The two remaining properties (mean value property, elimination principle) are obvious.
Axler and Ž. Čučković, Integral Equations Operator Theory14 (1991), 1-12; W. Rudin, "Function Theory in the Unit Ball of CN," Springer-Verlag, New York/Berlin, 1980]: If f is bounded and satifies the invariant version of the area mean value property, then f is harmonic.
For example, Zhang [11] studied the hybrid mean value properties of Cochrane sums and Kloosterman sums and proved that for any prime p > 3, we have the asymptotic formula ∑ h = 1 p − 1 K ( h, 1 ; p ) C ( h, p ) = − 1 2 π 2 p 2 + O ( p ⋅ exp ( 3 ln p ln ln p ) ), where exp ( y ) = e y, K ( m, n ; q ) = ∑ ′ a = 1 q e ( m a + n a ¯ q ).
Conversely, does this mean value property imply thatfis harmonic ?
Another important observation is that, due to the mean value property, means are locally bounded functions.
First observe that (widetilde{mathscr{M}}) obviously admits the mean value property.
(iii) Mean value property: For all (n in mathbb {N}) and for all ((x,lambda ) in I^{n} times W_{n}(R)) min(x_{1},dots,x_{n}) le mathscr{M} x,lambda)le max(x_{1},dots,x_{n}).
Mean value property: For all (n in mathbb {N}) and for all ((x,lambda ) in I^{n} times W_{n}(R)) min(x_{1},dots,x_{n}) le mathscr{M} x,lambda)le max(x_{1},dots,x_{n}).
The proof of this result is a consequence of a recasting of the classical mean value property for harmonic functions in terms of an identity (5.1) that links the Bochner-artinelli CF form on a sphere with the sphere's Euclidean surface measure.
The mean value property for analytic functions f n k ′ Open image in new window and f ′ yields f n k ′ ( z 1 ) − f ′ ( z 1 ) ≤ 4 Π ( 1 − | z 1 | ) 2 ∫ | z 1 − z | < 1 − | z | 2 × ( f n k ′ ( z ) − f ′ ( z ) ) dA ( z ).
The Lévy Laplacian ΔF = limN→∞N−1∑n = 1N 〈F″,en⊗ en〉 is shown to be equal to (i) ∝TF″s″(ξ t dt, where Fs″ is the singular part of F″, and (ii) 2limϱ→0ϱ−2(MF−F, where MF is the spherical mean of F. It is proved that regular polynomials are Δ-harmonic and possess the mean value property.
