Showing papers in "Journal of Geometric Analysis in 2021"

Weak Hardy-Type Spaces Associated with Ball Quasi-Banach Function Spaces II: Littlewood--Paley Characterizations and Real Interpolation

Abstract: Let X be a ball quasi-Banach function space on $${\mathbb R}^n$$ . In this article, assuming that the powered Hardy–Littlewood maximal operator satisfies some Fefferman–Stein vector-valued maximal inequality on X as well as it is bounded on both the weak ball quasi-Banach function space WX and the associated space, the authors establish various Littlewood–Paley function characterizations of $$WH_X({{\mathbb {R}}}^n)$$ under some weak assumptions on the Littlewood–Paley functions. The authors also prove that the real interpolation intermediate space $$(H_{X}({{\mathbb {R}}}^n),L^\infty ({{\mathbb {R}}}^n))_{\theta ,\infty }$$ , between the Hardy space associated with X, $$H_{X}({{\mathbb {R}}}^n)$$ , and the Lebesgue space $$L^\infty ({\mathbb R}^n)$$ , is $$WH_{X^{{1}/{(1-\theta )}}}({{\mathbb {R}}}^n)$$ , where $$\theta \in (0, 1)$$ . All these results are of wide applications. Particularly, when $$X:=M_q^p({{\mathbb {R}}}^n)$$ (the Morrey space), $$X:=L^{\vec {p}}({{\mathbb {R}}}^n)$$ (the mixed-norm Lebesgue space) and $$X:=(E_\Phi ^q)_t({{\mathbb {R}}}^n)$$ (the Orlicz-slice space), all these results are even new; when $$X:=L_\omega ^\Phi ({\mathbb R}^n)$$ (the weighted Orlicz space), the result on the real interpolation is new and, when $$X:=L^{p(\cdot )}({{\mathbb {R}}}^n)$$ (the variable Lebesgue space) and $$X:=L_\omega ^\Phi ({{\mathbb {R}}}^n)$$ , the Littlewood–Paley function characterizations of $$WH_X({{\mathbb {R}}}^n)$$ obtained in this article improves the existing results via weakening the assumptions on the Littlewood–Paley functions.

Two Weight Commutators on Spaces of Homogeneous Type and Applications

Abstract: In this paper, we establish the two weight commutator theorem of Calderon–Zygmund operators in the sense of Coifman–Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for $$A_2$$ weights and by proving the sparse operator domination of commutators. The main tool here is the Haar basis, the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutators) for the following Calderon–Zygmund operators: Cauchy integral operator on $${\mathbb {R}}$$ , Cauchy–Szego projection operator on Heisenberg groups, Szego projection operators on a family of unbounded weakly pseudoconvex domains, the Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (in one and several dimensions).

Atomic Decomposition for Mixed Morrey Spaces

Abstract: In this paper, we consider some norm estimates for mixed Morrey spaces considered by the first author. Mixed Lebesgue spaces are realized as a special case of mixed Morrey spaces. What is new in this paper is a new norm estimate for mixed Morrey spaces that is applicable to mixed Lebesgue spaces as well. An example shows that the condition on parameters is optimal. As an application, the Olsen inequality adapted to mixed Morrey spaces can be obtained.

Direct Methods for Pseudo-relativistic Schrödinger Operators

Abstract: In this paper, we establish various maximal principles and develop the direct moving planes and sliding methods for equations involving the physically interesting (nonlocal) pseudo-relativistic Schrodinger operators $$(-\Delta +m^{2})^{s}$$ with $$s\in (0,1)$$ and mass $$m>0$$ . As a consequence, we also derive multiple applications of these direct methods. For instance, we prove monotonicity, symmetry and uniqueness results for solutions to various equations involving the operators $$(-\Delta +m^{2})^{s}$$ in bounded or unbounded domains with certain geometrical structures (e.g., coercive epigraph and epigraph), including pseudo-relativistic Schrodinger equations, 3D boson star equations and the equations with De Giorgi-type nonlinearities. When $$m=0$$ and $$s=1$$ , equations with De Giorgi-type nonlinearities are related to De Giorgi conjecture connected with minimal surfaces and the scalar Ginzburg–Landau functional associated to harmonic map.

