Subexponential decay and regularity estimates for eigenfunctions of localization operators

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