We prove a $C^{1,1}$ estimate for solutions of complex Monge–Ampère equations on compact almost Hermitian manifolds. Using this $C^{1,1}$ estimate, we show the existence of $C^{1,1}$ solutions to the degenerate Monge–Ampère equations, the corresponding Dirichlet problems and the singular Monge–Ampère equations. We also study the singularities of the pluricomplex Green’s function. In addition, the proof of the above $C^{1,1}$ estimate is valid for a kind of complex Monge–Ampère-type equation. As a geometric application, we prove the $C^{1,1}$ regularity of geodesics in the space of Sasakian metrics.

We extend to the case of a $d$-dimensional compact connected oriented Riemannian manifold $\mathcal{ℳ}$ the theorem of A. Bondarenko, D. Radchenko and M. Viazovska (Ann. of Math. $\left(2\right)$ 178:2 (2013), 443–452) on the existence of $L$-designs consisting of $N$ nodes for any $N\ge C_{\mathcal{ℳ}}{L}^d$. For this, we need to prove a version of the Marcinkiewicz–Zygmund inequality for the gradient of diffusion polynomials.

The noncommutative $m$-hyperball, $m\in \mathbb{N}$, is defined by

$${\mathcal{𝒟}}_n^m\left(\mathcal{\mathscr{H}}\right):=\left\{X:=\left(X_1,\dots ,X_n\right)\in B{\left(\mathcal{\mathscr{H}}\right)}^n:{\left(id-{\mathrm{\Phi }}_X\right)}^k\left(I\right)\ge 0\text{for }1\le k\le m\right\},$$

where ${\mathrm{\Phi }}_X:B\left(\mathcal{\mathscr{H}}\right)\to B\left(\mathcal{\mathscr{H}}\right)$ is the completely positive map given by ${\mathrm{\Phi }}_X\left(Y\right):={\sum }_{i=1}^n{X}_{i}Y{X}_i^{\ast }$ for $Y\in B\left(\mathcal{\mathscr{H}}\right)$. Its right universal model is an $n$-tuple $\mathrm{\Lambda }=\left({\mathrm{\Lambda }}_1,\dots ,{\mathrm{\Lambda }}_n\right)$ of weighted right creation operators acting on the full Fock space $F^2\left(H_n\right)$ with $n$ generators. We prove that an operator $T\in B\left(F^2\left(H_n\right)\right)$ is a multi-Toeplitz operator with free pluriharmonic symbol on ${\mathcal{𝒟}}_n^m\left(\mathcal{\mathscr{H}}\right)$ if and only if it satisfies the Brown–Halmos-type equation

$${\text{Λ}}^{\prime \ast }T{\text{Λ}}^{\prime }=\underset{i=1}{\overset{n}{\oplus }}\left({\displaystyle \sum _{j=0}^{m-1}\left(\frac{m}{j+1}\right)}{\displaystyle \sum _{\alpha \in {𝔽}_n^{+},\left|\alpha \right|=j}{\text{Λ}}_{\alpha }T{\text{Λ}}_{\alpha }^{\ast }}\right),$$

where ${\mathrm{\Lambda }}^{\prime }$ is the Cauchy dual of $\mathrm{\Lambda }$ and ${\mathbb{𝔽}}_n^{+}$ is the free unital semigroup with $n$ generators. This is a noncommutative multivariable analogue of Louhichi and Olofsson characterization of Toeplitz operators with harmonic symbols on the weighted Bergman space $A_m\left(\mathbb{𝔻}\right)$, as well as Eschmeier and Langendörfer extension to the unit ball of ${ℂ}^n$.

