论文标题
定位运算符本征函数的次指数衰减和规律性估计值
Subexponential decay and regularity estimates for eigenfunctions of localization operators
论文作者
论文摘要
我们考虑使用符号$ a $ a_a^{φ_2} $的时频定位操作员，带有符号$ a $在广泛的加权调制空间中$ a $ $ a $ $ \ MATHCAL {s}^{\ left（1 \右）}（\ Mathbb {r}^{d}）$。如果所考虑的权重具有超高的增长，我们证明$ a_a^{φ_1，φ_2} $的特征函数在相位空间中具有适当的次代衰减，即它们属于gefand-shilov s-shilov $ \ s-shilov $ \ s-mathcal $ \ mathcal {s} s}}^$ nlo n were n were n w s}^{（γ） $γ\ geq 1 $与所考虑的重量的增长有关。 $τ$ -PseudOdifferential Operator $ \ Mathrm {op}_τ（σ）$扮演一个重要角色。在这个方向上，我们显示了在加权调制空间上作用时$ \ mathrm {op}_τ（σ）$的方便连续性属性。此外，当符号$σ$属于具有适当选择的权重功能的调制空间时，我们证明了$ \ mathrm {op}_τ（σ）$的特征functions的次指定衰减和规律性属性。作为一种工具，我们还证明了（准）Banach加权调制空间的新卷积关系。
We consider time-frequency localization operators $A_a^{φ_1,φ_2}$ with symbols $a$ in the wide weighted modulation space $ M^\infty_{w}(\mathbb{R}^{2d})$, and windows $ φ_1, φ_2 $ in the Gelfand-Shilov space $\mathcal{S}^{\left(1\right)}(\mathbb{R}^{d})$. If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of $A_a^{φ_1,φ_2}$ have appropriate subexponential decay in phase space, i.e. that they belong to the Gefand-Shilov space $ \mathcal{S}^{(γ)} (\mathbb{R}^{d}) $, where the parameter $γ\geq 1 $ is related to the growth of the considered weight. An important role is played by $τ$-pseudodifferential operators $\mathrm{Op}_τ(σ)$. In that direction we show convenient continuity properties of $\mathrm{Op}_τ(σ)$ when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of $\mathrm{Op}_τ(σ)$ when the symbol $σ$ belongs to a modulation space with appropriately chosen weight functions. As a tool we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.
