Generalized Volterra Companion Operators on Bergman Spaces over Convex Domains of Finite Type

Abstract

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.