The star transform is a generalized Radon transform mapping a function of two variables to its integrals along “star-shaped” trajectories, which consist of a finite number of rays emanating from a common vertex. Such operators appear in mathematical models of various imaging modalities based on scattering of elementary particles. The paper presents a comprehensive study of the inversion of the star

Given \(k\in \mathbb {R}\), v, \(D>0\), and \(n\in \mathbb {N}\), let \(\left\{ M_{\alpha }\right\} _{\alpha =1}^{\infty }\) be a Gromov–Hausdorff convergent sequence of Riemannian n–manifolds with sectional curvature \(\ge k\), volume \(>v\), and diameter \(\le D\). Perelman’s Stability Theorem implies that all but finitely many of the \(M_{\alpha }\)s are homeomorphic. The Diffeomorphism Stability

In this article, we establish scale-invariant Strichartz estimates for the Schrödinger equation on arbitrary compact globally symmetric spaces and some bilinear Strichartz estimates on products of rank-one spaces. As applications, we provide local well-posedness results for nonlinear Schrödinger equations on such spaces in both subcritical and critical regularities.

We consider the spectral problem for the Grushin Laplacian subject to homogeneous Dirichlet boundary conditions on a bounded open subset of \({\mathbb {R}}^N\). We prove that the symmetric functions of the eigenvalues depend real analytically upon domain perturbations and we prove an Hadamard-type formula for their shape differential. In the case of perturbations depending on a single scalar parameter

In this paper we investigate the property of engulfing for H-convex functions defined on the Heisenberg group \({\mathbb {H}^n}\). Starting from the horizontal sections introduced by Capogna and Maldonado (Proc Am Math Soc 134:3191–3199, 2006) , we consider a new notion of section, called \({\mathbb {H}^n}\)-section, as well as a new condition of engulfing associated to the \({\mathbb {H}^n}\)-sections

In this article we study the local estimates of Ricci flow when the curvature of the initial metric is of almost PIC1 and use it to study complete manifolds with weakly PIC1 and maximal volume growth.

The main purpose of this article is to present a generalization of Forelli’s theorem for the functions holomorphic along a general pencil of holomorphic discs. This generalizes the main result of Joo et al. (Math Ann 355(3):1171–1176, 2013) and the original Forelli’s theorem, and furthermore, answers one of the problems posed in Chirka (Proc Steklov Inst Math 253(2):212–220, 2006).

Given a contact 3-manifold, we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero set. However, we will see that resolving such

In this paper, we endow the space of continuous translation invariant valuations on convex sets generated by mixed volumes coupled with a suitable Radon measure on tuples of convex bodies with two appropriate norms. This enables us to construct a continuous extension of the convolution operator on smooth valuations to non-smooth valuations, which are in the completion of the spaces of valuations with

For each positive integer d, we prove a uniform \(l^2\)-decoupling inequality for the collection of all polynomials phases of degree at most d. Our result is intimately related to Biswas et al. (Proc Am Math Soc 148(5):1987–1997, 2020), but we use a different partition that is determined by the geometry of each individual phase function.

We prove a pair of sharp reverse isoperimetric inequalities for domains in nonpositively curved surfaces: (1) metric disks centered at the vertex of a Euclidean cone of angle at least \(2\pi \) have minimal area among all nonpositively curved disks of the same perimeter and the same total curvature; (2) geodesic triangles in a Euclidean (resp. hyperbolic) cone of angle at least \(2\pi \) have minimal

We lay the foundations of Fatou theory in one and several complex variables. We describe the main contributions contained in E. M. Stein’s book Boundary Behavior of Holomorphic Functions, published in 1972 and still a source of inspiration. We also give an account of his contributions to the study of the boundary behavior of harmonic functions. The point of this paper is not simply to exposit well-known

Let \((\phi _t)_{t \ge 0}\) be a semigroup of holomorphic self-maps of the unit disk \({{\,\mathrm{{\mathbb {D}}}\,}}\) with Denjoy–Wolff point \(\tau =1\). The angular derivative is \(\phi _t^{\prime }(1)= e^{-\lambda t}\), where \(\lambda \ge 0\) is the spectral value of \((\phi _t)\). If \(\lambda >0\) the semigroup is hyperbolic, otherwise it is parabolic. Suppose K is a compact non-polar subset

Motivated by Müller–Haslhofer results on the dynamical stability and instability of Ricci-flat metrics under the Ricci flow, we obtain dynamical stability and instability results for pairs of Ricci-flat metrics and vanishing 3-forms under the generalized Ricci flow.

