We introduce a convolution form, in terms of integration over the unit disc \(\mathbb {D},\) for operators on functions f in \(H(\mathbb {D})\), which correspond to Taylor expansion multipliers. We demonstrate advantages of the introduced integral representation in the study of mapping properties of such operators. In particular, we prove the Young theorem for Bergman spaces in terms of integrability of the kernel of the convolution. This enables us to refer to the introduced convolutions as Hadamard–Bergman convolution. Another, more important, advantage is the study of mapping properties of a class of such operators in Holder type spaces of holomorphic functions, which in fact is hardly possible when the operator is defined just in terms of multipliers. Moreover, we show that for a class of fractional integral operators such a mapping between Holder spaces is onto. We pay a special attention to explicit integral representation of fractional integration and differentiation.

我们在单元盘引入卷积形式，在一体化方面\（\ mathbb {d}，\）上的功能为运营商˚F在\（H（\ mathbb {d}）\），它对应于泰勒展开倍数。我们在研究此类算子的映射属性时证明了引入的积分表示的优势。特别是，我们根据卷积核的可积性证明了伯格曼空间的杨定理。这使我们能够将引入的卷积称为Hadamard-Bergman卷积。另一个更重要的优点是，在全纯函数的Holder类型空间中研究此类算子的映射属性，实际上，仅根据乘数来定义算子时，这几乎是不可能的。而且，我们表明对于一类分数阶积分算子，Holder空间之间的这种映射是存在的。我们特别注意分数积分和微分的显式积分表示。