Weighted Fréchet–Kolmogorov Theorem and Compactness of Vector-Valued Multilinear Operators

Abstract: In this paper, we establish a weighted version of the well-known Frechet–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 Calderon–Zygmund type operators, including the commutators of multilinear Littlewood–Paley type operators. It is worthy to pointing out that the commutators we considered contain almost all the commutators formerly studied in this literature.

The Light Ray transform in Stationary and Static Lorentzian geometries

Abstract: Given a Lorentzian manifold, the light ray transform of a function is its integrals along null geodesics. This paper is concerned with the injectivity of the light ray transform on functions and tensors, up to the natural gauge for the problem. First, we study the injectivity of the light ray transform of a scalar function on a globally hyperbolic stationary Lorentzian manifold and prove injectivity holds if either a convex foliation condition is satisfied on a Cauchy surface on the manifold or the manifold is real analytic and null geodesics do not have cut points. Next, we consider the light ray transform on tensor fields of arbitrary rank in the more restrictive class of static Lorentzian manifolds and show that if the geodesic ray transform on tensors defined on the spatial part of the manifold is injective up to the natural gauge, then the light ray transform on tensors is also injective up to its natural gauge. Finally, we provide applications of our results to some inverse problems about recovery of coefficients for hyperbolic partial differential equations from boundary data.

Spacelike Mean Curvature Flow

Abstract: We prove long-time existence and convergence results for spacelike solutions to mean curvature flow in the pseudo-Euclidean space $$\mathbb {R}^{n,m}$$ , which are entire or defined on bounded domains and satisfying Neumann or Dirichlet boundary conditions. As an application, we prove long-time existence and convergence of the $${{\,\mathrm{G}\,}}_2$$ -Laplacian flow in cases related to coassociative fibrations.

$$L^p$$-Bounds for Pseudo-differential Operators on Graded Lie Groups

Abstract: In this work we obtain sharp $$L^p$$ -estimates for pseudo-differential operators on arbitrary graded Lie groups. The results are presented within the setting of the global symbolic calculus on graded Lie groups by using the Fourier analysis associated to every graded Lie group which extends the usual one due to Hormander on $${\mathbb {R}}^n$$ . The main result extends the classical Fefferman’s sharp theorem on the $$L^p$$ -boundedness of pseudo-differential operators for Hormander classes on $${\mathbb {R}}^n$$ to general graded Lie groups, also adding the borderline $$\rho =\delta $$ case.

Configuration Sets with Nonempty Interior

Abstract: A theorem of Steinhaus states that if $$E\subset \mathbb {R}^d$$ has positive Lebesgue measure, then the difference set $$E-E$$ contains a neighborhood of 0. Similarly, if E merely has Hausdorff dimension $$\hbox {dim}_{{\mathcal {H}}}(E)>(d+1)/2,$$ a result of Mattila and Sjolin states that the distance set $$\varDelta (E)\subset \mathbb {R}$$ contains an open interval. In this work, we study such results from a general viewpoint, replacing $$E-E$$ or $$\varDelta (E)$$ with more general $$\varPhi $$ -configurations for a class of $$\varPhi :\mathbb {R}^d\times \mathbb {R}^d\rightarrow \mathbb {R}^k,$$ and showing that, under suitable lower bounds on $$\hbox {dim}_{{\mathcal {H}}}(E)$$ and a regularity assumption on the family of generalized Radon transforms associated with $$\varPhi ,$$ it follows that the set $$\varDelta _\varPhi (E)$$ of $$\varPhi $$ -configurations in E has nonempty interior in $$\mathbb {R}^k$$ . Further extensions hold for $$\varPhi $$ -configurations generated by two sets, E and F, in spaces of possibly different dimensions and with suitable lower bounds on $$\hbox {dim}_{{\mathcal {H}}}(E)+\hbox {dim}_{{\mathcal {H}}}(F).$$

Quasimode, Eigenfunction and Spectral Projection Bounds for Schrödinger Operators on Manifolds with Critically Singular Potentials

