×

zbMATH — the first resource for mathematics

A converse to the mean value theorem for harmonic functions. (English) Zbl 0808.31004
Let \(U\) be a bounded domain in \(\mathbb{R}^ d\), \(d\geq 1\), \(\rho(x)= \text{dist} (x,U^ c)\), and \(r:U\to \mathbb{R}\) a function satisfying \(0<r\leq\rho\). Let \(B^ x\), \(x\in U\), denote the open ball centered at \(x\) with radius \(r(x)\). A Lebesgue measurable function \(f\) on \(U\) satisfying \[ f(x)= {1\over {\lambda(B^ x)}} \int_{B^ x} f d\lambda \] for every \(x\in U\) (where \(\lambda\) denotes the Lebesgue measure), is said to be \(r\)-median. Results of the type under what conditions is an \(r\)-median function \(f\) actually harmonic, are usually known as a converse to the mean value theorem.
The main result of the present paper states that if \(f\) is \(r\)-median, continuous (on \(U\)) and \(h\)-bounded (i.e. \(| f|\leq h\) with \(h\) harmonic on \(U\)), then \(f\) is harmonic on \(U\). As a rather simple consequence of this result, the authors show that an \(r\)-median, \(h\)- bounded, Lebesgue measurable function \(f\) on \(U\) is harmonic provided that the function \(r\) is bounded away from zero on compact subsets of \(U\), thus improving the result of W. A. Veech [Ann. Math., II. Ser. 97, 189-216 (1973; Zbl 0282.60048)], where \(U\) was assumed to be a Lipschitz domain.
The proof of the results is analytic, but with the strong probabilistic flavor in the background. It uses the minimal fine topology of the Martin compactification of \(U\), an appropriate (transfinite) sweeping of measures, and certain properties of the Schrödinger equation \(\Delta u- \delta\rho^{-2} 1_ A=0\) on \(U\) (\(\delta>0\), \(A\) a suitable subset of \(U\)).

MSC:
31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions
31C35 Martin boundary theory
60J45 Probabilistic potential theory
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akcoglu, M. A. &Sharpe R. W., Ergodic theory and boundaries.Trans. Amer. Math. Soc., 132 (1968), 447–460. · Zbl 0162.19402 · doi:10.1090/S0002-9947-1968-0224770-7
[2] Baxter, J. R., Restricted mean values and harmonic functions.Trans. Amer. Math. Soc., 167 (1972), 451–463. · Zbl 0238.31006 · doi:10.1090/S0002-9947-1972-0293112-4
[3] –, Harmonic functions and mass cancellation.Trans. Amer. Math. Soc., 245 (1978), 375–384. · Zbl 0391.60065 · doi:10.1090/S0002-9947-1978-0511416-X
[4] Bliedtner, J. & Hansen, W.,Potential Theory–An Analytic and Probabilistic Approach to Balayage. Universitext, Springer, 1986. · Zbl 0706.31001
[5] Boukricha, A., Hansen, W., &Hueber, H., Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces.Exposition. Math., 5 (1987), 97–135. · Zbl 0659.35025
[6] Brunel, A., Propriété restreinte de valeur moyenne caractérisant les fonctions harmoniques bornées sur un ouvert dansR n (selon D. Heath et L. Orey). Exposé no XIV, inSéminaire, Goulaouic-Schwartz, Paris, 1971–1972.
[7] Burckel, R. B., A strong converse to Gauss’s mean value theorem.Amer. Math. Monthly, 87 (1980), 819–820. · Zbl 0475.31003 · doi:10.2307/2320795
[8] Doob, J. L.,Classical Potential Theory and its Probabilistic Counterpart. Grundlehren Math. Wiss., 262. Springer, 1984. · Zbl 0549.31001
[9] Feller, W., Boundaries induced by nonnegative matrices.Trans. Amer. Math. Soc., 83 (1956), 19–54. · Zbl 0071.34901 · doi:10.1090/S0002-9947-1956-0090927-3
[10] Fenton, P. C., Functions having the restricted mean value property.J. London Math. Soc., 14 (1976), 451–458. · Zbl 0344.31002 · doi:10.1112/jlms/s2-14.3.451
[11] – On sufficient conditions for harmonicity.Trans. Amer. Math. Soc., 253 (1979), 139–147. · Zbl 0368.31001 · doi:10.1090/S0002-9947-1979-0536939-X
[12] – On the restricted mean value property.Proc. Amer. Math. Soc., 100 (1987), 477–481. · Zbl 0634.31002 · doi:10.1090/S0002-9939-1987-0891149-1
[13] Gong, X., Functions with the restricted mean value property.J. Xiamen Univ. Natur. Sci., 27 (1988), 611–615. · Zbl 0693.31002
[14] de Guzmán, M.,Differentiation of Integrals in R n . Lecture Notes in Math., 481. Springer, 1975.
[15] Hansen, W., Valeurs propres pour l’opérateur de Schrödinger, inSéminaire de Théorie du Potentiel, Paris, No. 9. Lecture Notes in Math., 1393, pp. 117–134. Springer, 1989.
[16] Hansen, W. &Ma, Zh., Perturbations by differences of unbounded potentials.Math. Ann., 287 (1990), 553–569. · Zbl 0685.31005 · doi:10.1007/BF01446913
[17] Heath, D., Functions possessing restricted mean value properties.Proc. Amer. Math. Soc., 41 (1973), 588–595. · Zbl 0251.31004 · doi:10.1090/S0002-9939-1973-0333213-1
[18] Helms, L. L.,Introduction to Potential Theory. Wiley, 1969. · Zbl 0188.17203
[19] Huckemann, F., On the ’one circle’ problem for harmonic functions.J. London Math. Soc., 29 (1954), 491–497. · Zbl 0056.32601 · doi:10.1112/jlms/s1-29.4.491
[20] Kellogg, O. D., Converses of Gauss’s theorem on the arithmetic mean.Trans. Amer. Math. Soc., 36 (1934), 227–242. · Zbl 0009.11205
[21] Littlewood, J. E.,Some Problems in Real and Complex Analysis. Heath Math. Monographs. Lexington, Massachusetts, 1968. · Zbl 0185.11502
[22] Netuka, I., Harmonic functions and mean value theorems. (In Czech.)Časopis Pěst. Mat., 100 (1975), 391–409.
[23] Veech, W. A., A zero-one law for a class of random walks and a converse to Gauss’ mean value theorem.Ann. of Math., 97 (1973), 189–216. · Zbl 0282.60048 · doi:10.2307/1970845
[24] –, A converse to the mean value theorem for harmonic functions.Amer. J. Math., 97 (1975), 1007–1027. · Zbl 0324.31002 · doi:10.2307/2373685
[25] Veselý, J., Restricted mean value property in axiomatic potential theory.Comment. Math. Univ. Carolin., 23 (1982), 613–628. · Zbl 0513.31009
[26] Volterra, V., Alcune osservazioni sopra proprietà atte individuare una funzione.Atti della Reale Academia dei Lincei, 18 (1909), 263–266. · JFM 40.0453.01
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.