We study the mapping properties of metaplectic operators $\hat{S}\in \mathrm{Mp}(2d,\mathbb{R})$ on modulation spaces of the type ${\mathrm{M}}_m^{p,q}({\mathbb{R}}^d)$. Our main result is a full characterization of the pairs $(\hat{S},{\mathrm{M}}^{p,q}({\mathbb{R}}^d))$ for which the operator $\hat{S}:{\mathrm{M}}^{p,q}({\mathbb{R}}^d)\to {\mathrm{M}}^{p,q}({\mathbb{R}}^d)$ is (i) well-defined, (ii) bounded. It turns out that these two properties are equivalent, and they entail that $\hat{S}$ is a Banach space automorphism. For polynomially bounded weight functions, we provide a simple sufficient criterion to determine whether the well-definedness (boundedness) of $\hat{S}:{\mathrm{M}}^{p,q}({\mathbb{R}}^d)\to {\mathrm{M}}^{p,q}({\mathbb{R}}^d)$ transfers to $\hat{S}:{\mathrm{M}}_m^{p,q}({\mathbb{R}}^d)\to {\mathrm{M}}_m^{p,q}({\mathbb{R}}^d)$.

中文翻译：

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调制空间上的元波作用

我们研究元逻辑算子的映射性质$\widehat{S}\epsilon \mathrm{熔点}（2d,\mathbb{右}）$在类型的调制空间上${\mathrm{中号}}_{米}^{p,q}（{\mathbb{右}}^d）$。我们的主要结果是对这对的完整表征$（\widehat{S},{\mathrm{中号}}^{p,q}（{\mathbb{右}}^d））$对于哪个操作员$\widehat{S}:{\mathrm{中号}}^{p,q}（{\mathbb{右}}^d）\to {\mathrm{中号}}^{p,q}（{\mathbb{右}}^d）$(i)明确定义，(ii)有界。事实证明，这两个属性是等价的，并且它们意味着$\widehat{S}$是巴纳赫空间自同构。对于多项式有界权重函数，我们提供了一个简单的充分标准来确定$\widehat{S}:{\mathrm{中号}}^{p,q}（{\mathbb{右}}^d）\to {\mathrm{中号}}^{p,q}（{\mathbb{右}}^d）$转移到$\widehat{S}:{\mathrm{中号}}_{米}^{p,q}（{\mathbb{右}}^d）\to {\mathrm{中号}}_{米}^{p,q}（{\mathbb{右}}^d）$。