发表机构
Tianshui Normal University; Nanjing Forestry University（天水师范学院; 南京林业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文在有限型凸域的 Bergman 空间上刻画了绝对求和嵌入与Hankel算子，给出上下范数估计、显式逼近误差及Toeplitz谱渐近，并应用于复合与Volterra算子。
AI 中文摘要
设 $\Omega\subset\mathbb C^n$ 为光滑有界有限型凸域。我们刻画了其 Bergman 空间上的绝对可和性，并估计了求和范数中的逼近误差。对于 $1<p<\infty$ 和 $1\le q,r<\infty$，我们通过由 McNeal 球上的归一化质量构成的对角算子，刻画了绝对 $r$-求和 Carleson 嵌入 $A^p(\Omega)\to L^q(\mu)$。对于 $1<p,q<\infty$ 和 $1\le r<\infty$，我们利用到全纯函数的局部 $L^q$ 距离及适当的体积权重，得到了相应的大 Hankel 判别准则。两个准则均给出上下范数估计。证明结合了均匀局部核估计、核合成以及两个固定尺度上的解析分解。局部 Taylor 多项式给出了嵌入的显式秩和误差界。当 $p=2$ 或 $q=2$ 时，这些界可转移到一般 Hankel 算子。适当的原子嵌入与其对角模型具有相同的最佳逼近误差（至多相差常数）。一族 Hankel 算子包含对角理想的补副本，并在 Hilbert 情形下对幂序列具有匹配的逼近速率。对于有限型椭球上的正 Toeplitz 算子，我们在三种情形下计算了精确的谱渐近，包括临界对数因子和几何符号的主常数。这些渐近给出了求和范数中的最优逼近速率。对复合算子和 Volterra 算子的应用包括目标端点 $q=1$。
英文摘要
Let $Ω\subset\mathbb C^n$ be a smoothly bounded convex domain of finite type. We characterize absolute summability on its Bergman spaces and estimate approximation errors in the summing norm. For $1<p<\infty$ and $1\le q,r<\infty$, we characterize absolutely $r$-summing Carleson embeddings $A^p(Ω)\to L^q(μ)$ by diagonal operators formed from normalized masses on McNeal balls. For $1<p,q<\infty$ and $1\le r<\infty$, we obtain a corresponding big Hankel criterion using local $L^q$ distances to holomorphic functions, with the appropriate volume weights. Both criteria give upper and lower norm estimates. The proofs combine uniform local nuclear estimates, kernel synthesis, and an analytic decomposition at two fixed scales. Local Taylor polynomials give explicit rank and error bounds for the embeddings. These bounds transfer to general Hankel operators when $p=2$ or $q=2$. Suitable atomic embeddings have the same best approximation errors as their diagonal models, up to constants. A family of Hankel operators contains a complemented copy of the diagonal ideal and has matching approximation rates for power sequences in the Hilbert case. For positive Toeplitz operators on finite-type ellipsoids, we compute exact spectral asymptotics in three regimes, including the critical logarithmic factor and the leading constants for geometric symbols. These asymptotics give optimal approximation rates in the summing norm. Applications to composition and Volterra operators include the target endpoint $q=1$.
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