Abstract: We obtain quasimode, eigenfunction and spectral projection bounds for Schrodinger operators, $$H_V=-\Delta _g+V(x)$$ , on compact Riemannian manifolds (M, g) of dimension $$n\ge 2$$ , which extend the results of the third author (Sogge 1988) corresponding to the case where $$V\equiv 0$$ . We are able to handle critically singular potentials and consequently assume that $$V\in L^{\tfrac{n}{2}}(M)$$ and/or $$V\in {{\mathcal {K}}}(M)$$ (the Kato class). Our techniques involve combining arguments for proving quasimode/resolvent estimates for the case where $$V\equiv 0$$ that go back to the third author (Sogge 1988) as well as ones which arose in the work of Kenig et al. (1987) in the study of “uniform Sobolev estimates” in $${{\mathbb {R}}}^n$$ . We also use techniques from more recent developments of several authors concerning variations on the latter theme in the setting of compact manifolds. Using the spectral projection bounds we can prove a number of natural $$L^p\rightarrow L^p$$ spectral multiplier theorems under the assumption that $$V\in L^{\frac{n}{2}}(M)\cap {{\mathcal {K}}}(M)$$ . Moreover, we can also obtain natural analogs of the original Strichartz estimates (1977) for solutions of $$(\partial _t^2-\Delta +V)u=0$$ . We also are able to obtain analogous results in $${{\mathbb {R}}}^n$$ and state some global problems that seem related to works on absence of embedded eigenvalues for Schrodinger operators in $${{\mathbb {R}}}^n$$ (e.g., Ionescu and Jerison 2003; Jerison and Kenig 1985; Kenig and Nadirashvili 2000; Koch and Tataru 2002; Rodnianski and Schlag 2004).

A Multifractal Formalism for Hewitt–Stromberg Measures

Abstract: In the present work, we give a new multifractal formalism for which the classical multifractal formalism does not hold. We precisely introduce and study a multifractal formalism based on the Hewitt–Stromberg measures and that this formalism is completely parallel to Olsen’s multifractal formalism which is based on the Hausdorff and packing measures.

Martingale Musielak–Orlicz–Lorentz Hardy Spaces with Applications to Dyadic Fourier Analysis

Abstract: Let $$(\Omega ,{\mathcal {F}},{\mathbb {P}})$$ be a probability space, $$\varphi :\ \Omega \times [0,\infty )\rightarrow [0,\infty )$$ a Musielak–Orlicz function, and $$q\in (0,\infty ]$$ . In this article, the authors introduce five martingale Musielak–Orlicz–Lorentz Hardy spaces and prove that these new spaces have some important features such as atomic characterizations, the boundedness of $$\sigma $$ -sublinear operators, and martingale inequalities. This new scale of martingale Hardy spaces requires the introduction of the Musielak–Orlicz–Lorentz space $$L^{\varphi ,q}(\Omega )$$ . In particular, the authors show that this Lorentz type space has some fundamental properties including the completeness, the convergence, real interpolations, and the Fefferman–Stein vector-valued inequality for the Doob maximal operator. As applications, the authors prove that the maximal Fejer operator is bounded from the martingale Musielak–Orlicz–Lorentz Hardy space $$H_{\varphi ,q}[0,1)$$ to $$L^{\varphi ,q}[0,1)$$ , which further implies some convergence results of the Fejer means. Moreover, all the above results are new even for Musielak–Orlicz functions with particular structure such as weight, weight Orlicz, and double-phase growth. The main approach used in this article can be viewed as a combination of the stopping time argument in probability theory and the real-variable technique of function spaces in harmonic analysis.

On pointwise $\ell^r$-sparse domination in a space of homogeneous type

Abstract: We prove a general sparse domination theorem in a space of homogeneous type, in which a vector-valued operator is controlled pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination in which the usual $$\ell ^1$$ -sum in the sparse operator is replaced by an $$\ell ^r$$ -sum. This sparse domination theorem is applicable to various operators from both harmonic analysis and (S)PDE. Using our main theorem, we prove the $$A_2$$ -theorem for vector-valued Calderon–Zygmund operators in a space of homogeneous type, from which we deduce an anisotropic, mixed-norm Mihlin multiplier theorem. Furthermore, we show quantitative weighted norm inequalities for the Rademacher maximal operator, for which Banach space geometry plays a major role.

