有限型凸域上Bergman空间上的广义Volterra伴随算子
Generalized Volterra Companion Operators on Bergman Spaces over Convex Domains of Finite Type
AI总结：
本文研究有限型凸域上Bergman空间的广义Volterra伴随算子，给出有界性、紧性、本质范数及Schatten类判据，并证明奇异值指数衰减的幂律。
AI中文摘要：
我们研究了有限型光滑有界凸域上Bergman空间上的广义Volterra伴随算子。一个导数Carleson嵌入给出了自反Bergman空间之间的有界性和紧性判据。当目标指数小于源指数时，有界性已经蕴含紧性。在另一范围内，我们还得到了一个本质范数公式。McNeal多盘上的局部质量描述了这些判据。它们的范数估计需要一个额外的权重，因为径向导数在原点处消失。在Hilbert Bergman空间上，我们刻画了在Hilbert-Schmidt阈值处及以上的Schatten类隶属关系，并证明了低于该阈值的充分条件。我们还给出了一个Hilbert-Schmidt核检验。对于有界符号和具有相对紧像的自映射，奇异值以其指数的幂次指数衰减。一个椭球例子给出了由边界类型和维数控制的尖锐幂律。这些结果推广到正径向平移。
英文摘要：
We study generalized Volterra companion operators on Bergman spaces over smoothly bounded convex domains of finite type. A derivative Carleson embedding gives boundedness and compactness criteria between reflexive Bergman spaces. When the target exponent is smaller than the source exponent, boundedness already implies compactness. In the other range, we also obtain an essential-norm formula. Local masses on McNeal polydiscs describe these criteria. Their norm estimates require an extra weight because radial derivatives vanish at the origin. On the Hilbert Bergman space, we characterize Schatten-class membership at and above the Hilbert--Schmidt threshold and prove a sufficient condition below it. We also give a Hilbert--Schmidt kernel test. For bounded symbols and self-maps with relatively compact image, singular values decay exponentially in a power of their index. An ellipsoid example gives a sharp power law governed by boundary type and dimension. The results extend to positive radial shifts.
