We prove a C1,1 estimate for solutions of complex Monge–Ampère equations on compact almost Hermitian manifolds. Using this C1,1 estimate, we show the existence of C1,1 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

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

The noncommutative m-hyperball, m ∈ ℕ, is defined by 𝒟nm(ℋ) := {X := (X 1,…,Xn) ∈ B(ℋ)n : (id−Φ X)k(I) ≥ 0 for 1 ≤ k ≤ m}, where ΦX : B(ℋ) → B(ℋ) is the completely positive map given by ΦX(Y ) := ∑ i=1nXiY Xi∗ for Y ∈ B(ℋ). Its right universal model is an n-tuple Λ = (Λ1,…,Λn) of weighted right creation operators acting on the full Fock space F2(Hn) with n generators. We prove that an operator T ∈

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

Let f : 𝒟→ Ω be a complex analytic function. The Julia quotient is given by the ratio between the distance of f(z) to the boundary of Ω and the distance of z to the boundary of 𝒟. A classical Julia–Carathéodory-type theorem states that if there is a sequence tending to τ in the boundary of 𝒟 along which the Julia quotient is bounded, then the function f can be extended to τ such that f is nontangentially

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

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

Let π : 𝒩˜ →𝒩 be a Riemannian covering, with 𝒩, 𝒩˜ smooth compact connected Riemannian manifolds. If ℳ is an m-dimensional compact simply connected Riemannian manifold, 0 < s < 1 and 2 ≤ sp < m, we prove that every mapping u ∈ Ws,p(ℳ,𝒩 ) has a lifting in Ws,p; i.e., we have u = π ∘ũ for some mapping ũ ∈ Ws,p(ℳ, 𝒩˜). Combined with previous contributions of Bourgain, Brezis and Mironescu and Bethuel

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 ℬ of surface cubes (in the sense of Christ and David) near which E does not look approximately like a union of planes, then

In the theory of reproducing kernel Hilbert spaces, weak product spaces generalize the notion of the Hardy space H1 . For complete Nevanlinna–Pick spaces ℋ, we characterize all multipliers of the weak product space ℋ⊙ℋ. In particular, we show that if ℋ has the so-called column-row property, then the multipliers of ℋ and of ℋ⊙ℋ coincide. This result applies in particular to the classical Dirichlet space

Let Σ 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 Σ 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 c1(L) and the negative of normalized Donaldson–Futaki invariants of test configurations of (X,L). 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

Ansatze are constructed under which the solutions of 11-dimensional supergravity must be stationary points of a parabolic flow on a Riemannian manifold M10−p . This parabolic flow turns out to be the Ricci flow coupled to a scalar field, a (3−p)-form, and a 4-form. This allows the introduction of techniques from parabolic partial differential equations to the search of solutions to 11-dimensional supergravity

Consider the stationary Boltzmann equation in 2-dimensional convex domains with diffusive boundary condition. We establish the hydrodynamic limits while the boundary layers are present, and derive the steady Navier–Stokes–Fourier system with nonslip boundary conditions. Our contribution focuses on novel weighted W1,∞ estimates for the 𝜖-Milne problem with geometric correction. Also, we develop stronger

Let u denote a solution to a rotationally invariant Hessian equation F(D2u) = 0 on a bounded simply connected domain Ω ⊂ ℝ2, with constant Dirichlet and Neumann data on ∂Ω. We prove that if u is real analytic and not identically zero, then u is radial and Ω is a disk. The fully nonlinear operator F≢0 is of general type and, in particular, not assumed to be elliptic. We also show that the result is

We introduce Calderón–Zygmund theory for singular stochastic integrals with operator-valued kernel. In particular, we prove Lp-extrapolation results under a Hörmander condition on the kernel. Sparse domination and sharp weighted bounds are obtained under a Dini condition on the kernel, leading to a stochastic version of the solution to the A2-conjecture. The results are applied to obtain p-independence

Krein–de Branges spectral theory establishes a correspondence between the class of differential operators called canonical Hamiltonian systems and measures on the real line with finite Poisson integral. We further develop this area by giving a description of canonical Hamiltonian systems whose spectral measures have logarithmic integral converging over the real line. This result can be viewed as a

We consider the energy-critical heat equation in ℝn for n ≥ 6 ut = Δu + |u| 4 n−2 u in ℝn × (0,∞), u( ⋅,0) = u0 in ℝn, which corresponds to the L2-gradient flow of the Sobolev-critical energy J(u) = ∫ ℝne[u],e[u] := 1 2|∇u|2 −n − 2 2n |u| 2n n−2 . Given any k ≥ 2 we find an initial condition u0 that leads to sign-changing solutions with multiple blow-up at a single point (tower of bubbles) as t → +∞

