DISTORTION THEOREMS, LIPSCHITZ CONTINUITY AND THEIR APPLICATIONS FOR BLOCH TYPE MAPPINGS ON BOUNDED SYMMETRICDOMAINS IN Cn

摘要

Let BX be a bounded symmetric domain realized as the unit ball of an n-dimensional JB*-triple X= (Cn,X).In this paper, we give a new definition of Bloch type mappings on BXand give distortion theorems for Bloch type mappings on BX.When BX is the Euclidean unit ball in Cn,this new definition coincides with that given by Chen and Kalaj or by the author.As a corollary of the distortion theorem, we obtain the lower estimate for the radius of thelargest schlicht ball in the image of f centered at f(0)for -Bloch mappings f on BX.Next, as another corollary of the distortion theorem, we show the Lipschitz continuity of (det B(z,z))1}/2n|det Df(z)|1for Bloch type mappings f on BX with respect to the Kobayashi metric,where B(z,z) is the Bergman operator on X,and use it to give a sufficient condition for the composition operator C to be bounded from belowon the Bloch type space on BX,where is a holomorphic self mapping of BX.In the case BX = Bn, we also give a necessary condition for C to be bounded from belowwhich is a converse to the above result.Finally, as another application of the Lipschitz continuity,we obtaina result related tothe interpolating sequences for the Bloch type spaceon BX.