\documentclass{article} \font\twrm=cmr10 \font\twit=cmmi10 \font\twb=cmbx10 \font\tsl=cmsl10 \textheight43pc \textwidth28pc \begin{document} \twrm \begin{center} TITLE \vskip1cm AUTHOR \vskip1cm ABSTRACT \end{center} \rm \hskip1cm We investigate relationships between spaces of harmonic \hskip1cm functions corresponding to the meromorphic $Q_{p}^{\#}$ spaces \hskip1cm We give many analogues to the situations in the corres-- \hskip1cm ponding analytic and meromorphic spaces, and we give \hskip1cm some examples for which the behaviors are different in \hskip1cm the harmonic spaces. \vskip1cm \begin{center} {\twb 1. Introduction} \end{center} \vskip.5cm Let C denote the complex plane, let W denote the Riemann sphere, let D denote the unit disk $\{z \in$ C: $|z| <$ 1$\}$, and let $\Sigma$ denote the collection of all one-to-one conformal mappings of D onto itself. If f is a meromorphic function in D, we say that f is a normal function if the family $F = \{f(g(z))$: $g \in \Sigma \}$ is a normal family. We denote the family of all normal meromorphic functions by $N$. There is a related subfamily, the so-called "little normal functions", which is defined by \hskip1cm $N_{0} = \{f$: $f \mbox{ meromorphic in D and lim}_{|z| \to 1} (1 - |z|^{2} ) f^{\#}(z) = 0 \}$ \noindent where $f ^{\#}(z) = \frac{|f'(z)|}{1 + |f(z)|^{2}}$ is the spherical derivative of f. If u is a function which is harmonic and real valued in D, we say that u is a {\twit normal} {\twit harmonic} {\twit function} if the family F = $\{u(g(z)): g \in \Sigma \}$ is a normal family. It is consequence of this definition that if u is a normal harmonic function, and if f is the analytic function f(z) = u(z) + iv(z), where v(z) is a harmonic conjugate of u(z), then f is a normal (analytic) function). However, the converse is not true, since the elliptic modular function is a normal (analytic) function for which the real part is not a normal harmonic function. We denote by $N_{h}$ the family of all real harmonic normal functions. Let w $\in$ D and let $g(z,w) = \mbox{log }|\frac {1 - \bar{w}z}{z - w}|$ be the Green's function in D with logarithmic singularity at w, let $u^{\#}(z) = \frac {|\mbox{grad }\ u(z)|} {1 + |u(z)|^{2}}$, and let dm(z) denote the Euclidean element of area in C. In it was proved that a real valued function u, harmonic in D, is a normal function if and only if \hskip2cm $\mbox{sup}_{z \in D} (1 - |z| ) u^{\#} (z) < \infty$. In addition to $N_{h}$ , we will be considering the following classes of functions: \hskip1cm $UBC_{h} = \{u$: $u$ real harmonic in D and \hskip3.5cm $\mbox{sup}_{a \in D} \int \int_{D} (u^{\#}(z))^{2} g(z,a) dm(z) < \infty \}$, \hskip1cm $UBC_{h,0} = \{u$: $u$ real harmonic in D and \hskip3.5cm $\mbox{lim}_{|a| \to 1} \int \int_{D} (u^{\#} (z))^{2} g(z,a) dm(z) = 0 \}$, \hskip1cm $N_{h,0} = \{u$: $u$ real harmonic in D and \hskip3.5cm $\mbox{lim}_{|z| \to 1} (1 - |z|^{2} )u^{\#} (z) = 0 \}$, \hskip1cm $D_{h}^{\#} = \{u$: $u$ real harmonic in D and \hskip3.5cm $\int \int_{D} (u^{\#} (z))^{2} dm(z) < \infty \}$, \noindent and, for $0 < p < \infty$, \hskip1cm $Q_{h,p}^{\#} = \{u$: $u$ real harmonic in D and \hskip3.5cm $\mbox{sup}_{a \in D} \int\int_{D} (u^{\#} (z))^{2} (g(z,a))^{p} dm(z) < \infty \}$, \noindent and \hskip1cm $Q_{h,p,0}^{\#} = \{u$: $u$ real harmonic in D and \hskip3.5cm $\mbox{lim}_{|a| \to 1} \int\int_{D} (u^{\#} (z))^{2} (g(z,a))^{p} dm(z) = 0 \}$. For $0 < p < \infty$, the spaces $Q_{p} = \{f \mbox{: }f \mbox{ analytic in } D \mbox{ and sup}_{a \in D} \int \int_{D} |f'(z)|^{2} (g(z,a))^{p} dm(z) < \infty \}$ \noindent and the classes $Q_{p}^{\#} = \{f \mbox{: }f \mbox{ meromorphic in } D \mbox{ and}$ \hskip3.5cm $\mbox{sup}_{a \in D} \int \int_{D} (f^{\#} (z))^{2} (g(z,a))^{p} dm(z) < \infty \}$ \noindent were introduced in . It is possible to let $p = 0$ with the interpretation that $Q_{0} = \{f$: $f$ analytic in D and $\int \int_{D} |f'(z)|^{2} dm(z) < \infty \}$ \noindent and $Q_{0}^{\#} = \{f$: $f$ meromorphic in D and $\int \int_{D} (f^{\#} (z))^{2} dm(z) < \infty \}$. \noindent Under these interpretations, $Q_{0}$ is simply the usual Dirichlet space $D_{A}$ and $Q_{0}^{\#}$ is the spherical Dirichlet space $D_{A}^{\#}$ . We will use these interpretations in section 5. It has been shown that the $Q_{p}$ spaces and the $Q_{p}^{\#}$ classes have the nesting property that for $0 < p < q < \infty$ both $Q_{p} \subset Q_{q}$ and $Q_{p}^{\#} \subset Q_{q}^{\#}$). In section 4, we will give the corresponding property for the $Q_{h,p}^{\#}$ classes. In , it was proved that $D_{h}^{\#} \subset N_{h}$ (also see ). \vskip.5cm \noindent THEOREM 1. {\twit Let} $u$ {\twit be} {\twit a} {\twit real} {\twit harmonic} {\twit function} {\twit in} D, {\twit let} $0 < r < 1$, $2 < p < \infty$, {\twit and} $1 < q < \infty$. {\twit The} {\twit following} {\twit statements} {\twit are} {\twit equivalent}: \noindent (A) $u \in N_{h}$ , \noindent (B) $\mbox{sup}_{a \in D} \frac {1} {|D(a,r)|^{1-p/2}} \int \int_{D(a,r)}(u^{\#}(z))^{p} dm(z) < \infty$, \noindent (C) $\mbox{sup}_{a \in D} \int \int_{D(a,r)} (u^{\#} (z))^{p} (1 - |z|^{2} )^{p-2} dm(z) < \infty$, \noindent (D) $\mbox{sup}_{a \in D} \int \int_{D} (u^{\#} (z))^{p} (1 - |z|^{2} )^{p-2} (1 - |\phi_{a}(z)|^{2} )^{q} dm(z) < \infty$, \noindent (E) $\mbox{sup}_{a \in D} \int \int_{D} (u^{\#} (z))^{p} (1 - |z|^{2} )^{p-2} (g(z,a))^{q} dm(z) < \infty$, \noindent (F) $\mbox{sup}_{a \in D} \int \int_{D} (u^{\#} (z))^{p} (log \frac {1} {|z|})^{p} |\phi'_{a}(z)|^{2} dm(z) < \infty$ . \vskip.5cm \end{document}