We consider the singular set in the thin obstacle problem with weight |xn+1|a for a ∈ (−1,1), which arises as the local extension of the obstacle problem for the fractional Laplacian (a nonlocal problem). We develop a refined expansion of the solution around its singular points by building on the ideas introduced by Figalli and Serra to study the fine properties of the singular set in the classical

We initiate the study of (1+2)-dimensional wave maps on a curved space time in the low-regularity setting. Our main result asserts that in this context the wave maps equation is locally well-posed at almost critical regularity. As a key part of the proof of this result, we generalize the classical optimal bilinear L2 estimates for the wave equation to variable coefficients by means of wave packet decompositions

This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension d = 1 for 2 × 2 hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive

We prove global-in-time Strichartz-type estimates for the Schrödinger equation on manifolds of the form ℝn × 𝕋d , where 𝕋d is a d-dimensional torus. Our results generalize and improve a global space-time estimate for the Schrödinger equation on ℝ × 𝕋2 due to Z. Hani and B. Pausader. As a consequence we prove global existence and scattering in H1∕2 for small initial data for the quintic NLS on ℝ

We consider large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift (j 2) ω + jy + x mod 1 for irrational ω. We prove that the eigenvalue distribution of these matrices converges to the corresponding distribution from random matrix theory on the global scale, namely, the Wigner semicircle law for square matrices and the Marchenko–Pastur

We revisit Matui and Sato’s notion of property (SI) for C∗-algebras and C∗-dynamics. More specifically, we generalize the known framework to the case of C∗-algebras with possibly unbounded traces. The novelty of this approach lies in the equivariant context, where none of the previous work allows one to (directly) apply such methods to actions of amenable groups on highly nonunital C∗-algebras, in

We initiate the study of multilayer potential operators associated with any given homogeneous constant-coefficient higher-order elliptic system L in an open set Ω ⊆ ℝn satisfying additional assumptions of a geometric measure theoretic nature. We develop a Calderón–Zygmund-type theory for this brand of singular integral operators acting on Whitney arrays, starting with the case when Ω is merely of locally

We first establish short-time existence and a Shi-type estimate of second Ricci flow on complete noncompact Hermitian manifolds. As an application, we use the second Ricci flow to discuss the existence of a Kähler–Ricci metric on complete noncompact Hermitian manifolds.

We consider the problem of locating and reconstructing the geometry of a penetrable obstacle from time-domain measurements of causal waves. More precisely, we assume that we are given the scattered field due to point sources placed on a surface enclosing the obstacle and that the scattered field is measured on the same surface. From these multistatic scattering data we wish to determine the position

We study a class of Monge–Ampère-type functionals arising from the Lp dual Minkowski problem in convex geometry. As an application, we obtain some existence and nonuniqueness results for the problem.

Let A( ⋅) be an (n+1)×(n+1) uniformly elliptic matrix with Hölder continuous real coefficients and let ℰA(x,y) be the fundamental solution of the PDE divA( ⋅)∇u = 0 in ℝn+1 . Let μ be a compactly supported n-AD-regular measure in ℝn+1 and consider the associated operator Tμf(x) = ∫ ∇xℰA(x,y)f(y)dμ(y). We show that if Tμ is bounded in L2(μ), then μ is uniformly n-rectifiable. This extends the solution

In view of classical results of Masur and Veech almost every element in the moduli space of compact translation surfaces is recurrent; i.e., its Teichmüller positive semiorbit returns to a compact subset infinitely many times. We focus on the problem of recurrence for elements of smooth curves in the moduli space. We give an effective criterion for the recurrence of almost every element of a smooth

This paper establishes a theory of nonlinear spectral decompositions by considering the eigenvalue problem related to an absolutely one-homogeneous functional in an infinite-dimensional Hilbert space. This approach is motivated by works for the total variation, where interesting results on the eigenvalue problem and the relation to the total variation flow have been proven previously, and by recent

We estimate the L2 norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace–Beltrami operator on the standard flat torus 𝕋d, d ≥ 2. We reduce getting correct bounds to counting lattice points in the intersection of some ν-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In

We consider approximations introduced by Sacks and Uhlenbeck of the harmonic energy for maps from S2 into S2. We continue our previous analysis (J. Differential Geom. 116:2 (2020), 321–348) on limits of α-harmonic maps with uniformly bounded energy. Using a recent energy identity of Li and Zhu (Ann. Inst. H. Poincaré Anal. Non Linéaire 36:1 (2019), 103–118), we obtain an optimal gap theorem for the

We consider a family of nonlocal problems that model the effects of transport and vortex stretching in the incompressible Euler equations. Using modulation techniques, we establish stable self-similar blow-up near a family of known self-similar blow-up solutions.