Quantum Graphs as Quantum Relations

Abstract: The “noncommutative graphs” which arise in quantum error correction are a special case of the quantum relations introduced in Weaver (Quantum relations. Mem Am Math Soc 215(v–vi):81–140, 2012). We use this perspective to interpret the Knill–Laflamme error-correction conditions (Knill and Laflamme in Theory of quantum error-correcting codes. Phys Rev A 55:900-911, 1997) in terms of graph-theoretic independence, to give intrinsic characterizations of Stahlke’s noncommutative graph homomorphisms (Stahlke in Quantum zero-error source-channel coding and non-commutative graph theory. IEEE Trans Inf Theory 62:554–577, 2016) and Duan, Severini, and Winter’s noncommutative bipartite graphs (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovasz number. IEEE Trans Inf Theory 59:1164–1174, 2013), and to realize the noncommutative confusability graph associated to a quantum channel (Duan et al., op. cit. in Zero-error communication via quantum channels, noncommutative graphs, and a quantum Lovasz number. IEEE Trans Inf Theory 59:1164–1174, 2013) as the pullback of a diagonal relation. Our framework includes as special cases not only purely classical and purely quantum information theory, but also the “mixed” setting which arises in quantum systems obeying superselection rules. Thus we are able to define noncommutative confusability graphs, give error correction conditions, and so on, for such systems. This could have practical value, as superselection constraints on information encoding can be physically realistic.

Decay Estimates for Nonradiative Solutions of the Energy-Critical Focusing Wave Equation

Abstract: We consider the energy-critical focusing wave equation in space dimension $$N\ge 3$$ . The equation has a nonzero radial stationary solution W, which is unique up to scaling and sign change. It is conjectured (soliton resolution) that any radial, bounded in the energy norm solution of the equation, behaves asymptotically as a sum of modulated Ws, decoupled by the scaling, and a radiation term. A nonradiative solution of the equation is by definition a solution of which energy in the exterior $$\{|x|>|t|\}$$ of the wave cone vanishes asymptotically as $$t\rightarrow +\infty $$ and $$t\rightarrow -\infty $$ . In our previous work [9], we have proved that the only radial nonradiative solutions of the equation in three space dimensions are, up to scaling, 0 and $$\pm W$$ . This was crucial in the proof of soliton resolution in [9]. In this paper, we prove that the initial data of a radial nonradiative solution in odd space dimension have a prescribed asymptotic behavior as $$r\rightarrow \infty $$ . We will use this property for the proof of soliton resolution, for radial data, in all odd space dimensions. The proof uses the characterization of nonradiative solutions of the linear wave equation in odd space dimensions obtained by Lawrie, Liu, Schlag, and the second author in [15]. We also study the propagation of the support of nonzero radial solutions with compactly supported initial data and prove that these solutions cannot be nonradiative.

Bochner-Simons formulas and the rigidity of biharmonic submanifolds

Abstract: New integral formulas of Simons and Bochner type are found and then used to study biharmonic and biconservative submanifolds in space forms. This leads to new rigidity results and partial answers to conjectures on biharmonic submanifolds in spheres.

Vacuum Static Spaces with Vanishing of Complete Divergence of Weyl Tensor

Abstract: In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Weyl tensor with the non-negativity of the complete divergence of the Bach tensor implies the harmonicity of the metric, and we present examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. Second, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness of the metric. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.

Infinitesimal Hilbertianity of Locally $$\mathrm{CAT}(\kappa )$$-Spaces

Abstract: We show that, given a metric space $$(\mathrm{Y},\textsf {d} )$$ of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $$\mu $$ on $$\mathrm{Y}$$ giving finite mass to bounded sets, the resulting metric measure space $$(\mathrm{Y},\textsf {d} ,\mu )$$ is infinitesimally Hilbertian, i.e. the Sobolev space $$W^{1,2}(\mathrm{Y},\textsf {d} ,\mu )$$ is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at $$x\in \mathrm{Y}$$ is the tangent cone at x of $$\mathrm{Y}$$ . The conclusion then follows from the fact that for every $$x\in \mathrm{Y}$$ such a cone is a $$\mathrm{CAT}(0)$$ space and, as such, has a Hilbert-like structure.

