# Mean-value characterization

## Harmonic functions.

Let denote the sphere of radius and centre in and let be the normalized Lebesgue measure on . One version of the classical converse of Gauss' mean-value theorem for harmonic functions asserts that a function which satisfies

(a1) |

is harmonic in (cf. also Harmonic function). In fact, one need only require that (a1) holds for , where is an arbitrary positive function of . A corresponding "local" result holds for continuous functions defined on an arbitrary domain in .

Remarkably, for the harmonicity of it suffices that (a1) holds only for two distinct values of (and all ), so long as the radii are not related in a special way. Specifically, let

where is the Bessel function of the first kind of order (cf. also Bessel functions), and let be the set of positive quotients of zeros of . J. Delsarte proved that if (a1) holds for and and , then is harmonic in [a11], cf. [a20]. (In fact, , so any two distinct radii are sufficient in dimension .) In [a10], Delsarte's theorem is extended to non-compact irreducible symmetric spaces of rank . There is also a local version of this result [a9], [a21]. Let be the ball of radius centred at in . Now, if satisfies (a1) for () and such that , then is harmonic on so long as .

In this connection, one should also mention Littlewood's one-circle problem, solved by W. Hansen and N. Nadirashvili [a14]. Let be a bounded continuous function on the open unit disc in . Suppose that for each point in there exists an such that the mean-value condition of (a1) holds. Must be harmonic? The answer turns out to be "no" [a14]. On the other hand, the one-radius condition obtained by replacing the peripheral mean in (a1) by the (areal) average over the disc of radius does imply harmonicity [a13]. This last result extends to functions defined on arbitrary bounded domains in (and many unbounded domains as well); one can also weaken the boundedness assumption on to for some positive harmonic function . For a survey of these and related results, see [a12].

Interesting new phenomena arise when one allows the integration to extend over the full space on which is defined. Consider, for instance, functions integrable with respect to the (normalized) Lebesgue measure on the unit ball in . If is harmonic with respect to the invariant Laplacian [a17], 4.1, then

for every in . The converse holds if and only if [a1], cf. [a7] and, for a Euclidean analogue, [a6]. Asymptotic mean-value conditions for (non-integrable) functions on are studied in [a8]. Finally, for a detailed overview of the whole subject, see [a15].

### Generalization.

The extent to which mean-value theorems and their converses generalize to differential equations other than is explored in [a22]. There it is shown that if is a homogeneous polynomial, then is a (weak) solution of the differential equation if and only if it satisfies the generalized mean-value condition

(a2) |

where is an appropriate complex measure supported on the unit ball of and . (The choice corresponds to (a1).) The local version of this result requires that (a2) holds for all and all . Solutions of are also characterized by two-radius theorems of Delsarte type [a22], [a23], cf. [a19].

## Pluriharmonic and separately harmonic functions.

Mean-value characterizations of pluriharmonic functions (i.e., real parts of holomorphic functions, cf. also Pluriharmonic function) and separately harmonic functions (i.e., functions harmonic with respect to each variable , ) are studied in [a3]. Let

here , . If is a complete bounded Reinhardt domain with centre at the point and is separately harmonic in and continuous in , then

(a3) |

Take for the -circular ellipsoids with centre at the point ,

where , , and all . Then the following result holds. Let be such that for each the conditions obtained by setting in (a3) , , and hold. If no belongs to and if

then is separately harmonic in .

Similarly, if is a complete bounded circular (Cartan) domain with centre at the point (cf. also Reinhardt domain) and is pluriharmonic in and continuous in , then

(a4) |

Consider now circular ellipsoids with centre at the point :

Let () be the inverse matrix of for fixed. Let (; ) be the -matrix with entries

Then the following result holds. Let be such that for every the conditions (a4) hold for , , , ( conditions). If and are such that no belongs to , and , then is pluriharmonic.

Local versions of the above-mentioned results are given also in [a3], as well as mean-value characterizations of pluriharmonic functions and separately harmonic functions by integration over the distinguished boundaries of poly-discs.

## Holomorphic and pluriharmonic functions.

In certain situations, Temlyakov–Opial–Siciak-type mean-value theorems (see [a2], [a16], [a18]) can be used to characterize holomorphic and pluriharmonic functions. For -times continuously differentiable functions on , the integral representation under discussion can be written as

(a5) |

where is the unit simplex in the real Euclidean -dimensional space, , , , , , and . Let denote a certain differential operator of order , which will be specified separately for holomorphic functions, for pluriharmonic functions, and also for anti-holomorphic functions (that is, functions holomorphic with respect to ). More precisely,

with the first-order differential operator to be specified, as mentioned above.