We consider the following nonlinear Schrödinger equation of derivative type: i∂tu + ∂x2u + i|u|2∂ xu + b|u|4u = 0,(t,x) ∈ ℝ × ℝ,b ∈ ℝ. If b = 0, this equation is known as a standard derivative nonlinear Schrödinger equation (DNLS), which is mass-critical and completely integrable. The equation above can be considered as a generalized equation of DNLS while preserving mass-criticality and Hamiltonian

We develop a novel decomposition of the monomials in order to study the set of monomial convergence for spaces of holomorphic functions over ℓr for 1 < r ≤ 2. For Hb(ℓr), the space of entire functions of bounded type in ℓr, we prove that monHb(ℓr) is exactly the Marcinkiewicz sequence space mΨr, where the symbol Ψr is given by Ψr(n) := log(n + 1)1−1∕r for n ∈ ℕ0. For the space of m-homogeneous polynomials

For distributions, we build a theory of higher-order pointwise differentiability comprising, for order zero, Łojasiewicz’s notion of point value. Results include Borel regularity of differentials, higher-order rectifiability of the associated jets, a Rademacher–Stepanov-type differentiability theorem, and a Lusin-type approximation. A substantial part of this development is new also for zeroth order

We are concerned with the short-time observability constant of the heat equation from a subdomain ω of a bounded domain ℳ. The constant is of the form e𝔎∕T , where 𝔎 depends only on the geometry of ℳ and ω. Luc Miller (J. Differential Equations 204:1 (2004), 202–226) conjectured that 𝔎 is (universally) proportional to the square of the maximal distance from ω to a point of ℳ. We show in particular

We prove the global stability of the Minkowski space viewed as the trivial solution of the Einstein–Vlasov system. To estimate the Vlasov field, we use the vector field and modified vector field techniques we previously developed in 2017. In particular, the initial support in the velocity variable does not need to be compact. To control the effect of the large velocities, we identify and exploit several

We consider the following perturbed critical Dirichlet problem involving the Hardy–Schrödinger operator: −Δu − γ(u∕|x|2) − 𝜖u = |u| 4 N−2 u in Ω, u = 0  on ∂Ω, when 𝜖 > 0 is small, γ < (N−2)2 4 , and where Ω ⊂ ℝN , N ≥ 3, is a smooth bounded domain with 0 ∈ Ω. We show that there exists a sequence (γj)j=1∞ in (−∞,0] with limj→∞γj = −∞ such that, if γ≠γj for any j and γ ≤ (N−2)2 4 − 1, then the above

We study the regularity of minima of scalar variational integrals of p-growth, 1 < p < ∞, in the Heisenberg group and prove the Hölder continuity of horizontal gradient of minima.

We derive local C2 estimates for complete noncompact translating solitons of the Gauss curvature flow in ℝ3 which are graphs over a convex domain Ω. This is closely is related to deriving local C1,1 estimates for the degenerate Monge–Ampère equation. As a result, given a weakly convex bounded domain Ω, we establish the existence of a Cloc1,1 translating soliton. In particular, when the boundary ∂Ω

We consider the dispersive logarithmic Schrödinger equation in a semiclassical scaling. We extend the results of Carles and Gallagher (Duke Math. J. 167:9 (2018), 1761–1801) about the large-time behavior of the solution (dispersion faster than usual with an additional logarithmic factor and convergence of the rescaled modulus of the solution to a universal Gaussian profile) to the case with semiclassical

We establish optimal local and global Besov–Lipschitz and Triebel–Lizorkin estimates for the solutions to linear hyperbolic partial differential equations. These estimates are based on local and global estimates for Fourier integral operators that span all possible scales (and in particular both Banach and quasi-Banach scales) of Besov–Lipschitz spaces Bp,qs(ℝn) and certain Banach and quasi-Banach

Żuk proved that if a finitely generated group admits a Cayley graph such that the Laplacian on the links of this Cayley graph has a spectral gap > 1 2, then the group has property (T), or equivalently, every affine isometric action of the group on a Hilbert space has a fixed point. We prove that the same holds for affine isometric actions of the group on a uniformly curved Banach space (for example

