Sum of some product-type operators from mixed-norm spaces to weighted-type spaces on the unit ball

Abstract

Let $ u_{j} $ be the holomorphic functions on the open unit ball $ \mathbb{B} $ in $ \mathbb{C}^{n} $, $ j = \overline{0, m} $, $ \varphi $ a holomorphic self-map of $ \mathbb{B} $, and $ \Re^{j} $ the $ j $th iterated radial derivative operator. In this paper, the boundedness and compactness of the sum operator $ \mathfrak{S}^m_{\vec{u}, \varphi} = \sum_{j = 0}^m M_{u_j}C_\varphi\Re^j $ from the mixed-norm space $ H(p, q, \phi) $, where $ 0 < p, q < +\infty $, and $ \phi $ is normal, to the weighted-type space $ H^\infty_\mu $ are characterized. For the mixed-norm space $ H(p, q, \phi) $, $ 1\leq p < +\infty $, $ 1 < q < +\infty $, the essential norm estimate of the operator is given, and the Hilbert-Schmidt norm of the operator on the weighted Bergman space $ A^2_\alpha $ is also calculated.

Keywords

• product-type operator
• boundedness
• compactness
• mixed-norm space
• weighted-type space
• essential norm
• hilbert-schmidt norm