复球上的双线性Bochner-Riesz平均
Bilinear Bochner-Riesz Means on the Complex Sphere
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中文总结 AI 辅助
本文证明复球上双线性Bochner-Riesz平均在特定指数范围及光滑性参数下的$L^p$有界性，并推导了若干具有独立价值的解析估计。
中文摘要 AI 辅助
本文建立复球$\boldsymbol{\rm S}$上双线性Bochner-Riesz平均$\boldsymbol{\rm B}^{\boldsymbol{\rm \tiny \text{α}}}_R$的有界性。更确切地说，我们证明当$1/p_1+1/p_2=1/p$且$1 \boldsymbol{\rm \tiny \text{≤}} p_1,p_2 \boldsymbol{\rm \tiny \text{≤}} \boldsymbol{\rm \tiny \text{∞}}$时，$\boldsymbol{\rm B}^{\boldsymbol{\rm \tiny \text{α}}}_R$是从$L^{p_1}(\boldsymbol{\rm S}) \times L^{p_2}(\boldsymbol{\rm S})$到$L^p(\boldsymbol{\rm S})$的有界算子，其中指数的容许范围由$\boldsymbol{\rm S}$的拓扑维数描述所需的光滑性参数$\boldsymbol{\rm α}$。为便于证明，我们建立了若干解析估计，包括限制型估计、带大权重幂的加权Plancherel估计及双线性加权Plancherel估计，这些均在本文设定的框架下全新推导，可视为具有独立研究价值。
英文摘要
In this paper, we establish the boundedness of the bilinear Bochner-Riesz means $\mathcal{B}^α_R$ on the complex sphere $\mathbb{S}$. More precisely, we prove that $\mathcal{B}^α_R$ is bounded from $L^{p_1}(\mathbb{S}) \times L^{p_2}(\mathbb{S}) \to L^p(\mathbb{S})$ where $1/p_1+1/p_2=1/p$ and $1\leq p_1, p_2 \leq \infty$, for an admissible range of exponents, with the required smoothness parameter $α$ described in terms of the topological dimension of $\mathbb{S}$. To facilitate our proof, we establish several analytic estimates, including restriction-type estimates, weighted Plancherel estimates with large power of weights and bilinear weighted Plancherel estimates, which are derived from the ground up in our setting and may be considered of independent interest.