For Riemannian submersions with fibers of basic mean curvature, we compare the spectrum of the total space with the spectrum of a Schrödinger operator on the base manifold. Exploiting this concept, we study submersions arising from actions of Lie groups. In this context, we extend the state-of-the-art results on the bottom of the spectrum under Riemannian coverings. As an application, we compute the

Let \(G=(V,E)\) be a finite graph (here V and E denote the set of vertices and edges of G respectively) and \(M_G\) be the centered Hardy–Littlewood maximal operator defined there. We find the optimal value \(\mathbf{{C}}_{G,p}\) such that the inequality $$\begin{aligned} \mathrm{Var\,}_{p}M_{G}f\le \mathbf{C}_{G,p}\mathrm{Var\,}_{p}f \end{aligned}$$ holds for every \(f:V\rightarrow {\mathbb {R}},\)

We prove certain gradient and eigenvalue estimates, as well as the heat kernel estimates, for the Hodge Laplacian on (m, 0) forms, i.e., sections of the canonical bundle of Kähler manifolds, where m is the complex dimension of the manifold. Instead of the usual dependence on curvature tensor, our condition depends only on the Ricci curvature bound. The proof is based on a new Bochner type formula for

For a compact connected Riemannian n-manifold \((\Omega ,g)\) with smooth boundary, we explicitly calculate the first two coefficients \(a_0\) and \(a_1\) of the asymptotic expansion of \(\sum _{k=1}^\infty \mathrm{{e}}^{-t \tau _k^{\mp }}= a_0t^{-n/2} {\mp } a_1 t^{-(n-1)/2} +O(t^{1-n/2})\) as \(t\rightarrow 0^+\), where \(\tau ^-_k\) (respectively, \(\tau ^+_k\)) is the k-th Navier–Lamé eigenvalue

In this paper, we study areas (called p-areas) and volumes for parametric surfaces in the 3D-Heisenberg group \({\mathbb {H}}_1\), which is considered as a flat model of pseudo-hermitian manifolds. We derive the formulas of p-areas and volumes for parametric surfaces in \({\mathbb {H}}_1\) and show that the classical result of Pappus-Guldin theorems for surface areas and volumes hold if the surfaces

We endow the disc \(D=\{(x_1,x_2)\in {\mathbb {R}}^2: x_1^2+x_2^2<4\}\) with a Poincaré-type Randers metric \(F_\lambda \), \(\lambda \in [0,1]\) that ’linearly’ interpolates between the usual Riemannian Poincaré disc model (\(\lambda =0\), having constant sectional curvature \(-1\) and zero S-curvature) and the Finsler–Poincaré metric (\(\lambda =1\), having constant flag curvature \(-1/4\) and constant

We show asymptotic completeness for the charged Klein–Gordon equation in the exterior De Sitter–Reissner–Nordström spacetime when the product of the charge of the black with the charge of the Klein-Gordon field is small enough. We then interpret scattering as asymptotic transports along principal null geodesics in a Kaluza–Klein extension of the original spacetime.

