Cheeger-harmonic functions in metric measure spaces revisited (original) (raw)
Lipschitz Continuity of Solutions of Poisson Equations in Metric Measure Spaces
Potential Analysis, 2012
Let X be a pathwise connected metric space equipped with an Ahlfors Qregular measure µ, Q ∈ [1, ∞). Suppose that (X, µ) supports a (1, 2)-Poincaré inequality and a Sobolev-Poincaré type inequality for the corresponding "Gaussian measure". The author uses the heat equation to study the Lipschitz regularity of solutions of the Poisson equation D 2 u = f , where D denotes the Cheeger derivative and f ∈ L p . When p > Q, the local Lipschitz continuity of u is established.
Gradient estimates for heat kernels and harmonic functions
Journal of Functional Analysis, 2019
Let (X, d, µ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a "carré du champ". Assume that (X, d, µ, E) supports a scale-invariant L 2-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p ∈ (2, ∞]: (i) (G p): L p-estimate for the gradient of the associated heat semigroup; (ii) (RH p): L p-reverse Hölder inequality for the gradients of harmonic functions; (iii) (R p): L p-boundedness of the Riesz transform (p < ∞); (iv) (GBE): a generalised Bakry-Émery condition. We show that, for p ∈ (2, ∞), (i), (ii) (iii) are equivalent, while for p = ∞, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the L 2-Poincaré inequality. Our result gives a characterisation of Li-Yau's gradient estimate of heat kernels for p = ∞, while for p ∈ (2, ∞) it is a substantial improvement as well as a generalisation of earlier results by Auscher-Coulhon-Duong-Hofmann [7] and Auscher-Coulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and sub-Riemannian manifolds as well as to non-smooth spaces, and to degenerate elliptic/parabolic equations in these settings.
Gradient estimate for solutions to Poisson equations in metric measure spaces
Journal of Functional Analysis, 2011
Let (X, d) be a complete, pathwise connected metric measure space with locally Ahlfors Q-regular measure µ, where Q > 1. Suppose that (X, d, µ) supports a (local) (1, 2)-Poincaré inequality and a suitable curvature lower bound. For the Poisson equation ∆u = f on (X, d, µ), Moser-Trudinger and Sobolev inequalities are established for the gradient of u. The local Hölder continuity with optimal exponent of solutions is obtained. g ds 2000 Mathematics Subject Classification. 31C25; 31C45; 35B33; 35B65
2 O ct 2 01 7 Gradient estimates for heat kernels and harmonic functions
2018
Let (X, d, μ) be a doubling metric measure space endowed with a Dirichlet form E deriving from a “carré du champ”. Assume that (X, d, μ, E ) supports a scale-invariant L-Poincaré inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for p ∈ (2,∞]: (i) (Gp): L -estimate for the gradient of the associated heat semigroup; (ii) (RHp): L -reverse Hölder inequality for the gradients of harmonic functions; (iii) (Rp): L -boundedness of the Riesz transform (p < ∞); (iv) (GBE): a generalised Bakry-Émery condition. We show that, for p ∈ (2,∞), (i), (ii) (iii) are equivalent, while for p = ∞, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the L-Poincaré inequality. Our result gives a characterisation of Li-Yau’s gradient estimate of heat kernels for p = ∞, while for p ∈ (2,∞) it is a substantial improvement as well as a generalisation of earlier results by Auscher-...
Differentiability of p-Harmonic Functions on Metric Measure Spaces
Potential Analysis, 2013
We study p-harmonic functions on metric measure spaces, which are formulated as minimizers to certain energy functionals. For spaces supporting a p-Poincaré inequality, we show that such functions satisfy an infinitesmal Lipschitz condition almost everywhere. This result is essentially sharp, since there are examples of metric spaces and p-harmonic functions that fail to be locally Lipschitz continuous on them. As a consequence of our main theorem, we show that p-harmonic functions also satisfy a generalized differentiability property almost everywhere, in the sense of Cheeger's measurable differentiable structures.
Heliyon
In this paper local and global gradient estimates are obtained for positive solutions to the following nonlinear elliptic equation Δ + () + () = 0, on complete smooth metric measure spaces (, , −) with ∞-Bakry-Émery Ricci tensor bounded from below, where is an arbitrary real constant, () and () are smooth functions. As an application, Liouvilletype theorems for various special cases of the equation are recovered. Furthermore, we discuss nonexistence of smooth solution to Yamabe type problem on (, , −) with nonpositive weighted scalar curvature.
manuscripta mathematica, 2017
Given a complete, smooth metric measure space (M, g, e −f dv) with the Bakry-Émery Ricci curvature bounded from below, various gradient estimates for solutions of the following general f-heat equations ut = ∆ f u + au log u + bu + Au p + Bu −q and ut = ∆ f u + Ae pu + Be −pu + D are studied. As by-product, we obtain some Liouville-type theorems and Harnack-type inequalities for positive solutions of several nonlinear equations including the Schrödinger equation, the Yamabe equation, and Lichnerowicz-type equations as special cases.
A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces
Advances in Mathematics, 2015
On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo-Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber-Krahn inequalities or localized Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies-Gaffney estimates.
Poincaré inequality meets Brezis–Van Schaftingen–Yung formula on metric measure spaces
Journal of Functional Analysis
Let (X, ρ, µ) be a metric measure space of homogeneous type which supports a certain Poincaré inequality. Denote by the symbol C * c (X) the space of all continuous functions f with compact support satisfying that Lip f := lim sup r→0 sup y∈B(•,r) | f (•) − f (y)|/r is also a continuous function with compact support and Lip f = lim r→0 sup y∈B(•,r) | f (•) − f (y)|/r converges uniformly. Let p ∈ [1, ∞). In this article, the authors prove that, for any f ∈ C * c (X), sup λ∈(0,∞) λ p X µ y ∈ X : | f (x) − f (y)| > λρ(x, y)[V(x, y)] 1 p dµ(x) ∼ X [Lip f (x)] p dµ(x) with the positive equivalence constants independent of f , where V(x, y) := µ(B(x, ρ(x, y))). This generalizes a recent surprising formula of H. Brezis, J. Van Schaftingen, and P.-L. Yung from the n-dimensional Euclidean space R n to X. Applying this generalization, the authors establish new fractional Sobolev and Gagliardo-Nirenberg inequalities in X. All these results have a wide range of applications. Particularly, when applied to two concrete examples, namely, R n with weighted Lebesgue measure and the complete n-dimensional Riemannian manifold with non-negative Ricci curvature, all these results are new. The proofs of these results strongly depend on the geometrical relation of differences and derivatives in the metric measure space and the Poincaré inequality.