We consider Banach spaces of analytic functions in the unit disc which satisfy a weighted conformal invariance property, that is, for a fixed $\alpha >0$ and every conformal automorphism φ of the disc, $f\to f\circ \phi {({\phi }^{\prime })}^{\alpha }$ defines a bounded linear operator on the space in question, and the family of all such operators is uniformly bounded in operator norm. Many common examples of Banach spaces of analytic functions like Korenblum growth classes, Hardy spaces, standard weighted Bergman and certain Besov spaces satisfy this condition. The aim of the paper is to develop a general approach to the study of such spaces based on this property alone. We consider polynomial approximation, duality and complex interpolation, we identify the largest and the smallest as well as the “unique” Hilbert space satisfying this property for a given $\alpha >0$. We investigate the weighted conformal invariance of the space of derivatives, or anti-derivatives with the induced norm, and arrive at the surprising conclusion that they depend entirely on the properties of the (modified) Cesàro operator acting on the original space. Finally, we prove that this last result implies a John-Nirenberg type estimate for analytic functions g with the property that the integration operator $f\to {\int }_0^{z}f(t)g^{\prime }(t)dt$ is bounded on a Banach space satisfying the weighted conformal invariance property.

我们考虑单位圆盘中满足加权共形不变性的解析函数的Banach空间，即对于固定 $\alpha >0$光盘的每个共形自同构φ$F\to F\circ \phi {（{\phi }^{\prime }）}^{\alpha }$在所讨论的空间上定义了有界线性算子，并且所有此类算子的族在算子范数中统一有界。解析函数的Banach空间的许多常见示例（例如Korenblum增长类，Hardy空间，标准加权Bergman空间和某些Besov空间）都满足此条件。本文的目的是开发仅基于这种性质的研究此类空间的通用方法。我们考虑多项式逼近，对偶和复数插值，确定给定条件下满足该性质的最大和最小以及“唯一”希尔伯特空间$\alpha >0$。我们研究了带有诱导范数的导数空间或反导数空间的加权共形不变性，并得出了令人惊讶的结论，即它们完全取决于作用于原始空间的（修改的）Cesàro算子的性质。最后，我们证明这最后一个结果隐含解析函数g的John-Nirenberg类型估计，其性质为积分算符$F\to {\int }_0^{ž}F（Ť）G^{\prime }（Ť）dŤ$ 在满足加权共形不变性的Banach空间上有界。