Let $G\subset {\bf SU}(2,1)$ be a non-elementary complex hyperbolic Kleinian group. If $G$ preserves a complex line, then $G $ is $\mathbb {C}$-Fuchsian; if $ G $ preserves a Lagrangian plane, then $ G $ is $\mathbb {R}$-Fuchsian; $ G $ is Fuchsian if $ G $ is either $\mathbb {C}$-Fuchsian or $\mathbb {R}$-Fuchsian. In this paper, we prove that if the traces of all elements in $ G $ are real, then $ G $ is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an application of our main result, we show that $ G $ is conjugate to a subgroup of ${\bf S}(U(1)\times U(1,1))$ or ${\bf SO}(2,1)$ if each loxodromic element in $G $ is hyperbolic. Moreover, we show that the converse of our main result does not hold by giving a $\mathbb {C}$-Fuchsian group.
The relationship between weighted Lipschitz functions and analytic Bloch spaces has attracted much attention. In this paper, we define harmonic ω-α-Bloch space and characterize it in terms of \omega \left( {{{\left( {1 - {{\left| x \right|}^2}} \right)}^\beta }{{\left( {1 - {{\left| y \right|}^2}} \right)}^{\alpha - \beta }}} \right)\left| {\frac{{f\left( x \right) - f\left( y \right)}}{{x - y}}} \right| and \omega \left( {{{\left( {1 - {{\left| x \right|}^2}} \right)}^\beta }{{\left( {1 - {{\left| y \right|}^2}} \right)}^{\alpha - \beta }}} \right)\left| {\frac{{f\left( x \right) - f\left( y \right)}}{{\left| x \right|y - x'}}} \right| where ω is a majorant. Similar results are extended to harmonic little ω-α-Bloch and Besov spaces. Our results are generalizations of the corresponding ones in G.Ren, U.Kähler (2005)., Xi Fu, Bowen Lu., and Obsahuje seznam literatury