We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schrödinger operators. Such refinements quantify near-optimizers of the Strichartz estimate and play a pivotal part in the global theory of mass-critical NLS. On one hand, the harmonic analysis is quite subtle in the $L^2$-critical setting due to an enormous group of symmetries, while on the other hand, the space-time Fourier analysis employed by the existing approaches to the constant-coefficient equation are not adapted to nontranslation-invariant situations, especially with potentials as large as those considered in this article.

Using phase-space techniques, we reduce to proving certain analogues of (adjoint) bilinear Fourier restriction estimates. Then we extend Tao’s bilinear restriction estimate for paraboloids to more general Schrödinger operators. As a particular application, the resulting inverse Strichartz theorem and profile decompositions constitute a key harmonic analysis input for studying large-data solutions to the $L^2$-critical NLS with a harmonic oscillator potential in dimensions $\ge 2$. This article builds on recent work of Killip, Visan, and the author in one space dimension.

We consider the Cauchy problem for the defocusing power-type nonlinear wave equation in $\left(1+3\right)$-dimensions for energy subcritical powers $p$ in the superconformal range $3<p<5$. We prove that any solution is global-in-time and scatters to free waves in both time directions as long as its critical Sobolev norm stays bounded on the maximal interval of existence.

The goal of the paper is to prove an exact representation formula for the Laplacian of the distance (and more generally for an arbitrary $1$-Lipschitz function) in the framework of metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense (more precisely in essentially nonbranching $\mathsf{MCP}\left(K,N\right)$-spaces). Such a representation formula makes apparent the classical upper bounds together with lower bounds and a precise description of the singular part. The exact representation formula for the Laplacian of a general 1-Lipschitz function holds also (and seems new) in a general complete Riemannian manifold.

We apply these results to prove the equivalence of $\mathsf{CD}\left(K,N\right)$ and a dimensional Bochner inequality on signed distance functions. Moreover we obtain a measure-theoretic splitting theorem for infinitesimally Hilbertian, essentially nonbranching spaces satisfying $\mathsf{MCP}\left(0,N\right)$.

We consider the volume-preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long-time asymptotics approach round spheres. The proofs are based on a priori estimates on the inner and outer radii of the solutions.

Let $p\in \left(1,\infty \right)\setminus \left\{2\right\}$. We show that every homomorphism from a $C^{\ast }$-algebra $\mathcal{𝒜}$ into $B\left(l^p\left(J\right)\right)$ satisfies a compactness property where $J$ is any set. As a consequence, we show that a $C^{\ast }$-algebra $\mathcal{𝒜}$ is isomorphic to a subalgebra of $B\left(l^p\left(J\right)\right)$, for some set $J$, if and only if $\mathcal{𝒜}$ is residually finite-dimensional.

We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge–Ampère type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We derive a differential Harnack inequality for a special class of functions that solve the linearized problem. Using this Harnack inequality and certain techniques specific to mass transport, we control the oscillation in time of solutions to the parabolic equation, and obtain exponential convergence. Additionally, in the course of the proof, we present a connection with the pseudo-Riemannian framework introduced by Kim and McCann in the context of optimal transport, which is interesting in its own right.

We prove that $\mathcal{𝒵}$-stable, simple, separable, nuclear, nonunital $C^{\ast }$-algebras have nuclear dimension at most 1. This completes the equivalence between finite nuclear dimension and $\mathcal{𝒵}$-stability for simple, separable, nuclear, nonelementary $C^{\ast }$-algebras.

We revisit the question of regularity for minimizers of scalar autonomous integral functionals with so-called $\left(p,q\right)$-growth. In particular, we establish Lipschitz regularity under the condition $\frac{q}{p}<1+\frac{2}{n-1}$ for $n\ge 3$, improving a classical result due to Marcellini (J. Differential Equations 90:1 (1991), 1–30).