A Liouville theorem for weighted p−Laplace operator on smooth metric measure spaces

We prove that on a smooth metric measure space with m ?Bakry–Émery curvature bounded from below by ?(m ? 1)K for some constant K ≥0 (i.e., Ric_{f ,m} ≥?(m ? 1)K ), the following degenerate elliptic equation (0.1) has no nonconstant positive solution when p > 1 and constant λ _{f ,p} satisfies Our approach is based on the local Sobolev inequality and the Moser's iterative technique and is different from Cheng‐Yau's method, which was used by Wang‐Zhu in 2012 to derive a same Liouville theorem when 1 < p ≤2, Ric_{f ,m} ≥?(m ? 1)K and the sectional curvature is bounded from below. Copyright © 2016 John Wiley & Sons, Ltd.     

The Stepanov differentiability theorem in metric measure spaces

We extend Cheeger’s theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov’s condition. As a consequence, we obtain the analogue of Calderon’s differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincaré inequality. Communicated by Steven Krantz     

De Lellis-Topping type inequalities on smooth metric measure spaces

We obtain some De Lellis-Topping type inequalities on the smooth metric measure spaces, some of them are as generalization of De Lellis-Topping type inequality that was proved by X. Cheng [Ann. Global Anal. Geom., 2013, 43: 153-160].     

Universal inequalities on complete noncompact smooth metric measure spaces

In this paper, we obtain universal inequalities for the eigenvalues of the Dirichlet problem and clamped plate problem of drifting Laplacian on (\(n+1\))-dimensional (\(n\ge 4\)) complete noncompact simply connected smooth metric measure spaces which meet some conditions of the sectional curvature and radial weighted Ricci curvature.     

Liouville theorem for weighted p-Lichnerowicz equation on smooth metric measure space

In this paper, let $(M^n,g,d\mu )$ be n-dimensional noncompact metric measure space which satisfies Poincaré inequality with some Ricci curvature condition. We obtain a Liouville theorem for positive weak solutions to weighted p-Lichnerowicz equation

${△}_{p,f}v+c{v}^{\sigma }=0,$

where $c\le 0,m>n\ge 1,1<p<\frac{m?1+\sqrt{(m?1)(m+3)}}{2},\sigma \le p?1$ are real constants.     

A Ramsey theorem for metric spaces

We show the Ramsey property of the metric spaces $(V,d)$ where $1\le d(x,y)\le 3$ ($x\ne y$).     

A fixed point theorem in metric spaces

We discuss a fixed point theorem for a function f mapping a complete metric space X into itself. For all x ? X{x \in X} the iterates of f(x) are shown to converge to x_* = f(x_*){{x_{\star} = f(x_{\star})}} and an explicit estimate of the convergence rate is given.     

A mean value theorem for metric spaces

We present a form of the Mean Value Theorem (MVT) for a continuous function f between metric spaces, connecting it with the possibility to choose the relation of f in a homeomorphic way. We also compare our formulation of the MVT with the classic one when the metric spaces are open subsets of Banach spaces. As a consequence, we derive a version of the Mean Value Propriety for measure spaces that also possesses a compatible metric structure.     

A splitting theorem for the weighted measure

Let M be an n-dimensional complete noncompact Riemannian manifold, h be a smooth function on M and dμ = e ^h dV be the weighted measure. In this article, we prove that when the spectrum of the weighted Laplacian \triangle_m{\triangle_{\mu}} has a positive lower bound λ₁(M) > 0 and the m(m > n)-dimensional Bakry-émery curvature is bounded from below by -\fracm-1m-2l₁(M){-\frac{m-1}{m-2}\lambda_1(M)}, then M splits isometrically as R × N whenever it has two ends with infinite weighted volume, here N is an (n − 1)-dimensional compact manifold.     

Parabolic flow on metric measure spaces

We present parabolic equations on metric measure spaces. We prove existence and uniqueness of solutions. Under some assumptions the existence of global in time solution is proved. Moreover, regularity and qualitative property of the solutions are shown.     

Coercive inequalities on metric measure spaces

In this paper we study coercive inequalities on finite dimensional metric spaces with probability measures which do not have the volume doubling property.     

A differentiable structure for metric measure spaces

The main result of this paper is the provision of conditions under which a metric measure space admits a differentiable structure. This differentiable structure gives rise to a finite-dimensional L^∞ cotangent bundle over the given metric measure space and then to a Sobolev space H^1,p over the given metric measure space, the latter which is reflexive for p>1. This extends results of Cheeger (Geom. Funct. Anal. 9 (1999) (3) 428) to a wider collection of metric measure spaces.     

A curvature-dimension condition for metric measure spaces

We present a curvature-dimension condition $\mathit{CD}(K,N)$ for metric measure spaces $(M,\mathsf{d},m)$. In some sense, it will be the geometric counterpart to the Bakry–Émery [D. Bakry, M. Émery, Diffusions hypercontractives, in: Séminaire de Probabilités XIX, in: Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177–206. [1]] condition for Dirichlet forms. For Riemannian manifolds, it holds if and only if $\dim (M)?N$ and ${\mathrm{Ric}}_M(\xi ,\xi )?K?{|\xi |}^2$ for all $\xi \in \mathit{TM}$. The curvature bound introduced in [J. Lott, C. Villani, Ricci curvature for metric-measure spaces via optimal transport, Annals of Math., in press. [4]; K.T. Sturm, Generalized Ricci bounds and convergence of metric measure spaces, C. R. Acad. Sci. Paris, Ser. I 340 (2005) 235–238. [6]; K.T. Sturm, On the geometry of metric measure spaces. I, Acta Math., in press. [7]] is the limit case $\mathit{CD}(K,\infty )$.Our curvature-dimension condition is stable under convergence. Furthermore, it entails various geometric consequences e.g. the Bishop–Gromov theorem and the Bonnet–Myers theorem. In both cases, we obtain the sharp estimates known from the Riemannian case. To cite this article: K.-T. Sturm, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

Band-dominated operators on metric measure spaces

We generalise the notion of band-dominated operators originally introduced for the space ℓ_p(ℤⁿ) to the setup of metric measure spaces and show various algebraic properties of this space of operators. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A best approximation theorem in hyperconvex metric spaces

In this paper, we present a best approximation theorem for set-valued mappings in hyperconvex metric spaces, which generalize the well-known result of Kirk, Sims and Yuan [W.A. Kirk, B. Sims, X.Z. Yuan, The Knaster–Kuratowski and Mazurkiewicz theory in hyperconvex metric spaces and some of its applications, Nonlinear Anal. 39 (2000) 611–627].     

Diffusion polynomial frames on metric measure spaces

We construct a multiscale tight frame based on an arbitrary orthonormal basis for the L² space of an arbitrary sigma finite measure space. The approximation properties of the resulting multiscale are studied in the context of Besov approximation spaces, which are characterized both in terms of suitable K-functionals and the frame transforms. The only major condition required is the uniform boundedness of a summability operator. We give sufficient conditions for this to hold in the context of a very general class of metric measure spaces. The theory is illustrated using the approximation of characteristic functions of caps on a dumbell manifold, and applied to the problem of recognition of hand-written digits. Our methods outperforms comparable methods for semi-supervised learning.