In [a4], the following criteria are proved for functions that are -times continuously differentiable on .

A function is holomorphic in if and only if (a5) holds with

A function is anti-holomorphic on if and only if (a5) holds with

A function is pluriharmonic on if and only if (a5) holds with

These results remain true without the assumption of smoothness; in this case, derivatives being understood in the distributional sense [a5].

#### References

[a1] | P. Ahern, M. Flores, W. Rudin, "An invariant volume-mean-value property" J. Funct. Anal. , 11 (1993) pp. 380–397 |

[a2] | L.A. Aizenberg, "Pluriharmonic functions" Dokl. Akad. Nauk. SSSR , 124 (1959) pp. 967–969 (In Russian) |

[a3] | L.A. Aizenberg, C.A. Berenstein, L. Wertheim, "Mean-value characterization of pluriharmonic and separately harmonic functions" Pacific J. Math. , 175 (1996) pp. 295–306 |

[a4] | L. Aizenberg, E. Liflyand, "Mean-value characterization of holomorphic and pluriharmonic functions" Complex Variables , 32 (1997) pp. 131–146 |

[a5] | L. Aizenberg, E. Liflyand, "Mean-value characterization of holomorphic and pluriharmonic functions, II" Complex Variables , 39 (1999) pp. 381–390 |

[a6] | Y. Ben Natan, Y. Weit, "Integrable harmonic functions on " J. Funct. Anal. , 150 (1997) pp. 471–477 |

[a7] | Y. Ben Natan, Y. Weit, "Integrable harmonic functions and symmetric spaces of rank one" J. Funct. Anal. , 160 (1998) pp. 141–149 |

[a8] | Y. Benyamini, Y. Weit, "Functions satisfying the mean value property in the limit" J. Anal. Math. , 52 (1989) pp. 167–198 |

[a9] | C.A. Berenstein, R. Gay, "A local version of the two-circles theorem" Israel J. Math. , 55 (1986) pp. 267–288 |

[a10] | C.A. Berenstein, L. Zalcman, "Pompeiu's problem on symmetric spaces" Comment. Math. Helv. , 55 (1980) pp. 593–621 |

[a11] | J. Delsarte, "Lectures on topics in mean periodic functions and the two-radius theorem" , Tata Institute, Bombay (1961) |

[a12] | W. Hansen, "Restricted mean value property and harmonic functions" J. Kral (ed.) et al. (ed.) , Potential Theory–ICPT 94 (Proc. Intern. Conf., Konty) , de Gruyter (1996) pp. 67–90 |

[a13] | W. Hansen, N. Nadirashvili, "A converse to the mean value theorem for harmonic functions" Acta Math. , 171 (1993) pp. 139–163 |

[a14] | W. Hansen, N. Nadirashvili, "Littlewood's one circle problem" J. London Math. Soc. , 50 (1994) pp. 349–360 |

[a15] | I. Netuka, J. Vesely, "Mean value property and harmonic functions" K. GowriSankaran (ed.) et al. (ed.) , Classical and Modern Potential Theory and Applications , Kluwer Acad. Publ. (1994) pp. 359–398 |

[a16] | Z. Opial, J. Siciak, "Integral formulas for function holomorphic in convex circular domains" Zeszyty Nauk. Uniw. Jagiello. Prace Mat. , 9 (1963) pp. 67–75 |

[a17] | W. Rudin, "Function theory in the unit ball of " , Springer (1980) |

[a18] | A.A. Temlyakov, "Integral representation of functions of two complex variables" Izv. Akad. Nauk. SSSR Ser. Mat. , 21 (1957) pp. 89–92 (In Russian) |

[a19] | V.V. Volchkov, "New theorems on the mean for solutions of the Helmholtz equation" Russian Acad. Sci. Sb. Math. , 79 (1994) pp. 281–286 |

[a20] | V.V. Volchkov, "New two-radii theorems in the theory of harmonic functions" Russian Acad. Sci. Izv. Math. , 44 (1995) pp. 181–192 |

[a21] | V.V. Volchkov, "The final version of the mean value theorem for harmonic functions" Math. Notes , 59 (1996) pp. 247–252 |

[a22] | L. Zalcman, "Offbeat integral geometry" Amer. Math. Monthly , 87 (1980) pp. 161–175 |

[a23] | L. Zalcman, "Mean values and differential equations" Israel J. Math. , 14 (1973) pp. 339–352 |

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Mean-value characterization.

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