All our results are proved in the more general setting of noncommutative polyhyperballs ${\mathit{D}}_{\mathit{n}}^{\mathit{m}}\left(\mathcal{\mathscr{H}}\right)$, $\mathit{n},\mathit{m}\in {\mathbb{N}}^k$, and are used to characterize the bounded free $k$-pluriharmonic functions with operator coefficients on polyhyperballs and to solve the associated Dirichlet extension problem. In particular, the results hold for the reproducing kernel Hilbert space with kernel

$${\kappa }_m(z,w):={\displaystyle \prod _{i=1}^k\frac{1}{{\left(1-{\overline{z}}_i{w}_i\right)}^{m_i}}},z,w\in {\mathbb{𝔻}}^k,$$

where $m_i\ge 1$. This includes the Hardy space, the Bergman space, and the weighted Bergman space over the polydisk.

The presented splitting lemma extends the techniques of Gromov and Forstnerič to glue local sections of a given analytic sheaf, a key step in the proof of all Oka principles. The novelty on which the proof depends is a lifting lemma for transition maps of coherent sheaves, which yields a reduction of the proof to the work of Forstnerič. As applications we get shortcuts in the proofs of Forster and Ramspott’s Oka principle for admissible pairs and of the interpolation property of sections of elliptic submersions, an extension of Gromov’s results obtained by Forstnerič and Prezelj.

Let $f:\mathcal{𝒟}\to \mathrm{\Omega }$ be a complex analytic function. The Julia quotient is given by the ratio between the distance of $f\left(z\right)$ to the boundary of $\mathrm{\Omega }$ and the distance of $z$ to the boundary of $\mathcal{𝒟}$. A classical Julia–Carathéodory-type theorem states that if there is a sequence tending to $\tau$ in the boundary of $\mathcal{𝒟}$ along which the Julia quotient is bounded, then the function $f$ can be extended to $\tau$ such that $f$ is nontangentially continuous and differentiable at $\tau$ and $f\left(\tau \right)$ is in the boundary of $\mathrm{\Omega }$. We develop an extended theory when $\mathcal{𝒟}$ and $\mathrm{\Omega }$ are taken to be the upper half-plane which corresponds to averaged boundedness of the Julia quotient on sets of controlled tangential approach, so-called $\lambda$-Stolz regions, and higher-order regularity, including but not limited to higher-order differentiability, which we measure using $\gamma$-regularity. Applications are given, including perturbation theory and moment problems.

This note concerns the asymptotics of the expected total Betti numbers of the nodal set for an important class of Gaussian ensembles of random fields on Riemannian manifolds. By working with the limit random field defined on the Euclidean space we were able to obtain a locally precise asymptotic result, though due to the possible positive contribution of large percolating components this does not allow us to infer a global result. As a by-product of our analysis, we refine the lower bound of Gayet and Welschinger for the important Kostlan ensemble of random polynomials and its generalisation to Kähler manifolds.

We study the quantization of coupled Kähler–Einstein (CKE) metrics, namely we approximate CKE metrics by means of the canonical Bergman metrics, called “balanced metrics”. We prove the existence and weak convergence of balanced metrics for the negative first Chern class, while for the positive first Chern class, we introduce an algebrogeometric obstruction which interpolates between the Donaldson–Futaki invariant and Chow weight. Then we show the existence and weak convergence of balanced metrics on CKE manifolds under the vanishing of this obstruction. Moreover, restricted to the case when the automorphism group is discrete, we also discuss approximate solutions and a gradient flow method towards the smooth convergence.

Let $\pi :\widetilde{\mathcal{𝒩}}\to \mathcal{𝒩}$ be a Riemannian covering, with $\mathcal{𝒩}$, $\widetilde{\mathcal{𝒩}}$ smooth compact connected Riemannian manifolds. If $\mathcal{ℳ}$ is an $m$-dimensional compact simply connected Riemannian manifold, and , we prove that every mapping $u\in W^{s,p}\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ has a lifting in $W^{s,p}$; i.e., we have $u=\pi \circ ũ$ for some mapping $ũ\in W^{s,p}\left(\mathcal{ℳ},\widetilde{\mathcal{𝒩}}\right)$. Combined with previous contributions of Bourgain, Brezis and Mironescu and Bethuel and Chiron, our result settles completely the question of the lifting in Sobolev spaces over covering spaces.

