Duals of variable exponent Hörmander spaces ( $$0< p^- \le p^+ \le 1$$ 0 < p - ≤ p + ≤ 1 ) and some applications

Abstract

In this paper we characterize the dual B c p(•) (Ω) of the variable exponent Hörmander space B c p(•) (Ω) when the exponent p(•) satisfies the conditions 0 < p − ≤ p + ≤ 1, the Hardy-Littlewood maximal operator M is bounded on L p(•)/p 0 for some 0 < p 0 < p − and Ω is an open set in R n. It is shown that the dual B c p(•) (Ω) is isomorphic to the Hörmander space B loc ∞ (Ω) (this is the p + ≤ 1 counterpart of the isomorphism B c p(•) (Ω) B loc p (•) (Ω), 1 < p − ≤ p + < ∞, recently proved by the authors) and hence the representation theorem B c p(•) (Ω) l N ∞ is obtained. Our proof relies heavily on the properties of the Banach envelopes of the steps of B c p(•) (Ω) and on the extrapolation theorems in the variable Lebesgue spaces of entire analytic functions obtained in a precedent paper. Other results for p(•) ≡ p, 0 < p < 1, are also given (e.g. B c p (Ω) does not contain any infinite-dimensional q-Banach subspace with p < q ≤ 1 or the quasi-Banach space B p ∩ E (Q) contains a copy of l p when Q is a cube in R n). Finally, a question on complex interpolation (in the sense of Kalton) of variable exponent Hörmander spaces is proposed. Keywords Variable exponent • Hardy-Littlewood maximal operator • Banach envelope • L p(•)-spaces of entire analytic functions, Hörmander spaces Mathematics Subject Classification (2000) 46F05 (46E50) • 46A16 • 42B25

Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

“…For a thorough discussion of these spaces and their history, see [1,2]. Our paper lies in this field of variable exponent function spaces and is a continuation of [3] (see also [4,5]). In [5] the (nonweighted) variable exponent Hörmander spaces (⋅) , (⋅) (Ω), and loc (⋅) (Ω) were introduced (recall that the classical Hörmander spaces , , , (Ω), and loc , (Ω) play a crucial role in the theory of linear partial differential operators (see, e.g., [6][7][8][9][10])) and there, extending a Hörmander result [6, Chapter XV] to our context, the dual of (⋅) (Ω) (when 1 < − ≤ + < ∞) was calculated (as a consequence some results on sequence space representation of variable exponent Hörmander spaces were obtained).…”

“…In [5] the (nonweighted) variable exponent Hörmander spaces (⋅) , (⋅) (Ω), and loc (⋅) (Ω) were introduced (recall that the classical Hörmander spaces , , , (Ω), and loc , (Ω) play a crucial role in the theory of linear partial differential operators (see, e.g., [6][7][8][9][10])) and there, extending a Hörmander result [6, Chapter XV] to our context, the dual of (⋅) (Ω) (when 1 < − ≤ + < ∞) was calculated (as a consequence some results on sequence space representation of variable exponent Hörmander spaces were obtained). In [3] the dual ( (⋅) (Ω)) was calculated when 0 < − ≤ + ≤ 1 (with techniques necessarily different from those used in [5]) and a number of applications were given. In the current article we show that the dual ( …”

“…The techniques used in the article (also in [3]) are based on the inequalities of the extrapolation theorems obtained by the authors in [4] and on the properties of the Banach envelopes of the 0 -Banach local spaces of loc (⋅) (Ω). Finally, three questions on duality and on sequence space representation of variable exponent Hörmander spaces are proposed.…”