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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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