A Note on Variable Exponent Hörmander Spaces

Abstract

In this paper we introduce the variable exponent Hörmander spaces and we study some of their properties. In particular, it is shown that B c p(•) (Ω) is isomorphic to B loc p (•) (Ω) (Ω open set in R n , p − > 1 and the Hardy-Littlewood maximal operator M is bounded in L p(•)) extending a Hörmander's result to our context. As a consequence, a number of results on sequence space representations of variable exponent Hörmander spaces are given.

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

“…[1,2] and the books of Diening et al [8] and Cruz-Uribe and Fiorenza [6]). In [17] we studied the properties of the (non-weighted) variable exponent Hörmander (Ω ) play a crucial role in the theory of linear partial differential operators (see e.g. [9])).…”

“…In the present paper we extend this duality to exponents p(•) satisfying the conditions 0 < p − ≤ p + ≤ 1 and such that the Hardy-Littlewood maximal operator M is bounded on L p(•)/p 0 for some 0 < p 0 < p − . The techniques used are different from those used in [17] since if p + < 1 then the dual of L p(•) is trivial and the steps B p(•) ∩ E (K) are quasi-Banach spaces instead of Banach spaces. A number of applications of this duality are also given.…”

Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

“…The second part of lemma is obvious taking into account that ‖ ⋅ ‖ * , is a -norm and that [5 Proof. Let ‖ ⋅ ‖ be the quasi-norm on which generates the topology of .…”

“…On the other hand, the bilinear mapping ×( ∩E ( )) → ∩E ( ) : ( , ) → is continuous (see [5])). Finally,…”

“…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 ( …”