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TITLE 

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         AUTHOR 

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ABSTRACT
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\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.


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\begin{center} {\twb 1. Introduction}
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     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 ). 

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\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) $\mbox{sup}_{a \in D} \int \int_{D} (u^{\#} (z))^{p} (log
\frac {1} {|z|})^{p}  |\phi'_{a}(z)|^{2}  dm(z) < \infty$ .

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