Related papers: $\varepsilon$-approximability and Quantitative Fat…

In this paper, we prove the Lipschitz regularity of continuous harmonic maps from an finite dimensional Alexandrov space to a compact smooth Riemannian manifold. This solves a conjecture of F. H. Lin in \cite{lin97}. The proof extends the…

In this paper, we investigate some properties on harmonic functions and solutions to Poisson equations. First, we will discuss the Lipschitz type spaces on harmonic functions. Secondly, we establish the Schwarz-Pick lemma for harmonic…

We give a definition of convergence of differential of Lipschitz functions with respect to measured Gromov-Hausdorff topology. As their applications, we give a characterization of harmonic functions with polynomial growth on asymptotic…

In this paper we consider finite dimensional dynamical systems generated by a Lipschitz function. We prove a version of the Whitney's Extension Theorem on compact manifolds to obtain a version of the well-known Lambda Lemma for Lipschitz…

We define self-adjoint extensions of the Hodge Laplacian on Lipschitz domains in Riemannian manifolds, corresponding to either the absolute or the relative boundary condition, and examine regularity properties of these operators' domains…

Via a functor from certain Lorentzian to Riemannian manifolds, we obtain a finiteness result.

In this article, we extend a result of L. Loomis and W. Rudin, regarding boundary behavior of positive harmonic functions on the upper half space $\R_+^{n+1}$. We show that similar results remain valid for more general approximate…

We prove some Liouville-type theorems for positive harmonic functions on compact Riemannian manifolds with nonnegative Ricci curvature and strictly convex boundary, thereby confirming some cases of Wang's conjecture (J. Geom. Anal. 31,…

We prove that a compact Riemannian manifold of dimension $m \geq 3$ with harmonic curvature and $\lfloor\frac{m-1}{2}\rfloor$-positive curvature operator has constant sectional curvature, extending the classical Tachibana theorem for…

We prove a generalized version of the Quantum Ergodicity Theorem on smooth compact Riemannian manifolds without boundary. We apply it to prove some asymptotic properties on the distribution of typical eigenfunctions of the Laplacian in…

In this paper, we prove some splitting results for manifolds supporting a non-constant infinity harmonic function which has at most linear growth on one side. Manifolds with non-negative Ricci or sectional curvature are considered. In…

We prove an approximation theorem on a class of domains in $\mathbb{C}^n$ on which the $\overline{\partial}$-problem is solvable in $L^{\infty}$. Furthermore, as a corollary, we obtain a version of the Axler-\v{C}u\v{c}kovi\'c-Rao Theorem…

For a bounded domain equipped with a piecewise Lipschitz continuous Riemannian metric g, we consider harmonic map from $(\Omega, g)$ to a compact Riemannian manifold $(N,h)\subset\mathbb R^k$ without boundary. We generalize the notion of…

We prove a version of the Fatou theorem for bounded functions with bounded d_J-bar diferential on wedge-type domains in an almost complex manifold.

We study the Lusin approximation problem for real-valued measurable functions on Carnot groups. We prove that k-approximate differentiability almost everywhere is equivalent to admitting a Lusin approximation by $C^{k}_{\mathbb{G}}$ maps.…

The classical Fatou theorem identifies bounded harmonic functions on the unit disk with bounded measurable functions on the boundary circle. We extend this theorem to bounded harmonic maps.

We study the asymptotic Dirichlet problem for $\mathcal{A}$-harmonic functions on a Cartan-Hadamard manifold whose radial sectional curvatures outside a compact set satisfy an upper bound $$ K(P)\le - \frac{1+\varepsilon}{r(x)^2 \log r(x)}…

We extend the result of Lavrentiev which asserts that the harmonic measure and the arc-length measure are $A_\infty$ equivalent in a chord-arc Jordan domain. By using this result we extend the classical result of Lindel\"of to the class of…

It is developed the theory of the Dirichlet problem for harmonic functions. On this basis, for the nondegenerate Beltrami equations in the quasidisks and, in particular, in the smooth domains, it is proved the existence of regular solutions…

We generalize the classical Blaschke Rolling Theorem to convex domains in Riemannian manifolds of bounded sectional curvature and arbitrary dimension. Our results are sharp and, in this sharp form, are new even in the model spaces of…