AbstractIn this paper we continue the study of the spaces $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$
O
M
,
ω
(
R
N
)
and $${\mathcal O}_{C,\omega }({\mathbb R}^N)$$
O
C
,
ω
(
R
N
)
undertaken in Albanese and Mele (J Pseudo-Differ Oper Appl, 2021). We determine new representations of such spaces and we give some structure theorems for their dual spaces. Furthermore, we show that $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$
O
C
,
ω
′
(
R
N
)
is the space of convolutors of the space $${\mathcal S}_\omega ({\mathbb R}^N)$$
S
ω
(
R
N
)
of the $$\omega $$
ω
-ultradifferentiable rapidly decreasing functions of Beurling type (in the sense of Braun, Meise and Taylor) and of its dual space $${\mathcal S}'_\omega ({\mathbb R}^N)$$
S
ω
′
(
R
N
)
. We also establish that the Fourier transform is an isomorphism from $${\mathcal O}'_{C,\omega }({\mathbb R}^N)$$
O
C
,
ω
′
(
R
N
)
onto $${\mathcal O}_{M,\omega }({\mathbb R}^N)$$
O
M
,
ω
(
R
N
)
. In particular, we prove that this isomorphism is topological when the former space is endowed with the strong operator lc-topology induced by $${\mathcal L}_b({\mathcal S}_\omega ({\mathbb R}^N))$$
L
b
(
S
ω
(
R
N
)
)
and the last space is endowed with its natural lc-topology.
The Fourier Transform on Quantum Euclidean Space