We consider a compact manifold with a piece isometric to a (finite-length) cylinder. By making the length of the cylinder tend to infinity, we obtain an asymptotic gluing formula for the zeta determinant of the Hodge Laplacian and an asymptotic expansion of the torsion of the corresponding long exact sequence of cohomology equipped with L2-metrics. As an application, we give a purely analytic proof

We study the local propagation of conormal singularities for solutions of semilinear wave equations □u = P(y,u), where P(y,u) is a polynomial of degree N ≥ 3 in u with C∞(ℝy3) coefficients. We know from the work of Melrose and Ritter and Bony that if u is conormal to three waves which intersect transversally at point q, then after the triple interaction u(y) is a conormal distribution with respect

We establish interior Schauder estimates for kinetic equations with integrodifferential diffusion. We study equations of the form ft + v ⋅∇xf = ℒvf + c, where ℒv is an integrodifferential diffusion operator of order 2s acting in the v-variable. Under suitable ellipticity and Hölder continuity conditions on the kernel of ℒv, we obtain an a priori estimate for f in a properly scaled Hölder space.

Internal waves describe the (linear) response of an incompressible stably stratified fluid to small perturbations. The inclination of their group velocity with respect to the vertical is completely determined by their frequency. Therefore the reflection on a sloping boundary cannot follow Descartes’ laws, and it is expected to be singular if the slope has the same inclination as the group velocity

We construct Green’s functions for elliptic operators of the form ℒu = −div(A∇u + bu) + c∇u + du in domains Ω ⊆ ℝn , under the assumption d ≥ divb or d ≥ divc. We show that, in the setting of Lorentz spaces, the assumption b − c ∈ Ln,1(Ω) is both necessary and optimal to obtain pointwise bounds for Green’s functions. We also show weak-type bounds for the Green’s function and its gradients. Our estimates

We complete the microlocal study of the geodesic x-ray transform on Riemannian manifolds with Anosov geodesic flow initiated by Guillarmou (J. Differential Geom. 105:2 (2017), 177–208) and pursued by Guillarmou and Lefeuvre in (Ann. of Math. (2) 190:1 (2019), 321–344). We prove new stability estimates and clarify some properties of the operator Πm — the generalized x-ray transform. These estimates

This paper studies the phenomenon of invasion for heterogeneous reaction-diffusion equations in periodic domains with monostable and combustion reaction terms. We give an answer to a question rised by Berestycki, Hamel and Nadirashvili in [5] concerning the connection between the speed of invasion and the speed of fronts. To do so, we extend the classical Freidlin-Gartner formula to such equations

We present an alternative approach to the theory of free Gibbs states with convex potentials developed in several papers of Guionnet, Shlyakhtenko, and Dabrowski. Instead of solving SDE's, we combine PDE techniques with a notion of asymptotic approximability by trace polynomials for a sequence of functions on $M_N(\mathbb{C})_{sa}^m$ to prove the following. Suppose $\mu_N$ is a probability measure

We demonstrate a parametrix construction, together with associated pseudodifferential operator calculus, for an operator of sum-of-squares type with semiclassical parameter. The form of operator we consider includes the generator of kinetic Brownian motion on the cosphere bundle of a Riemannian manifold. Regularity estimates in semiclassical Sobolev spaces are proven by establishing mapping properties

In this paper we study some properties of propagation of regularity of solutions of the dispersive generalized Benjamin-Ono (BO) equation. This model defines a family of dispersive equations, that can be seen as a dispersive interpolation between Benjamin-Ono equation and Korteweg-de Vries (KdV) equation. Recently, it has been showed that solutions of the KdV equation and BenjaminOno equation, satisfy

Regularity estimates in time and space for solutions to the porous medium equation are shown in the scale of Sobolev spaces. In addition, higher spatial regularity for powers of the solutions is obtained. Scaling arguments indicate that these estimates are optimal. In the linear limit, the proven regularity estimates are consistent with the optimal regularity of the linear case.

We develop refined Strichartz estimates at $L^2$ regularity for a class of time-dependent Schr\"{o}dinger operators. Such refinements begin to characterize the 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

We consider the Cauchy problem for the defocusing power type nonlinear wave equation in $(1+3)$-dimensions for energy subcritical powers $p$ in the 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 non-branching MCP(K,N)-spaces). Such a representation formula makes apparent the classical upper bounds and

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 apriori estimates on the inner and outer radii of the solutions.