Carleson measures and some classes of meromorphic functions

Abstract:

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