Fock空间的尖锐高斯混合范数理论
Sharp Gaussian Mixed-Norm Theory for Fock Spaces
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中文总结 AI 辅助
本文发展高斯混合范数Fock空间的结构理论，证明最优圆周均值估计，刻画嵌入、原子分解、Carleson测度及对偶，并给出定量重中心定理。
中文摘要 AI 辅助
我们发展了高斯混合范数Fock空间$\mathcal F_α^{p,q}$的结构理论，其中$α>0$且$0<p,q\leq\infty$，该空间将角向$L^p$均值与径向高斯$L^q$可和性分离。我们首先证明了尖锐的圆周均值估计$M_p(f,r)\leq C_{α,q}(1+r)^{-\frac{1}{q}} e^{\fracα{2}r^2}\lVert f\rVert_{\mathcal F_α^{p,q}}$，$r\geq0$，其中$M_p(f,r)$表示$f$在圆周$\lvert z\rvert=r$上的角向$L^p$均值，约定$\frac{1}{\infty}=0$，并证明指数$-\frac{1}{q}$是最优的。证明基于凸Laplace型指数的局部质量原理，该原理也给出了混合高斯范数的环形离散化。利用该离散化，我们刻画了混合范数Fock空间之间的连续嵌入，并获得了以归一化Fock核表示的原子分解，其系数属于相应的环形混合序列空间。我们进一步刻画了Carleson测度和消失Carleson测度，并确定了Banach、拟Banach和端点情形下的连续对偶。对于$p<1$，原子分解依赖于一个定量的重中心定理，该定理控制归一化Fock核在其中心扰动下的展开。该定理即使对于Hilbert Fock空间$\mathcal F_α^{2}$也具有独立意义。
英文摘要
We develop a structural theory of the Gaussian mixed-norm Fock spaces $\mathcal F_α^{p,q}$, where $α>0$ and $0<p,q\leq\infty$, which separate angular $L^p$-means from radial Gaussian $L^q$-summability. We first prove the sharp circle-mean estimate $M_p(f,r)\leq C_{α,q}(1+r)^{-\frac{1}{q}} e^{\fracα{2}r^2}\lVert f\rVert_{\mathcal F_α^{p,q}}$, $r\geq0$, where $M_p(f,r)$ denotes the angular $L^p$-mean of $f$ on the circle $\lvert z\rvert=r$ with the convention $\frac{1}{\infty}=0$, and show that the exponent $-\frac{1}{q}$ is optimal. The proof is based on a local mass principle for convex Laplace-type exponents, which also yields an annular discretization of the mixed Gaussian norm. Using this discretization, we characterize the continuous embeddings between mixed-norm Fock spaces and obtain an atomic decomposition in terms of normalized Fock kernels, with coefficients belonging to a corresponding annular mixed sequence space. We further characterize Carleson and vanishing Carleson measures and identify the continuous duals in the Banach, quasi-Banach, and endpoint regimes. For $p<1$, the atomic decomposition relies on a quantitative re-centering theorem that controls expansions of normalized Fock kernels under perturbations of their centers. This theorem is of independent interest even for the Hilbert Fock space $\mathcal F_α^{2}$.
发表机构
- National Yang Ming Chiao Tung University（国立阳明交通大学）
- VNU University of Science, Vietnam National University, Hanoi（越南国立大学河内科技大学）
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