We prove the uniform estimates for the resolvent ${(\mathrm{\Delta }-\alpha )}^{-1}$ as a map from $L^q$ to $L^{q^{\prime }}$ on real hyperbolic space ${ℍ}^n$, where $\alpha \in ℂ\setminus [{(n-1)}^2∕4,\infty )$ and . In contrast with analogous results on Euclidean space ${ℝ}^n$, the exponent $q$ here can be arbitrarily close to $2$. This striking improvement is due to two non-Euclidean features of hyperbolic space: the Kunze–Stein phenomenon and the exponential decay of the spectral measure. In addition, we apply this result to the study of eigenvalue bounds of the Schrödinger operator with a complex potential. The improved Sobolev inequality results in a better long-range eigenvalue bound on ${ℍ}^n$ than that on ${ℝ}^n$.

We obtain existence and global regularity estimates for gradients of solutions to quasilinear elliptic equations with measure data whose prototypes are of the form $-\text{div}(|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u)=\delta |\mathrm{\nabla }u{|}^q+\mu$ in a bounded domain $\mathrm{\Omega }\subset {ℝ}^n$ potentially with nonsmooth boundary. Here either $\delta =0$ or $\delta =1$, $\mu$ is a finite signed Radon measure in $\mathrm{\Omega }$, and $q\ge 1$. Our main concern is to extend earlier results to the strongly singular case . In particular, in the case $\delta =1$ which corresponds to a Riccati-type equation, we settle the question of solvability that has been raised for some time in the literature.

Here we are concerned with the compactness of metrics on the disk with prescribed Gaussian and geodesic curvatures. We consider a blowing up sequence of metrics and give a precise description of its asymptotic behavior. In particular, the metrics blow-up at a unique point on the boundary and we are able to give necessary conditions on its location. It turns out that such conditions depend locally on the Gaussian curvatures but they depend on the geodesic curvatures in a nonlocal way. This is a novelty with respect to the classical Nirenberg problem where the blow-up conditions are local, and this new aspect is driven by the boundary condition.

We show that for any Carnot group $G$ there exists a natural number $D_G$ such that for any the metric space $(G,d_G^{1-𝜀})$ admits a bi-Lipschitz embedding into ${ℝ}^{D_G}$ with distortion $O_G({𝜀}^{-1∕2})$. We do this by building on the approach of T. Tao (Rev. Mat.Iberoam. 37:1 (2021), 1–44), who established the above assertion when $G$ is the Heisenberg group using a new variant of the Nash–Moser iteration scheme combined with a new extension theorem for orthonormal vector fields. Beyond the need to overcome several technical issues that arise in the more general setting of Carnot groups, a key point where our proof departs from that of Tao is in the proof of the orthonormal vector field extension theorem, where we incorporate the Lovász local lemma and the concentration of measure phenomenon on the sphere in place of Tao’s use of a quantitative homotopy argument.

A basic property and useful tool in the theory of Sobolev spaces is the density of smooth compactly supported functions in the space $W^{k,p}({ℝ}^n)$ (i.e., the functions with weak derivatives of orders $0$ to $k$ in $L^p$). On Riemannian manifolds, it is well known that the same property remains valid under suitable geometric assumptions. However, on a complete noncompact manifold it can fail to be true in general, as we prove here. This settles an open problem raised for instance by E. Hebey (Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lect. Notes Math. 5 (1999), 48–49).

We show that for any there exist $\alpha$-Hölder continuous weak solutions of the three-dimensional incompressible Euler equation, which satisfy the local energy inequality and strictly dissipate the total kinetic energy. The proof relies on the convex integration scheme and the main building blocks of the solution are various Mikado flows with disjoint supports in space and time.

We provide quantitative inductive estimates for Green’s functions of matrices with (sub-)exponentially decaying off-diagonal entries in arbitrary dimensions. Together with Cartan’s estimates and discrepancy estimates, we establish explicit bounds for the large-deviation theorem for non-self-adjoint Toeplitz operators. As applications, we obtain the modulus of continuity of the integrated density of states with explicit bounds and the pure point spectrum property for analytic quasiperiodic operators. Moreover, our inductions are self-improved and work for perturbations with low-complexity interactions.