Abstract
We extend the algebraic construction of finite dimensional varying exponent L^{p(cdot )} space norms, defined in terms of Cauchy polynomials to a more general setting, including varying exponent L^{p(cdot )} 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(cdot )} space norms with p(cdot ) unbounded.
Highlights
The authors of Anatriello et al [1] considered the variable Lebesgue space L p(·)( ) where ⊂ Rn is a set of positive Lebesgue measure and the symbol p(·) is a simple function of the typeA
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 k
Summary
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. 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(·). 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; it is natural to expect that for general exponents p(·) the series should be an integral. More generally, from Talponen [14], where norms of variable Lebesgue spaces are considered, built as solutions of suitable ODEs
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