Some embedding theorems for Hörmander–Beurling spaces

Abstract

In this paper we prove a number of results on sequence space representations and embedding theorems of Hörmander-Beurling spaces. As a consequence and using sharp results of Meise, Taylor and Vogt, a result of Kaballo on short sequences and hypoelliptic operators is extended to ω-hypoelliptic differential operators and to the vector-valued setting.

A Note on Variable Exponent Hörmander Spaces

“…In [16,Section 3] the former isomorphism is extended to Hörmander spaces in the sense of Beurling and Björck. A number of applications of this duality (to sequence space representations of several ultradistributions spaces and to linear partial differential operators) are also given in [16] and [17]. In this section we extend the former isomorphism to variable exponent Hörmander spaces.…”

“…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.…”

“…The sequence space representation B loc 1,k (Ω) l N 1 was established by Vogt in [24] (Ω an open set in R n and k a temperate weight function on R n ). In [6] and [16] (see also [15], [17]) the sequence space representations…”