论文标题
凯奇家族的价值变化
The values of a family of Cauchy transforms
论文作者
论文摘要
凯奇（Cauchy）的家族转变\ [c_ {g}（z，w）= - \ frac {1}π\ int _ {\ mathbb {c}} \ frac {g（u）} {\ edroce {u-w}（u-z）（u-z）（u-z）}} $ ge compact $ compact g \ leq 1，$且适合所有复杂$ z，w，$与希尔伯特太空运营商的理论密切相关。基于这些连接，可以在此处使用基本方法来得出不等式\ [\ vert 1- \ exp c {g}（z，w）\ vert \ leq 1。\]。该方法涉及对凸的积分家族的详细研究\ [i_ {g} = - \ frac {1}π\ int _ {\ mathbb {c}} \ frac {g（u）} {g（u）} {\ edimelline {\ edimelline {U+1}（u+1}（u -1}（u -1}（u -1}（u -1）} da（u -1）满足$ 0 \ leq g \ leq 1. $这些积分被使用平面的参数转换为可拖动的形式，使用圆圈的家族通过$+1，-1。$这些圆盘界限的特性函数通过这些圆圈的特征函数来减去真实轴，这些圆圈的特征功能是这些圆圈的特征，是集成量的convex convex集合的独特点。
The family of Cauchy transforms \[C_{g}(z,w) = -\frac{1}π\int_{\mathbb{C} } \frac{g(u)}{\overline{u-w} (u-z) } da(u ),\] where the measurable function $g$ with compact (essential) support satisfies $0 \leq g\leq 1,$ and suitably defined for all complex $z, w,$ is closely connected to the theory of Hilbert space operators with one-dimensional self-commutators. Based on these connections one can derive the inequality \[\vert 1-\exp C{g}(z,w)\vert\leq 1. \] Here, using elementary methods, a direct proof of this inequality is given. The approach involves a detailed study of the convex family of integrals \[I_{g}= -\frac{1}π\int_{\mathbb{C} } \frac{g(u)}{\overline{u+1} (u-1) } da(u),\] where $g$ varies over the set of measurable functions with compact support satisfying $0 \leq g\leq 1.$ These integrals are transformed to a tractable form using a parametriztion of the plane minus the real axis using the family of circles passing though the points $+1,-1.$ The characeristic functions of discs bounded by these circles are unique points in the boundary of the convex set of values of the family of integrals.
