View source for Privalov operators

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''Privalov parameters''

Operators that allow one to express the condition of harmonicity of a function without using partial derivatives (cf. [[Harmonic function|Harmonic function]]). Let  $  u $
be a locally integrable function in a bounded domain  $  D $
of a Euclidean space  $  \mathbf R  ^ {n} $, 
$  n \geq  2 $; 
let  $  \omega ( h) $
denote the volume of the ball  $  B ( x;  h) $
of radius  $  h $
with centre  $  x \in D $, 
lying in  $  D $; 
and let

$$ 
\Delta _ {h} u ( x)  = \ 
{
\frac{1}{\omega ( h) }
 }
\int\limits _ {B ( x;  h) }
u ( y)  dy - u ( x).
$$

The upper and lower Privalov operators  $  \overline \Delta \; {}  ^ {*} u ( x) $
and  $  \underline \Delta   ^ {*} u ( x) $
are defined, respectively, by the formulas

$$ 
\overline \Delta \; {}  ^ {*} u ( x)  = \ 
\overline{\lim\limits}\; _ {h \rightarrow 0 } 2( n+ 
\frac{2)}{h  ^ {2} }

\Delta _ {h} u( x) ,
$$

$$ 
\underline \Delta   ^ {*} u ( x)  =  \lim\limits _ {\overline{ {h \rightarrow 0 }}\; } 2( n+ 
\frac{2)}{h  ^ {2} }
 \Delta _ {h} u( x) .
$$

If the upper and lower Privalov operators coincide, then the Privalov operator  $  \Delta  ^ {*} u ( x) $
is defined by

$$ 
\Delta  ^ {*} u ( x)  = \ 
\overline \Delta \; {}  ^ {*} u ( x)  = \ 
\underline \Delta   ^ {*} u ( x)  = \ 
\lim\limits _ {h \rightarrow 0 }
2( n+ 
\frac{2)}{h  ^ {2} }
 \Delta _ {h} u( x) .
$$

If the function  $  u $
has continuous partial derivatives up to and including the second order at  $  x \in D $, 
then the Privalov operator  $  \Delta  ^ {*} u( x) $
exists at  $  x $
and is equal to the value of the [[Laplace operator|Laplace operator]]:  $  \Delta  ^ {*} u ( x) = \Delta u ( x) $. 
Privalov's theorem says: If a function  $  u $, 
continuous in a domain  $  D $, 
satisfies everywhere in  $  D $
the conditions

$$ 
\underline \Delta   ^ {*} u ( x)  \leq  \ 
0  \leq   \overline \Delta \; {}  ^ {*} u ( x),
$$

then  $  u $
is harmonic in  $  D $. 
This implies that a function  $  u $, 
continuous in  $  D $, 
is harmonic if and only if at every point  $  x \in D $
one has  $  \Delta _ {h} u ( x) = 0 $, 
from some sufficiently small  $  h $
onwards, or, in other words, if and only if

$$ 
u ( x)  = \ 
{
\frac{1}{\omega ( h) }
 }
\int\limits _ {B ( x;  h) }
u ( y)  dy.
$$

The average value over the volume of a sphere can be replaced by that over the surface area.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.I. Privalov,   ''Mat. Sb.'' , '''32'''  (1925)  pp. 464–471</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.I. Privalov,   "Subharmonic functions" , Moscow-Leningrad  (1937)  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Brélot,   "Eléments de la théorie classique du potentiel" , Sorbonne Univ. Centre Doc. Univ. , Paris  (1969)</TD></TR></table>

====Comments====
More generally, if  $  u > - \infty $
is lower semi-continuous, then  $  u $
is hyperharmonic if and only if  $  \underline \Delta   ^ {*} u \leq  0 $
on  $  \{ u < \infty \} $(
the theorem of Blaschke–Privalov).

Similar results hold if the average value over the surface area is used for the operators and  $  2( n+ 2) $
is replaced by  $  2n $.