The proof relies on an a priori estimate of the oscillations of $W^{s,p}$ maps with and , in dimension $1$. Our argument also leads to the existence of a lifting when and , provided there is no topological obstruction on $u$; i.e., $u=\pi \circ ũ$ holds in this range provided $u$ is in the strong closure of $C^{\infty }\left(\mathcal{ℳ},\mathcal{𝒩}\right)$.

However, when , $sp=1$ and $m\ge 2$, we show that an (analytical) obstruction still arises, even in the absence of topological obstructions. More specifically, we construct some map $u\in W^{s,p}\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ in the strong closure of $C^{\infty }\left(\mathcal{ℳ},\mathcal{𝒩}\right)$ such that $u=\pi \circ ũ$ does not hold for any $ũ\in W^{s,p}\left(\mathcal{ℳ},\widetilde{\mathcal{𝒩}}\right)$.

We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and, in particular, is not necessarily Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors $d$-regular set $E$, if we consider the set $\mathcal{ℬ}$ of surface cubes (in the sense of Christ and David) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $\mathcal{ℬ}$ satisfies a Carleson packing condition, that is, for any surface cube $R$,

$$\sum _{\genfrac{}{}{0ex}{}{Q\subseteq R}{Q\in \mathcal{ℬ}}}{\left(diamQ\right)}^d\lesssim {\left(diamR\right)}^d.$$

We show that, for lower content regular sets that aren’t necessarily Ahlfors regular, if ${\beta }_E\left(R\right)$ denotes the square sum of $\beta$-numbers over subcubes of $R$ as in the traveling salesman theorem for higher-dimensional sets, presented by Azzam and Schul, then

$${\mathcal{\mathscr{H}}}^d\left(R\right)+\sum _{\genfrac{}{}{0ex}{}{Q\subseteq R}{Q\in \mathcal{ℬ}}}{\left(diamQ\right)}^d\sim {\beta }_E\left(R\right).$$

We prove similar results for other uniform rectifiability criteria, such as the local symmetry, local convexity, and generalized weak exterior convexity conditions.

En route, we show how to construct a corona decomposition of any lower content regular set by Ahlfors regular sets, similar to the classical corona decomposition of UR sets by Lipschitz graphs developed by David and Semmes.

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space $H^1$. For complete Nevanlinna–Pick spaces $\mathcal{\mathscr{H}}$, we characterize all multipliers of the weak product space $\mathcal{\mathscr{H}}\odot \mathcal{\mathscr{H}}$. In particular, we show that if $\mathcal{\mathscr{H}}$ has the so-called column-row property, then the multipliers of $\mathcal{\mathscr{H}}$ and of $\mathcal{\mathscr{H}}\odot \mathcal{\mathscr{H}}$ coincide. This result applies in particular to the classical Dirichlet space and to the Drury–Arveson space on a finite-dimensional ball. As a key device, we exhibit a natural operator space structure on $\mathcal{\mathscr{H}}\odot \mathcal{\mathscr{H}}$, which enables the use of dilations of completely bounded maps.

Let $\mathrm{\Sigma }$ be a complete Riemannian manifold with the volume-doubling property and the uniform Neumann–Poincaré inequality. We show that any positive minimal graphic function on $\mathrm{\Sigma }$ is constant.

Let $X$ be a compact Kähler manifold with a given ample line bundle $L$. Donaldson proved an inequality between the Calabi energy of a Kähler metric in $c_1\left(L\right)$ and the negative of normalized Donaldson–Futaki invariants of test configurations of $\left(X,L\right)$. He also conjectured that the bound is sharp.

We prove a metric analogue of Donaldson’s conjecture; we show that if we enlarge the space of test configurations to the space of geodesic rays in ${\mathcal{ℰ}}^2$ and replace the Donaldson–Futaki invariant by the radial Mabuchi K-energy $\mathit{M}$, then a similar bound holds and the bound is indeed sharp. Moreover, we construct explicitly a minimizer of $\mathit{M}$. On a Fano manifold, a similar sharp bound for the Ricci–Calabi energy is also derived.