In this paper we continue our study of local rigidity for maps of CR submanifolds of the complex space. We provide a linear sufficient condition for local rigidity of finitely nondegenerate maps between minimal CR manifolds. Furthermore, we show higher order infinitesimal conditions can be used to give a characterization of local rigidity.

We introduce the so-called doubling metric on the collection of non-empty bounded open subsets of a metric space. Given an open subset $\mathbb{U}$ of a metric space $X$, the predecessor $\mathbb{U}_\ast$ of $\mathbb{U}$ is defined by doubling the radii of all open balls contained inside $\mathbb{U}$, and taking their union. The predecessor of $\mathbb{U}$ is an open set containing $\mathbb{U}$. The directed doubling distance between $\mathbb{U}$ and another subset $\mathbb{V}$ is the number of times that the predecessor operation needs to be applied to $\mathbb{U}$ to obtain a set that contains $\mathbb{V}$. Finally, the doubling distance between open sets $\mathbb{U}$ and $\mathbb{V}$ is the maximum of the directed distance between $\mathbb{U}$ and $\mathbb{V}$ and the directed distance between $\mathbb{V}$ and $\mathbb{U}$.

We discuss the Kodaira problem for uniruled Kähler spaces. Building on a construction due to Voisin, we give an example of a uniruled Kähler space $X$ such that every run of the $K_X$-MMP immediately terminates with a Mori fibre space, yet $X$ does not admit an algebraic approximation. Our example also shows that for a Mori fibration, approximability of the base does not imply approximability of the total space.

Let $\mathfrak{g}$ be the Witt algebra or the positive Witt algebra. It is well known that the enveloping algebra $U(\mathfrak{g})$ has intermediate growth and thus infinite Gelfand–Kirillov (GK-) dimension. We prove that the GK-dimension of $U(\mathfrak{g})$ is just infinite in the sense that any proper quotient of $U(\mathfrak{g})$ has polynomial growth. This proves a conjecture of Petukhov and the second named author for the positive Witt algebra. We also establish the corresponding results for quotients of the symmetric algebra $S(\mathfrak{g})$ by proper Poisson ideals.

In fact, we prove more generally that any central quotient of the universal enveloping algebra of the Virasoro algebra has just infinite GK-dimension. We give several applications. In particular, we easily compute the annihilators of Verma modules over the Virasoro algebra.

Let ψ be a conformal map on $\mathbb{D}$ with $\psi \left(0\right)=0$ and let ${F_{\alpha }}=\left\{z\in \mathbb{D}:\left|\psi \left(z\right)\right|=\alpha \right\}$ for $\alpha > 0$. Denote by ${H^{p}}\left(\mathbb{D}\right)$ the classical Hardy space with exponent $p > 0$ and by $\mathtt{h}\left(\psi \right)$ the Hardy number of ψ. Consider the limits

\[L:=\underset{\alpha \to +\infty }{\lim }\left(\log {\omega _{\mathbb{D}}}{\left(0,{F_{\alpha }}\right)^{-1}}\Big/ \log \alpha \right),\hspace{1em}\mu :=\underset{\alpha \to +\infty }{\lim }\left({d_{\mathbb{D}}}\left(0,{F_{\alpha }}\right)\big/ \log \alpha \right),\]

where ${\omega _{\mathbb{D}}}\left(0,{F_{\alpha }}\right)$ denotes the harmonic measure at 0 of ${F_{\alpha }}$ and ${d_{\mathbb{D}}}\left(0,{F_{\alpha }}\right)$ denotes the hyperbolic distance between 0 and ${F_{\alpha }}$ in $\mathbb{D}$. We study a problem posed by P. Poggi-Corradini. What is the relation between L, μ and $\mathtt{h}\left(\psi \right)$? Motivated by the result of Kim and Sugawa that $\mathtt{h}\left(\psi \right)={\liminf _{\alpha \to +\infty }}(\log {\omega _{\mathbb{D}}}{\left(0,{F_{\alpha }}\right)^{-1}}{}\log \alpha )$, we show that $\mathtt{h}\left(\psi \right)={\liminf _{\alpha \to +\infty }}\left({d_{\mathbb{D}}}\left(0,{F_{\alpha }}\right)\big/\log \alpha \right)$. We also provide conditions for the existence of L and μ and for the equalities $L=\mu =\mathtt{h}\left(\psi \right)$. Poggi-Corradini proved that $\psi \notin {H^{\mu }}\left(\mathbb{D}\right)$ for a wide class of conformal maps ψ. We present an example of ψ such that $\psi \in {H^{\mu}}\left(\mathbb{D}\right)$.

The classical Morse theory proceeds by considering sublevel sets $f^{-1} (-\infty, a]$ of a Morse function $f : M \to \mathbb{R}$, where $M$ is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets $f^{-1} (a)$ and give conditions under which the topology of $f^{-1} (a)$ changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse functions, the topology of a regular level $f^{-1} (a)$ always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold $M$. When $f$ is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the base space. (Counter-)examples and applications to celestial mechanics are also discussed.

A condition on a Banach function space $X$ is given under which the Coifman–Fefferman and Fefferman–Stein inequalities on $X$ are equivalent.

The aim of this paper is to study the Lelong number, the integrability index and the Monge–Ampère mass at the origin of an $S^1$-invariant plurisubharmonic function on a balanced domain in $\mathbb{C}^n$ under the Schwarz symmetrization. We prove that $n$ times the integrability index is exactly the Lelong number of the symmetrization, and if the function is further toric with a single pole at the origin, then the Monge–Ampère mass is always decreasing under the symmetrization.

For the class of graph-directed self-similar measures on $\mathbf{R}$, which could have overlaps but are essentially of finite type, we set up a framework for deriving a closed formula for the spectral dimension of the Laplacians defined by these measures. For the class of finitely ramified graph-directed self-similar sets, the spectral dimension of the associated Laplace operators has been obtained by Hambly and Nyberg [6]. The main novelty of our results is that the graph-directed self-similar measures we consider do not need to satisfy the graph open set condition.

We investigate so-called Laplace–Carleson embeddings for large exponents. In particular, we extend some results by Jacob, Partington, and Pott. We also discuss some related results for Sobolev– and Besov spaces, and mapping properties of the Fourier transform. These variants of the Hausdorff–Young theorem appear difficult to find in the literature. We conclude the paper with an example related to an open problem.