In this survey, we explore how superorthogonality amongst functions in a sequence \(f_1,f_2,f_3,\ldots \) results in direct or converse inequalities for an associated square function. We distinguish between three main types of superorthogonality, which we demonstrate arise in a wide array of settings in harmonic analysis and number theory. This perspective gives clean proofs of central results, and

In this paper, we study the limiting flow of conical Kähler–Ricci flow modified by a holomorphic vector field on a compact Kähler manifold M which carries an effective \({\mathbb {R}}\)-ample divisor D with simple normal crossing support. By smooth approximation, we prove the existence and uniqueness of the flow with cusp singularity when the twisted canonical bundle \(K_M+D\) is ample. At last, we

Let \((X, d, \mu )\) be a space of homogeneous type with a metric d and a doubling measure \(\mu \). Assume that \(\rho \) is a critical function on X which has an associated class of weights containing the Muckenhoupt weights as a proper subset. In this paper, we prove the quantitative weighted estimates for certain singular integrals corresponding to the new class of weights. It is important to note

Given a compact set \(K\subset {\mathbb {R}}^n\) of positive volume, and fixing a hyperplane H passing through its centroid, we find a sharp lower bound for the ratio \(\mathrm {vol}(K^{-})/\mathrm {vol}(K)\), depending on the concavity nature of the function that gives the volumes of cross-sections (parallel to H) of K, where \(K^{-}\) denotes the intersection of K with a halfspace bounded by H. When

In this paper, we prove a classification theorem for the zero sets of real analytic Beltrami fields. Namely, we show that the zero set of a real analytic Beltrami field on a real analytic, connected 3-manifold without boundary is either empty after removing its isolated points or can be written as a countable, locally finite union of differentiably embedded, connected 1-dimensional submanifolds with

We prove that in a metric measure space X, if for some \(p \in (1,\infty )\) there are uniform bounds (independent of the measure) for the weak type (p, p) of the centered maximal operator, then X satisfies a certain geometric condition, the Besicovitch intersection property, which in turn implies the uniform weak type (1, 1) of the centered operator. Thus, the following characterization is obtained:

In this paper, we are interested in a coupled Schrödinger system with Stein–Weiss type convolution part. Firstly we study the existence and nonexistence of the solutions by variational methods. Second, by changing the system into an equivalent integral form, we study the symmetry, regularity and asymptotic behaviors of the solutions by moving plane arguments.

We consider a nondegenerate case of generic singularity when the limit cylinder of the rescaled mean curvature flow is \({\mathbb {R}}^3 \times {\mathbb {S}}^1\). We obtain a detailed description of a possibly small, but fixed, neighborhood of the singularity, up to (and including) the blowup time, and find that it is mean convex and the singularity is isolated. In the considered nondegenerate cases

We consider embedded, smooth curves in the plane which are either closed or asymptotic to two lines. We study their behaviour under curve shortening flow with a global forcing term. We prove an analogue to Huisken’s distance comparison principle for curve shortening flow for initial curves whose local total curvature does not lie below \(-\pi \) and show that this condition is sharp. With that, we

Infinite-order differential operators appear in different fields of mathematics and physics and in the past decade they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrödinger equation. Inspired by the operators arising in quantum mechanics, in this paper, we investigate the continuity of a class of infinite-order differential operators

For a closed Riemannian orbifold O, we compare the spectra of the Laplacian, acting on functions or differential forms, to the Neumann spectra of the orbifold with boundary given by a domain U in O whose boundary is a smooth manifold. Generalizing results of several authors, we prove that the metric of O can be perturbed to ensure that the first N eigenvalues of U and O are arbitrarily close to one

We show that the weak limit of a quasiminimizing sequence is a quasiminimal set. This generalizes the notion of weak limit of a minimizing sequences introduced by De Lellis, De Philippis, De Rosa, Ghiraldin, and Maggi. This result is also analogous to the limiting theorem of David in local Hausdorff convergence. The proof is based on the construction of suitable deformations and is not limited to the

In this paper, we study left invariant Lorentz algebraic Ricci solitons on nilpotent Lie groups. Based on the concept of Lorentz datum and the technique of double extensions, we are able to construct Lorentz Ricci nilsolitons from any Riemannian Ricci nilsoliton. Conversely, we prove that any Lorentz Ricci nilsoliton with degenerate center is a double extension of a Riemannian Ricci nilsoliton with

For a compact smooth manifold \((M,g_0)\) with a boundary, we study the conformal rigidity and non-rigidity of the scalar curvature in the conformal class. It is known that the sign of the first eigenvalue for a linearized operator of the scalar curvature by a conformal change determines the rigidity/non-rigidity of the scalar curvature by conformal changes when the scalar curvature \(R_{g_0}\) is

We correct an error in Lemma 2.3 in our paper with the above title, published in J. Geom. Anal. 30: 223–247.