A Békollè–Bonami Class of Weights for Certain Pseudoconvex Domains

Abstract: We prove the weighted $$L^p$$ regularity of the ordinary Bergman projection on certain pseudoconvex domains where the weight belongs to an appropriate generalization of the Bekolle–Bonami class. The main tools used are estimates on the Bergman kernel obtained by McNeal and Bekolle’s original approach of proving a good-lambda inequality.

Geometric Characterizations of Embedding Theorems: For Sobolev, Besov, and Triebel–Lizorkin Spaces on Spaces of Homogeneous Type—via Orthonormal Wavelets

Abstract: It was well known that geometric considerations enter in a decisive way in many questions. The embedding theorem arises in several problems from partial differential equations, analysis, and geometry. The purpose of this paper is to provide a deep understanding of analysis and geometry with a particular focus on embedding theorems for spaces of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure $$\mu $$ satisfies the doubling property only. We prove that embedding theorems hold on spaces of homogeneous type if and only if geometric conditions, namely the measures of all balls have lower bounds, hold. We make no additional geometric assumptions on the quasi-metric or the doubling measure, and thus, the results of this paper extend to the full generality of all related previous ones, in which the extra geometric assumptions were made on both the quasi-metric d and the measure $$\mu .$$ As applications, our results provide new and sharp previous related embedding theorems for the Sobolev, Besov, and Triebel–Lizorkin spaces. The crucial tool used in this paper is the remarkable orthonormal wavelet basis constructed recently by Auscher–Hytonen on spaces of homogeneous type in the sense of Coifman and Weiss.

Geometric Pluripotential Theory on Sasaki Manifolds

Abstract: We extend profound results in pluripotential theory on Kahler manifolds (Darvas in arXiv:1902.01982 , 2019) to Sasaki setting via its transverse Kahler structure. As in Kahler case, these results form a very important piece to solve the existence of Sasaki metrics with constant scalar curvature in terms of properness of $$\mathcal {K}$$ -energy, considered by the first named author in He ( arXiv:1802.03841 , 2019). One main result is to generalize Darvas’ theory on the geometric structure of the space of Kahler potentials in Sasaki setting. Along the way we extend most of corresponding results in pluripotential theory to Sasaki setting via its transverse Kahler structure.

Singular Doubly Nonlocal Elliptic Problems with Choquard Type Critical Growth Nonlinearities

Abstract: The theory of elliptic equations involving singular nonlinearities is well-studied topic but the interaction of singular type nonlinearity with nonlocal nonlinearity in elliptic problems has not been investigated so far. In this article, we study the very singular and doubly nonlocal singular problem $$(P_\lambda )$$ (See below). Firstly, we establish a very weak comparison principle and the optimal Sobolev regularity. Next using the critical point theory of nonsmooth analysis and the geometry of the energy functional, we establish the global multiplicity of positive weak solutions.

Geometric Invariants of Spectrum of the Navier–Lamé Operator

Abstract: 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–Lame eigenvalue on $$\Omega $$ with Dirichlet (respectively, Neumann) boundary condition. These two coefficients provide precise information for the volume of the elastic body $$\Omega $$ and the surface area of the boundary $$\partial \Omega $$ in terms of the spectrum of the Navier–Lame operator. This gives an answer to an interesting and open problem mentioned by Avramidi in (Non-Laplace type operators on manifolds with boundary, analysis, geometry and topology of elliptic operators. World Sci. Publ., Hackensack, pp. 107–140, 2006). As an application, we show that an n-dimensional ball is uniquely determined by its Navier–Lame spectrum among all bounded elastic bodies with smooth boundary.

Asymptotic Behaviours in Fractional Orlicz–Sobolev Spaces on Carnot Groups

Abstract: In this article, we define a class of fractional Orlicz–Sobolev spaces on Carnot groups, and in the spirit of the celebrated results of Bourgain–Brezis–Mironescu and of Maz’ya–Shaposhnikova, we study the asymptotic behaviour of the Orlicz functionals when the fractional parameter goes to 1 and 0.

Projective Embedding of Log Riemann Surfaces and K-Stability

Abstract: Given a smooth polarized Riemann surface (X, L) endowed with a hyperbolic metric $$\omega $$ that has standard cusp singularities along a divisor D, we show the $$L^2$$ projective embedding of (X, D) defined by $$L^k$$ is asymptotically almost balanced in a weighted sense. The proof depends on sufficiently precise understanding of the behavior of the Bergman kernel in three regions, with the most crucial one being the neck region around D. This is the first step towards understanding the algebro-geometric stability of extremal Kahler metrics with singularities.

