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Gradient estimate for solutions to Poisson equations in metric measure spaces. (English) Zbl 1255.58008
This work presents gradient estimates for solutions of the Poisson equation \(\Delta u = f\) on a complete, path-wise connected metric measure space \((X, d, \mu)\) with locally Ahlfors \(Q\)-regular measure \(\mu\), where \(Q>1\), such that \((X, d, \mu)\) supports a (local) (1,2)-Poincaré inequality (see equation (1.4)) and a suitable curvature lower bound (see equation (1.6)). Further discussion of these conditions is contained in the second section of the paper. The first main result of the paper is Theorem 1.1, a Moser-Trudinger inequality for the gradient of solutions to the Poisson equation, and the second main result of the paper is Theorem 1.2, a Sobolev inequality for solutions to the Poisson equation. Corollary 1.1 which follows from Theorem 1.2 is the Hölder continuity estimate with optimal exponent. These results are proven using the semi-group approach as in [L. A. Caffarelli and C. E. Kenig, Am. J. Math. 120, No. 2, 391–439 (1998; Zbl 0907.35026)] and [P. Koskela, K. Rajala and N. Shanmugalingam, J. Funct. Anal. 202, No. 1, 147–173 (2003; Zbl 1027.31006)]. In §2, preliminary definitions and results for Cheeger derivatives, Dirichlet forms, and Orlicz spaces as well as auxiliary results for Poisson equations are demonstrated. The heat semi-group is used in §3 together with generalized Riesz potentials to obtain a pointwise estimate for the gradient of the solution to an auxiliary equation. This estimate is used to prove Theorems 1.1 and 1.2 for solutions of the auxiliary equations in §4, and in §5 density arguments and the theory of Cheeger-harmonic functions is used to prove Theorems 1.1 and 1.2 and Corollary 1.1 for general solutions of the Poisson equation.

MSC:
58J05 Elliptic equations on manifolds, general theory
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
58J35 Heat and other parabolic equation methods for PDEs on manifolds
54E35 Metric spaces, metrizability
54E50 Complete metric spaces
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References:
[1] Astala, K.; Iwaniec, T.; Koskela, P.; Martin, G., Mappings of BMO-bounded distortion, Math. ann., 317, 703-726, (2000) · Zbl 0954.30009
[2] Bakry, D., On Sobolev and logarithmic inequalities for Markov semigroups, (), 43-75
[3] Bakry, D.; Emery, M., Diffusions hypercontractives, Sémin. de probabilités, XIX, 177-206, (1983/1984)
[4] Biroli, M.; Mosco, U., Sobolev inequalities for Dirichlet forms on homogeneous spaces, (), 305-311 · Zbl 0820.35035
[5] Biroli, M.; Mosco, U., A Saint-Venant type principle for Dirichlet forms on discontinuous media, Ann. mat. pura appl., 169, 125-181, (1995) · Zbl 0851.31008
[6] A. Björn, J. Björn, Nonlinear Potential Theory on Metric Spaces, EMS Tracts in Math., European Mathematical Society, Zurich, in press.
[7] Buser, P., A note on the isoperimetric constant, Ann. sci. école norm. sup. (4), 15, 213-230, (1982) · Zbl 0501.53030
[8] Caffarelli, L.A.; Kenig, C.E., Gradient estimates for variable coefficient parabolic equations and singular perturbation problems, Amer. J. math., 120, 391-439, (1998) · Zbl 0907.35026
[9] Cheeger, J., Differentiability of Lipschitz functions on metric measure spaces, Geom. funct. anal., 9, 428-517, (1999) · Zbl 0942.58018
[10] Cheng, S.Y.; Yau, S.T., Differential equations on Riemannian manifolds and their geometric applications, Comm. pure appl. math., 28, 3, 333-354, (1975) · Zbl 0312.53031
[11] Franchi, B.; Hajłasz, P.; Koskela, P., Definitions of Sobolev classes on metric spaces, Ann. inst. Fourier (Grenoble), 49, 1903-1924, (1999) · Zbl 0938.46037
[12] Fukushima, M.; Oshima, Y.; Takeda, M., Dirichlet forms and symmetric Markov processes, De gruyter stud. math., vol. 19, (1994), Walter de Gruyter & Co. Berlin · Zbl 0838.31001
