4.5 Article

ON A NEW PRODUCT-TYPE OPERATOR ON THE UNIT BALL

Journal

JOURNAL OF MATHEMATICAL INEQUALITIES
Volume 16, Issue 4, Pages 1675-1692

Publisher

ELEMENT
DOI: 10.7153/jmi-2022-16-109

Keywords

Product-type operator; boundedness; compactness; logarithmic Bloch space; weighted-type space

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This paper investigates the polynomial differentiation composition operator and provides necessary and sufficient conditions for the boundedness and compactness of the operator from the logarithmic Bloch spaces to weighted-type spaces of holomorphic functions on B.
Let m E N, uj , j = 1,m, be holomorphic functions on the open unit ball B C Cn , phi be a holomorphic self-map of B, and Dl be the partial derivative operator in the l th variable lE {1,2, ... ,n}. We introduce here the following polynomial differentiation composition operator PmD,phi f := m n-ary sumation j=1 ujC phi Dlj center dot center dot center dot Dl1 f and give some necessary and sufficient conditions for the boundedness and compactness of the operator from the logarithmic Bloch spaces to weighted-type spaces of holomorphic functions on B.

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