Killing forms on $2$-step nilmanifolds

Abstract: We study left-invariant Killing k-forms on simply connected 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For $$k=2,3$$ , we show that every left-invariant Killing k-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing 2-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing 3-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, $$k=2$$ or $$k=3$$ , we show that the space of left-invariant Killing k-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.

Two-Phase Heat Conductors with a Surface of the Constant Flow Property

Abstract: We consider a two-phase heat conductor in $${\mathbb {R}}^N$$ with $$N \ge 2$$ consisting of a core and a shell with different constant conductivities. We study the role played by radial symmetry for overdetermined problems of elliptic and parabolic type. First of all, with the aid of the implicit function theorem, we give a counterexample to radial symmetry for some two-phase elliptic overdetermined boundary value problems of Serrin type. Afterwards, we consider the following setting for a two-phase parabolic overdetermined problem. We suppose that, initially, the conductor has temperature 0 and, at all times, its boundary is kept at temperature 1. A hypersurface in the domain has the constant flow property if at every of its points the heat flux across surface only depends on time. It is shown that the structure of the conductor must be spherical, if either there is a surface of the constant flow property in the shell near the boundary or a connected component of the boundary of the heat conductor is a surface of the constant flow property. Also, by assuming that the medium outside the conductor has a possibly different conductivity, we consider a Cauchy problem in which the conductor has initial inside temperature 0 and outside temperature 1. We then show that a quite similar symmetry result holds true.

On a Class of Orientation-Preserving Maps of $$\pmb {\mathbb {R}}^4$$ R 4

Abstract: The purpose of this paper is to present several new, sometimes surprising, results concerning a class of hyperholomorphic functions over quaternions, the so-called slice regular functions. The concept of slice regular function is a generalization of the one of holomorphic function in one complex variable. The results we present here show that such a generalization is multifaceted and highly non-trivial. We study the behavior of the Jacobian matrix $$J_f$$ of a slice regular function f proving in particular that $$\det (J_f)\ge 0$$ , i.e., f is orientation-preserving. We give a complete characterization of the fibers of f making use of a new notion we introduce here, the one of wing of f. We investigate the singular set $$N_f$$ of f, i.e., the set in which $$J_f$$ is singular. The singular set $$N_f$$ turns out to be equal to the branch set of f, i.e., the set of points y such that f is not a homeomorphism locally at y. We establish the quasi-openness properties of f. As a consequence we deduce the validity of the Maximum Modulus Principle for f in its full generality. Our results are sharp as we show by explicit examples.

Valuations on Log-Concave Functions

Abstract: A classification of $${\text {SL}}(n)$$ and translation covariant Minkowski valuations on log-concave functions is established. The moment vector and the recently introduced level set body of log-concave functions are characterized. Furthermore, analogs of the Euler characteristic and volume are characterized as $${\text {SL}}(n)$$ and translation invariant valuations on log-concave functions.

Stability of the Spacetime Positive Mass Theorem in Spherical Symmetry

Abstract: The rigidity statement of the positive mass theorem asserts that an asymptotically flat initial data set for the Einstein equations with zero ADM mass, and satisfying the dominant energy condition, must arise from an embedding into Minkowski space. In this paper, we address the question of what happens when the mass is merely small. In particular, we formulate a conjecture for the stability statement associated with the spacetime version of the positive mass theorem, and give examples to show how it is basically sharp if true. This conjecture is then established under the assumption of spherical symmetry in all dimensions. More precisely, it is shown that a sequence of asymptotically flat initial data satisfying the dominant energy condition, without horizons except possibly at an inner boundary, and with ADM masses tending to zero must arise from isometric embeddings into a sequence of static spacetimes converging to Minkowski space in the pointed volume preserving intrinsic flat sense. The difference of second fundamental forms coming from the embeddings and initial data must converge to zero in $$L^p$$ , $$1\le p<2$$ . In addition some minor tangential results are also given, including the spacetime version of the Penrose inequality with rigidity statement in all dimensions for spherically symmetric initial data, as well as symmetry inheritance properties for outermost apparent horizons.