[13] Gigli, N.; Kuwada, K.; Ohta, S.I., Heat flow on Alexandrov spaces · Zbl 1267.58014
[14] Gilbarg, D.; Trudinger, N.S., Elliptic partial differential equations of second order, (2001), Springer-Verlag Berlin, xiv+517 pp · Zbl 0691.35001
[15] Gross, L., Logarithmic Sobolev inequalities, Amer. J. math., 97, 1061-1083, (1975) · Zbl 0318.46049
[16] Hajłasz, P.; Koskela, P., Sobolev meets Poincaré, C. R. acad. sci. Paris Sér. I math., 320, 10, 1211-1215, (1995) · Zbl 0837.46024
[17] Hajłasz, P.; Koskela, P., Sobolev met Poincaré, Mem. amer. math. soc., 145, 688, (2000) · Zbl 0954.46022
[18] Heinonen, J.; Koskela, P., Quasiconformal maps in metric spaces with controlled geometry, Acta math., 181, 1-61, (1998) · Zbl 0915.30018
[19] Jiang, R., Lipschitz continuity of solutions of Poisson equations in metric measure spaces
[20] Keith, S., Modulus and the Poincaré inequality on metric measure spaces, Math. Z., 245, 255-292, (2003) · Zbl 1037.31009
[21] Keith, S., A differentiable structure for metric measure spaces, Adv. math., 183, 271-315, (2004) · Zbl 1077.46027
[22] Keith, S.; Zhong, X., The Poincaré inequality is an open ended condition, Ann. of math. (2), 167, 575-599, (2008) · Zbl 1180.46025
[23] Koskela, P.; Rajala, K.; Shanmugalingam, N., Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces, J. funct. anal., 202, 147-173, (2003) · Zbl 1027.31006
[24] P. Koskela, Y. Zhou, Geometry and analysis of Dirichlet forms, in preparation. · Zbl 1253.53035
[25] Li, P.; Yau, S.T., On the parabolic kernel of the Schrödinger operator, Acta math., 156, 153-201, (1986)
[26] Lott, J.; Villani, C., Ricci curvature for metric-measure spaces via optimal transport, Ann. of math. (2), 169, 903-991, (2009) · Zbl 1178.53038
[27] Ni, L.; Shi, Y.G.; Tam, L.F., Poisson equation, Poincaré-Lelong equation and curvature decay on complete Kähler manifolds, J. differential geom., 57, 339-388, (2001) · Zbl 1046.53025
[28] Rao, M.; Ren, Z., Theory of Orlicz spaces, (1991), Dekker New York · Zbl 0724.46032
[29] Saloff-Coste, L., A note on Poincaré, Sobolev, and Harnack inequalities, Int. math. res. not. IMRN, 2, 27-38, (1992) · Zbl 0769.58054
[30] Savaré, G., Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds, C. R. math. acad. sci. Paris, 345, 151-154, (2007) · Zbl 1125.53064
[31] Shanmugalingam, N., Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. mat. iberoam., 16, 243-279, (2000) · Zbl 0974.46038
[32] Stein, E.M.; Weiss, G., Introduction to Fourier analysis on Euclidean spaces, (1971), Princeton University Press Princeton, NJ · Zbl 0232.42007
[33] Sturm, K.T., Analysis on local Dirichlet spaces. I. recurrence, conservativeness and \(L^p\)-Liouville properties, J. reine angew. math., 456, 173-196, (1994) · Zbl 0806.53041
[34] Sturm, K.T., Analysis on local Dirichlet spaces. III. the parabolic Harnack inequality, J. math. pures appl. (9), 75, 3, 273-297, (1996) · Zbl 0854.35016
[35] Sturm, K.T., On the geometry of metric measure spaces I, Acta math., 196, 65-131, (2006) · Zbl 1105.53035
[36] Sturm, K.T., On the geometry of metric measure spaces II, Acta math., 196, 133-177, (2006) · Zbl 1106.53032
[37] Wang, F.Y., Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. theory related fields, 109, 417-424, (1997) · Zbl 0887.35012
[38] Wang, F.Y., Equivalence of dimension-free Harnack inequality and curvature condition, Integral equations operator theory, 48, 547-552, (2004) · Zbl 1074.47020
[39] Wong, B.; Zhang, Q.S., Refined gradient bounds, Poisson equations and some applications to open Kähler manifolds, Asian J. math., 7, 337-364, (2003) · Zbl 1110.53057
[40] Zhang, H.C.; Zhu, X.-P., On a new definition of Ricci curvature on Alexandrov spaces, Acta math. sci. ser. B engl. ed., 30, 1949-1974, (2010) · Zbl 1240.53073
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