Hermite-Sobolev空间中的乘积与卷积
Products and Convolutions in Hermite-Sobolev spaces
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中文总结 AI 辅助
该数学研究证明了Hermite-Sobolev空间中函数乘积与卷积的空间归属性质，推导得到特定空间对的乘积结果，并证实平移算子在该空间上的一致有界性。
中文摘要 AI 辅助
本文证明：对于依赖于维数d的p，两个属于Hermite-Sobolev空间$\boldsymbol{\text{S}}_p(\boldsymbol{\text{R}}^d)$的函数的乘积（等价于卷积）仍属于该空间。由此可得，对该p，$\boldsymbol{\text{S}}_p(\boldsymbol{\text{R}}^d)$中的$\boldsymbol{\text{\textbackslash phi}}$与$\boldsymbol{\text{S}}_{-p}(\boldsymbol{\text{R}}^d)$中的$\boldsymbol{\text{\textbackslash psi}}$的乘积（等价于卷积）属于$\boldsymbol{\text{S}}_{-p}(\boldsymbol{\text{R}}^d)$。进一步可得，$\boldsymbol{\text{R}}^d$中平移x的算子在$\boldsymbol{\text{S}}_p(\boldsymbol{\text{R}}^d)$上关于x一致有界。
英文摘要
In this paper, we show that the product, or equivalently the convolutions of two functions in the Hermite-Sobolev spaces $\mathcal{S}_p(\mathbb{R}^d)$ is again in the same space, for $p$ depending on the dimension $d$. As a consequence, for such $p$ we show that the product, or equivalently the convolutions of $ϕ\in \mathcal{S}_p(\mathbb{R}^d)$ and $ψ\in \mathcal{S}_{-p}(\mathbb{R}^d)$ is in $\mathcal{S}_{-p}(\mathbb{R}^d)$. As a further consequence, we show that the operators of translation by $x$ on $\mathcal{S}_p(\mathbb{R}^d)$ are bounded uniformly in $x \in \mathbb{R}^d$.
