Weighted conformal invariance of Banach spaces of analytic functions

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