测度代数中的范数控制逆问题
Norm-Controlled Inversion in Measure Algebras
- University of Warsaw（华沙大学）
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中文总结 AI 辅助
本文解决了局部紧阿贝尔群测度代数中Nikolski的范数控制逆问题，证明了1/2是尖锐阈值，并给出了傅里叶代数及Toeplitz算子的相关结果。
中文摘要 AI 辅助
我们解决了局部紧阿贝尔群的测度代数中Nikolski的范数控制逆问题。对于每个$\delta>1/2$，存在一个不依赖于群$G$的常数$C_M(\delta)$，使得$\\|\mu\\|_{M(G)}\le1$且$\inf_{\gamma\in\widehat G}|\widehat\mu(\gamma)|\ge\delta$蕴含$\mu$在$M(G)$中可逆且$\\|\mu^{-1}\\|_{M(G)}\le C_M(\delta)$。结合Nikolski的否定结果，这表明$1/2$是尖锐的普适阈值。主要步骤是紧阿贝尔群的傅里叶代数$A(K)$的一致逆定理；其证明基于剖面分解和预解式论证。我们还获得了具有非零维纳符号且卷绕数为零的Toeplitz算子的范数控制逆，并给出了直接端点构造，表明在$\delta=1/2$时，对于每个无限紧阿贝尔群$K$，$A(K)$中范数控制失效，且对于每个非离散局部紧阿贝尔群$G$，$L^1(G)$的单位化中范数控制失效。
英文摘要
We solve Nikolski's norm-controlled inversion problem for measure algebras of locally compact Abelian groups. For every $δ>1/2$ there is a constant $C_M(δ)$, independent of the group $G$, such that $\|μ\|_{M(G)}\le1$ and $\inf_{γ\in\widehat G}|\widehatμ(γ)|\geδ$ imply that $μ$ is invertible in $M(G)$ and $\|μ^{-1}\|_{M(G)}\le C_M(δ)$. Together with Nikolski's negative results, this shows that $1/2$ is the sharp universal threshold. The main step is a uniform inversion theorem for Fourier algebras $A(K)$ of compact Abelian groups; its proof is based on a profile decomposition and a resolvent argument. We also obtain norm-controlled inversion for Toeplitz operators with non-vanishing Wiener symbols of winding number zero and give direct endpoint constructions showing failure of norm control at $δ=1/2$ in $A(K)$ for every infinite compact Abelian group $K$, and in the unitization of $L^1(G)$ for every nondiscrete locally compact Abelian group $G$.
