Generalizing algebraically defined norms
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Generalizing algebraically defined norms Alberto Fiorenza1,2 · Jarno Talponen3 Received: 4 August 2020 / Accepted: 11 September 2020 © The Author(s) 2020
Abstract We extend the algebraic construction of finite dimensional varying exponent L p(·) space norms, defined in terms of Cauchy polynomials to a more general setting, including varying exponent L p(·) 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 L p(·) space norms with p(·) unbounded. Keywords Modular spaces · Musielak–Orlicz spaces · Variable exponent Lebesgue spaces · Fixed point · Non-linear integral equation Mathematics Subject Classification 46E30 · 45G10
1 Introduction The authors of Anatriello et al. [1] considered the variable (exponent) Lebesgue space L p(·) () where ⊂ Rn is a set of positive Lebesgue measure and the symbol p(·) is a simple function of the type
B
Alberto Fiorenza [email protected] Jarno Talponen [email protected]
1
Dipartimento di Architettura, Università di Napoli Federico II, Via Monteoliveto, 3, 80134 Napoli, Italy
2
Istituto per le Applicazioni del Calcolo “Mauro Picone”, sezione di Napoli, Consiglio Nazionale delle Ricerche, Via Pietro Castellino, 111, 80131 Napoli, Italy
3
Department of Mathematics and Statistics, University of Helsinki, P.O. Box 64, Gustaf Hällströmin katu 2, 00014 Helsinki, Finland
123
A. Fiorenza, J. Talponen
p(x) =
k
jχ j (x), =
j=1
k
j , | j | > 0 ∀ j = 1, . . . , k,
j=1
j pairwise disjoint and observed that the usual Luxemburg norm of a function f ∈ L p(·) () is the unique positive root of a polynomial of degree k, k being the number of values of p(·). 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 P(x) = x − k
k
a j x k− j
(1.1)
j=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 positi
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