In this paper, we establish a weighted version of the well-known Fréchet–Kolmogorov theorem, which holds for weights beyond \(A_\infty \). This weighted theory extends the previous known unsatisfactory results in the terms of relaxing the index to the natural range. As applications, we obtain the weighted compactness theory for the commutators of multilinear vector-valued Calderón–Zygmund type operators

We show under natural assumptions that stable solutions to the abelian Yang–Mills–Higgs equations on Hermitian line bundles over the round 2-sphere actually satisfy the vortex equations, which are a first-order reduction of the (second-order) abelian Yang–Mills–Higgs equations. We also obtain a similar result for stable solutions on a flat 2-torus. Our method of proof comes from the work of Bourguignon–Lawson

We find the first three most general Minkowski or Hsiung–Minkowski identities relating the total mean curvatures \(H_i\), of degrees \(i=0,1,2,3\), of a closed hypersurface N immersed in a given orientable Riemannian manifold M endowed with any given vector field P. Then we specialize the three identities to the case when P is a position vector field. We further obtain that the classical Minkowski

In this paper, we prove the Miyaoka–Yau inequality for compact Kähler manifolds with semi-positive canonical bundles. The key point of the proof is the estimate for the \(L^2\)-norm of the scalar curvature along the Kähler–Ricci flow.

In this paper, we prove that a flat free boundary minimal n-disk, \(n\ge 3\), in the unit Euclidean ball \({\mathbb {B}}^{n+1}\) is the unique compact free boundary minimal hypersurface whose squared norm of the second fundamental form is less than either \(\frac{n^2}{4}\) or \(\frac{(n-2)^2}{4|x|^2}\). Moreover, we prove analogous results for compact free boundary minimal hypersurfaces in annular

We establish new local regularity results for the harmonic map and Yang–Mills heat flows on Riemannian manifolds of dimension greater than 2 and 4, respectively, obtaining criteria for the smooth local extensibility of these flows. As a corollary, we obtain new characterisations of singularity formation and use this to obtain a local estimate on the Hausdorff measure of the singular sets of these flows

We present recent results on elliptic boundary value problems where the theory of Hardy spaces associated with operators plays a key role.

We prove a modified form of the classical Morrey-Kohn-Hörmander identity, adapted to pseudoconcave boundaries. Applying this result to an annulus between two bounded pseudoconvex domains in \({{\mathbb {C}}}^n\), where the inner domain has \({\mathcal {C}}^{1,1}\) boundary, we show that the \(L^2\) Dolbeault cohomology group in bidegree (p, q) vanishes if \(1\le q\le n-2\) and is Hausdorff and infinite-dimensional

4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied harmonic map equation which are both given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. In this article we prove that finite energy solutions of both 4-harmonic

As an extension of the classical John ellipsoid and the \(L_{p}\)-John ellipsoids due to Lutwak–Yang–Zhang, this paper studies (p, q)-John ellipsoids. We consider an optimization problem about the (p, q)-mixed volumes, whose solution is uniquely existed for all \(0

We establish a new approach of treating elliptic boundary value problems (BVPs) on manifolds with boundary and regular corners, up to singularity order 2. Ellipticity and parametrices are obtained in terms of symbols taking values in algebras of BVPs on manifolds of corresponding lower singularity orders. Those refer to Boutet de Monvel’s calculus of operators with the transmission property, see Boutet

In this paper, we consider the boundary rigidity problem on a cylindrical domain in \({\mathbb {R}}^{1+n}\), \(n\ge 2\), equipped with a stationary (time-invariant) Lorentzian metric. We show that the time separation function between pairs of points on the boundary of the cylindrical domain determines the stationary spacetime, up to some time-invariant diffeomorphism, assuming that the metric satisfies

