Mean value extension theorems and microlocal analysis

Quinto, Eric

doi:10.1090/S0002-9939-03-06926-0

Mean value extension theorems and microlocal analysis
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by Eric Todd Quinto

Proc. Amer. Math. Soc. 131 (2003), 3267-3274

DOI: https://doi.org/10.1090/S0002-9939-03-06926-0

Published electronically: February 12, 2003

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Abstract:

We use microlocal analysis to prove new mean value theorems for harmonic functions on harmonic manifolds and for solutions to more general differential equations. The equations we consider all satisfy spherical mean value equalities, at least locally. Microlocal analysis and the mean value property in a small set allows us to show that the solution to the differential equation in a small set is also a solution in a much larger set.

References

Mark L. Agranovsky and Eric Todd Quinto, Injectivity sets for the Radon transform over circles and complete systems of radial functions, J. Funct. Anal. 139 (1996), no. 2, 383–414. MR 1402770, DOI 10.1006/jfan.1996.0090
Mark L. Agranovsky and Eric Todd Quinto, Injectivity of the spherical mean operator and related problems, Complex analysis, harmonic analysis and applications (Bordeaux, 1995) Pitman Res. Notes Math. Ser., vol. 347, Longman, Harlow, 1996, pp. 12–36. MR 1402020
L. A. Aizenberg, C. A. Berenstein, and L. Wertheim, Mean-value characterization of pluriharmonic and separately harmonic functions, Pacific J. Math. 175 (1996), no. 2, 295–306. MR 1432833
D. H. Armitage and Ü. Kuran, The convergence of the Pizzetti series in potential theory, J. Math. Anal. Appl. 171 (1992), no. 2, 516–531. MR 1194098, DOI 10.1016/0022-247X(92)90362-H
Carlos Berenstein, Der-Chen Chang, Daniel Pascuas, and Lawrence Zalcman, Variations on the theorem of Morera, The Madison Symposium on Complex Analysis (Madison, WI, 1991) Contemp. Math., vol. 137, Amer. Math. Soc., Providence, RI, 1992, pp. 63–78. MR 1190970, DOI 10.1090/conm/137/1190970
Carlos Berenstein and Daniel Pascuas, Morera and mean-value type theorems in the hyperbolic disk, Israel J. Math. 86 (1994), no. 1-3, 61–106. MR 1276131, DOI 10.1007/BF02773674
Carlos A. Berenstein and Lawrence Zalcman, Pompeiu’s problem on spaces of constant curvature, J. Analyse Math. 30 (1976), 113–130. MR 438254, DOI 10.1007/BF02786707
Carlos A. Berenstein and Lawrence Zalcman, Pompeiu’s problem on symmetric spaces, Comment. Math. Helv. 55 (1980), no. 4, 593–621. MR 604716, DOI 10.1007/BF02566709
Jürgen Berndt, Franco Tricerri, and Lieven Vanhecke, Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lecture Notes in Mathematics, vol. 1598, Springer-Verlag, Berlin, 1995. MR 1340192, DOI 10.1007/BFb0076902
Marc Chamberland, Mean value integral equations and the Helmholtz equation, Results Math. 30 (1996), no. 1-2, 39–44. MR 1402423, DOI 10.1007/BF03322178
R. Courant and D. Hilbert, Methods of mathematical physics. Vol. II: Partial differential equations, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. (Vol. II by R. Courant.). MR 0140802
J. Delsarte and J.-L. Lions, Moyennes généralisées, Comment. Math. Helv. 33 (1959), 59–69 (French). MR 102672, DOI 10.1007/BF02565907
Eric L. Grinberg and Eric Todd Quinto, Morera theorems for complex manifolds, J. Funct. Anal. 178 (2000), no. 1, 1–22. MR 1800789, DOI 10.1006/jfan.2000.3656
Sigurdur Helgason, Geometric analysis on symmetric spaces, Mathematical Surveys and Monographs, vol. 39, American Mathematical Society, Providence, RI, 1994. MR 1280714, DOI 10.1090/surv/039
Lars Hörmander, The analysis of linear partial differential operators. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 256, Springer-Verlag, Berlin, 1983. Distribution theory and Fourier analysis. MR 717035, DOI 10.1007/978-3-642-96750-4
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
P. Pizzetti, Sulla media dei valori che una funzione del punti dello spazio assume alla superficie di una sfera, Rend. Lincei ser. 5, 18(1909), 182-185.
—, Sul significato geometrico del secundo parametra differenziale di una funcione sopra una superficie qualunque, Rend. Lincei ser. 5, 18(1909), 309-316.
H. Poritsky, Generalizations of the Gauss law of the spherical mean, Trans. Amer. Math. Soc., 43(1938), 199-225.
Robert S. Strichartz, Mean value properties of the Laplacian via spectral theory, Trans. Amer. Math. Soc. 284 (1984), no. 1, 219–228. MR 742422, DOI 10.1090/S0002-9947-1984-0742422-0
V. V. Volchkov, New mean-value theorems for solutions of the Helmholtz equation, Mat. Sb. 184 (1993), no. 7, 71–78 (Russian, with Russian summary); English transl., Russian Acad. Sci. Sb. Math. 79 (1994), no. 2, 281–286. MR 1235290, DOI 10.1070/SM1994v079n02ABEH003500
V.V. Volchkov, Spherical means on symmetric spaces Dokl. NAN Ukraine (2002), 15-19.
Morgan Ward and R. P. Dilworth, The lattice theory of ova, Ann. of Math. (2) 40 (1939), 600–608. MR 11, DOI 10.2307/1968944
T. J. Willmore, An extension of Pizzetti’s formula to Riemannian manifolds, Analysis on manifolds (Conf., Univ. Metz, Metz, 1979) Astérisque, vol. 80, Soc. Math. France, Paris, 1980, pp. 3, 53–56 (English, with French summary). MR 620169
T. J. Willmore, Riemannian geometry, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993. MR 1261641
Lawrence Zalcman, Mean values and differential equations, Israel J. Math. 14 (1973), 339–352. MR 335835, DOI 10.1007/BF02764713
Lawrence Zalcman, Offbeat integral geometry, Amer. Math. Monthly 87 (1980), no. 3, 161–175. MR 562919, DOI 10.2307/2321600
L. Zalcman, A bibliographic survey of the Pompeiu problem, Approximation by solutions of partial differential equations (Hanstholm, 1991) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 365, Kluwer Acad. Publ., Dordrecht, 1992, pp. 185–194. MR 1168719
—, Supplementary bibliography to “A bibliographic survey of the Pompeiu problem”, Contemporary Math., 278(2001), 69-76.