AbstractWe consider time-frequency localization operators $$A_a^{\varphi _1,\varphi _2}$$
A
a
φ
1
,
φ
2
with symbols a in the wide weighted modulation space $$ M^\infty _{w}({\mathbb {R}^{2d}})$$
M
w
∞
(
R
2
d
)
, and windows $$ \varphi _1, \varphi _2 $$
φ
1
,
φ
2
in the Gelfand–Shilov space $$\mathcal {S}^{\left( 1\right) }(\mathbb {R}^d)$$
S
1
(
R
d
)
. If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of $$A_a^{\varphi _1,\varphi _2}$$
A
a
φ
1
,
φ
2
have appropriate subexponential decay in phase space, i.e. that they belong to the Gelfand–Shilov space $$ \mathcal {S}^{(\gamma )} (\mathbb {R^{d}}) $$
S
(
γ
)
(
R
d
)
, where the parameter $$\gamma \ge 1 $$
γ
≥
1
is related to the growth of the considered weight. An important role is played by $$\tau $$
τ
-pseudodifferential operators $$Op_{\tau } (\sigma )$$
O
p
τ
(
σ
)
. In that direction we show convenient continuity properties of $$Op_{\tau } (\sigma )$$
O
p
τ
(
σ
)
when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of $$Op_{\tau } (\sigma )$$
O
p
τ
(
σ
)
when the symbol $$\sigma $$
σ
belongs to a modulation space with appropriately chosen weight functions. As an auxiliary result we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.
The Grossmann–Royer Transform, Gelfand–Shilov Spaces, and Continuity Properties of Localization Operators on Modulation Spaces