weight generating the measure given by The assumption regarding the boundedness of p(·) is not completely innocent. These issues are discussed in the next section.

Proof of Proposition 3.2
We first prove the statement in the special case where both p(·) and f (·) are simple functions. In this case the integrals involved are actually finite sums.
According to the considerations above, we will find the unique value of λ by means of differentiation.
This is equivalent to an equality having a familiar form: corresponds to the unique value K where the minimum in the definition of the Amemiya norm is attained.
Recall that the Musielak-Orlicz norms and the Amemiya norms coincide (see e.g. [7]), so that we may consider which ever suits us. Our observation is then extended for all measurable exponents by taking limits of Musielak-Orlicz norms with simple exponents. Finally, we approximate with simple functions f to obtain the full statement.

Varying-exponent Lp functions whose norm admits an algebraic representation
The second named author introduced in [14] a method of defining var-expo L p(·) [0, 1] norms by means of weak solutions to suitable first-order ODEs. The measure space (in this case the Lebesgue measure on [0, 1]) is required to be linearly ordered in order for the ODE to be defined. There the following natural subspaces of Musielak-Orlicz L p(·) spaces were studied as well: Clearly, if p(·) is essentially bounded, then L p(·) 0 = L p(·) . As observed in [14], the ODE-determined class L p(·) need not be a linear space for an essentially unbounded exponent p(·), but nevertheless the following facts hold: • If p(·) is essentially bounded or non-decreasing, then the corresponding ODEdetermined class L p(·) is a Banach space in the natural way and its norm is equivalent to the Luxemburg norm of the Musielak-Orlicz space by an isomorphism constant 2. • The class L p(·) 0 is in any case a Banach space and its norm is equivalent to the Luxemburg norm of the Musielak-Orlicz space by an isomorphism constant 2.
We note that the last condition (3) may hold even if the integral in (1) is infinite for every 0 < λ < 1.

Proof of Theorem 3.3
Let us make the assumption of the theorem. Then it is clear that the first condition implies the last one.
The fact that is decreasing for f (x) ≡ 0 and the Monotone Convergence Theorem, we obtain that in (1) the mapping is continuous from the right. It easily follows from the continuity of the mapping in condition (1) from the left that Then condition (2) follows easily. Finally, let us check the last implication, namely, that condition (2) implies the first one. If p(·) is essentially bounded, then all the conditions (1)-(3) are readily true, so assume that p(·) is essentially unbounded. Let f ∈ L p(·) 0 ( , , μ). According to the Dominated Convergence Theorem it suffices to verify that there is 0 < λ < 1 such that To achieve this, we apply the observations in [15] to find a linear isometry where the domain is the Musielak-Orlicz space over ⊂ R n with the Luxemburg norm and the target space is a similar space, except over the unit interval. This is obtained by a weighted composition operator. The composition is given by the lifting of the Boolean isomorphism between the measure algebras, that is, a measure-preserving bijection → [0, 1], where μ is normalized to be a probability measure. Here p is obtained from p by applying the lifting L : → [0, 1] as follows: p (t) := p• L −1 (t). Then, by using the inequalities established in the same paper, we find an isomorphism by a measure-preserving rearrangement, where p becomes a non-decreasing rearrangement of p .
Here L p (·) 0 [0, 1] = L p (·) [0, 1] can be regarded as an ODE-determined Banach function space introduced in [14]. The equality of the spaces is due to the fact that p (·) is non-decreasing. The norm of the ODE-determined function space norm is equivalent to the Luxemburg norm in question.
Finally, by using the fact that a simultaneous rearrangement of g(·) and p (·) does not affect the value of the above integral, we obtain that By recalling the construction of T we obtain that This concludes the proof.
Although the Luxemburg-norm associated to the Musielak-Orlicz space is equivalent to the ODE-determined norm, these norms have some dissimilarities. In the ODE-determined class one can approximate the value f ODE,L p(·) for any f in the class as follows: one may restrict the support of f to obtain a sequence of functions f n := 1 C n f with | f n (t)| p(t) essentially bounded such that f n ODE,L p(·) → f ODE,L p(·) and f n → f a.e. as n → ∞. (see [14]). The similar statement does not hold true for · MO,L p(·) norms. Indeed, to see this, one can construct an example such that for all λ < 1.
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