On sequence space representations of Hörmander–Beurling spaces

Abstract

It is shown that B lock Beurling-Björck weight) extending a Hörmander's result (the proof we give is valid in the vector-valued case, too). As a consequence, and using Vogt's representation theorems and weighted L p -spaces of entire analytic functions, a number of results on sequence space representations of Hörmander-Beurling are given.

A Note on Variable Exponent Hörmander Spaces

“…In Section 3 we introduce the variable exponent Hörmander spaces B p(•) , B c p(•) (Ω) and B loc p(•) (Ω) and we study some of their properties when the Hardy-Littlewood maximal operator M is bounded in L p(•)/p 0 for some 0 < p 0 < p − (convolution, density, completeness, embedding theorems, multiplication operators) and, by using Fourier multipliers and a result of Diening [2], we obtain a sequence space representation of the space B c p(•) (]a, b[) (see Theorem 3.5/5). We also extend a result of Hörmander [8,Chapter XV,15.2] (Ω) (see also [8,Chapter XV] and [16]) and another result on sequence space representation is given. Finally, two questions on complex interpolation and on sequence space representation of variable exponent Hörmander spaces are proposed.…”

“…We begin with the variable exponent (and weight k ≡ 1) counterpart of [8, Definition 10.1.6] (see also [8,Sections 10 and 15] and [16], [17], [24]).…”

“…We introduce the variable exponent Hörmander spaces B p(•) , B c p(•) (Ω) and B loc p(•) (Ω) (it is well known that the Hörmander spaces B p,k , B c p,k (Ω) and B loc p,k (Ω) play a crucial role in the theory of linear partial differential operators (see e.g. [8], [24], [6], [15], [16], [17])) and we study some of their properties. We also give a number of results on sequence space representations of the introduced spaces.…”

Fréchet Envelopes of Nonlocally Convex Variable Exponent Hörmander Spaces

“…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 [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.…”

On the mutually non isomorphic ℓp(ℓq) spaces

“…In 10 (see also 11), the following question was posed: are the Banach spaces ℓ ∞ (ℓ 1 ) and ℓ 1 (ℓ ∞ ) isomorphic? The question had arisen in the study of certain function spaces but it has its roots in [13, footnote of page 242], where Triebel said that he had learnt from Pełczyński the following result: Let 1 ≤ p 0 , p 1 ≤ ∞ and 1 < q 0 , q 1 < ∞.…”