Privalov operators

Privalov parameters

Operators that allow one to express the condition of harmonicity of a function without using partial derivatives (cf. 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: $ \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.

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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 $.