Just as we study energy density e(u), energy E(u) of harmonic maps, extend them in Wei (An extrinsic average variational method, American Mathematical Society, Providence, 1989) to \(\Phi \)-energy density \(e_{\Phi }(u)\), \(\Phi \)-energy \(E_{\Phi }(u)\) of a map, and \(\Phi \)-harmonic map \(\big (\)from the view point of the second symmetric function \(\sigma _2\) of a pullback (0, 2)-tensor\(\big

Herein, the theory of Bergman kernel is developed to the weighted case. A general form of weighted Bergman reproducing kernel is obtained, by which we can calculate concrete Bergman kernel functions for specific weights and domains.

We continue our research on Fourier restriction for hyperbolic surfaces, by studying local perturbations of the hyperbolic paraboloid \(z=xy\) which are of the form \(z=xy+h(y),\) where h(y) is a smooth function which is flat at the origin. The case of perturbations of finite type had already been handled before, but the flat case imposes several new obstacles. By means of a decomposition into intervals

We investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and \(\infty \)-Laplacian, and mean curvature-type

We study the top-degree cohomology for the \({\bar{\partial }_b}\) operator defined on a generic submanifold of the complex space as well as for the differential complex associated with a locally integrable structure \({\mathcal {V}}\) over a smooth manifold. The main assumptions are that \({\mathcal {V}}\) is hypocomplex and that the differential complex is locally solvable in degree one. One of the

A brief biographical account of Guido Weiss’ life; his accomplishments, mathematics, and diverse interests; and several anecdotes of his interactions with family, friends, colleagues, and students.

We construct convex functions on \(\mathbb {R}^3\) and \(\mathbb {R}^4\) that are smooth solutions to the Monge–Ampère equation $$\begin{aligned} \det D^2u = 1 \end{aligned}$$ away from compact one-dimensional singular sets, which can be Y-shaped or form the edges of a convex polytope. The examples solve the equation in the Alexandrov sense away from finitely many points. Our approach is based on solving

We establish sharp lower bounds for the first nonzero eigenvalue of the weighted p-Laplacian with \(1< p< \infty \) on a compact Bakry–Émery manifold \((M^n,g,f)\) satisfying \({\text {Ric}}+\nabla ^2 f \ge \kappa \, g\), provided that either \(1

Let \(\Omega \) be a domain which belongs to a class of bounded weakly pseudoconvex domains of finite type in \({\mathbb {C}}^n\), let \(d\lambda \) be the Monge–Ampère boundary measure on \(b\Omega \) and \(\varrho \ge 0\) be a non-decreasing function. The aim of this paper is to establish the characterizations of boundedness and compactness for the commutator operators of Cauchy–Fantappiè type integrals

We analyze the problem of reconstruction of a bandlimited function f from the space–time samples of its states \(f_t=\phi _t*f\) resulting from the convolution with a kernel \(\phi _t\). It is well-known that, in natural phenomena, uniform space–time samples of f are not sufficient to reconstruct f in a stable way. To enable stable reconstruction, a space–time sampling with periodic nonuniformly spaced

We characterize the Schauder and the unconditional basis properties for the Haar system in the Triebel–Lizorkin spaces \(F^s_{p,q}({{\mathbb {R}}}^d)\), at the endpoint cases \(s=1\), \(s=d/p-d\), and \(p=\infty \). Together with the earlier results in Garrigós et al. (J Fourier Anal Appl 24(5):1319–1339, 2018) and Seeger and Ullrich (Math Z 285:91–119, 2017), this completes the picture for such properties

Breakthrough work of Bourgain, Demeter, and Guth recently established that decoupling inequalities can prove powerful results on counting integral solutions to systems of Diophantine equations. In this note we demonstrate that in appropriate situations this implication can also be reversed. As a first example, we observe that a count for the number of integral solutions to a system of Diophantine equations