Carleson measures and some classes of meromorphic functions

Abstract

For | a | > 1 |a|>1 let φ a \varphi _{a} be the Möbius transformation defined by φ a ( z ) = a − z 1 − a ¯ z \varphi _{a}(z)=\frac {a-z}{1-\bar az} , and let g ( z , a ) = log ⁡ | 1 − a ¯ z z − a | g(z,a)=\log |\frac {1-\bar az}{z-a}| be the Green’s function of the unit disk D \mathcal {D} . We construct an analytic function f f belonging to M p # = { f : M_{p}^{\#} = \{\, f : {} f f meromorphic in D \mathcal {D} and sup a ∈ D ∬ D ( f # ( z ) ) 2 ( 1 − | φ a ( z ) | 2 ) p d A ( z ) > ∞ } \sup _{a\in \mathcal {D}} \iint _{\mathcal {D}}(f^{\#}(z))^{2}(1-|\varphi _{a}(z)|^{2})^{p}\,dA(z)>\infty \, \} for all p p , 0 > p > ∞ 0>p>\infty , but not belonging to Q p # = { f : f Q_{p}^{\#}=\{\,f:f meromorphic in D \mathcal {D} and sup a ∈ D ∬ D ( f # ( z ) ) 2 ( g ( z , a ) ) p d A ( z ) > ∞ } \sup _{a\in \mathcal {D}}\iint _{\mathcal {D}}(f^{\#}(z))^{2}(g(z,a))^{p}\,dA(z)>\infty \,\} for any p p , 0 > p > ∞ 0>p>\infty . This gives a clear difference as compared to the analytic case where the corresponding function spaces ( M p M_{p} and Q p Q_{p} ) are same.