A converse to the mean value theorem for harmonic functions

Akcoglu, M. A. &Sharpe R. W., Ergodic theory and boundaries.Trans. Amer. Math. Soc., 132 (1968), 447–460. Baxter, J. R., Restricted mean values and harmonic functions.Trans. Amer. Math. Soc., 167 (1972), 451–463. —, Harmonic functions and mass cancellation.Trans. Amer. Math. Soc., 245 (1978), 375–384. Bliedtner, J. & Hansen, W.,Potential Theory—An Analytic and Probabilistic Approach to Balayage. Universitext, Springer, 1986. Boukricha, A., Hansen, W., &Hueber, H., Continuous solutions of the generalized Schrödinger equation and perturbation of harmonic spaces.Exposition. Math., 5 (1987), 97–135. 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. Burckel, R. B., A strong converse to Gauss's mean value theorem.Amer. Math. Monthly, 87 (1980), 819–820. Doob, J. L.,Classical Potential Theory and its Probabilistic Counterpart. Grundlehren Math. Wiss., 262. Springer, 1984. Feller, W., Boundaries induced by nonnegative matrices.Trans. Amer. Math. Soc., 83 (1956), 19–54. Fenton, P. C., Functions having the restricted mean value property.J. London Math. Soc., 14 (1976), 451–458. — On sufficient conditions for harmonicity.Trans. Amer. Math. Soc., 253 (1979), 139–147. — On the restricted mean value property.Proc. Amer. Math. Soc., 100 (1987), 477–481. Gong, X., Functions with the restricted mean value property.J. Xiamen Univ. Natur. Sci., 27 (1988), 611–615. de Guzmán, M.,Differentiation of Integrals in R n. Lecture Notes in Math., 481. Springer, 1975. 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. Hansen, W. &Ma, Zh., Perturbations by differences of unbounded potentials.Math. Ann., 287 (1990), 553–569. Heath, D., Functions possessing restricted mean value properties.Proc. Amer. Math. Soc., 41 (1973), 588–595. Helms, L. L.,Introduction to Potential Theory. Wiley, 1969. Huckemann, F., On the ‘one circle’ problem for harmonic functions.J. London Math. Soc., 29 (1954), 491–497. Kellogg, O. D., Converses of Gauss's theorem on the arithmetic mean.Trans. Amer. Math. Soc., 36 (1934), 227–242. Littlewood, J. E.,Some Problems in Real and Complex Analysis. Heath Math. Monographs. Lexington, Massachusetts, 1968. Netuka, I., Harmonic functions and mean value theorems. (In Czech.)Časopis Pěst. Mat., 100 (1975), 391–409. 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. —, A converse to the mean value theorem for harmonic functions.Amer. J. Math., 97 (1975), 1007–1027. Veselý, J., Restricted mean value property in axiomatic potential theory.Comment. Math. Univ. Carolin., 23 (1982), 613–628. Volterra, V., Alcune osservazioni sopra proprietà atte individuare una funzione.Atti della Reale Academia dei Lincei, 18 (1909), 263–266.