"A priori" estimates for solutions of singular elliptic inequalities and applications

Harnack inequalities and Bôcher-type theorems for conformally invariant fully nonlinear degenerate elliptic equations

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions.

Inequalities for eigenvalues of fourth-order elliptic operators in divergence form on complete Riemannian manifolds

We prove some inequalities of Payne-Pólya-Weinberger-Yang type for eigenvalues of fourth-order elliptic operators in weighted divergence form on complete Riemannian manifolds which generalizes the corresponding result for the clamped plate problem. We also prove estimates for lower order eigenvalues that contain some of the estimates from the literature. As an application of our results, we obtain eigenvalues estimates for the bi-drifted Cheng-Yau operator.

Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part IV: Riesz transforms on manifolds and weights

This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume property and Gaussian upper bounds.

Grassmann流形上的几何性质与距离不等式

Geometric Properties and Distance Inequalities on Grassmannians

本文研究了Grassmann流形上元素的几何平均不等式，揭示了其作为黎曼流形的椭圆几何性质，包含半平行四边形定律、余弦定律和测地三角形不等式。

Weighted norm inequalities, off-diagonal estimates and elliptic operators

We give an overview of the generalized Calderón-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. $L^p-L^q$ off-diagonal estimates when $p\le q$ play an important role and we present them. They are particularly well suited to the semigroups generated by second order elliptic operators and the range of exponents $(p,q)$ rules the $L^p$ theory for many operators constructed from the semigr…

渐近非负Ricci曲率、椭圆Kato常数与等周不等式

Asymptotically non-negative Ricci curvature, elliptic Kato constant and isoperimetric inequalities

本文拓展ABP方法，证明含少量负曲率流形的等周不等式，通过椭圆Kato常数控制负曲率小性，并改进相关技术得到全流形Green函数新估计。

Spectral inequalities for elliptic pseudo-differential operators on closed manifolds

Let $(M,g)$ be a closed Riemannian manifold. The aim of this work is to prove the Lebeau-Robbiano spectral inequality for a positive elliptic pseudo-differential operator $E(x,D)$ on $M,$ of order $ν>0,$ in the Hörmander class $Ψ^ν_{ρ,δ}(M).$ In control theory this has been an open problem prior to this work. As an application of this fundamental result, we establish the null-controllability of the (fractional) heat equation associated with $E(x,D).$ The sensor $ω\subset M$ in the observability inequality is an open subset of $M.$ The obtained results (that are, the corresponding spectral ineq…