Journal de Mathematiques Pures et Appliquees最新文献

For a Lorentzian space measured by $m$ in the sense of Kunzinger, Sämann, Cavalletti, and Mondino, we introduce and study synthetic notions of timelike lower Ricci curvature bounds by $K\in R$ and upper dimension bounds by $N\in [1,\infty )$, namely the timelike curvature-dimension conditions ${\mathrm{TCD}}_p(K,N)$ and ${\mathrm{TCD}}_p^{⁎}(K,N)$ in weak and strong forms, where $p\in (0,1)$, and the timelike measure-contraction properties $\mathrm{TMCP}(K,N)$ and ${\mathrm{TMCP}}^{⁎}(K,N)$. These are formulated by convexity properties of the Rényi entropy with respect to $m$ along ${\ell }_p$-geodesics of probability measures.

We show many features of these notions, including their compatibility with the smooth setting, sharp geometric inequalities, stability, equivalence of the named weak and strong versions, local-to-global properties, and uniqueness of chronological ${\ell }_p$-optimal couplings and chronological ${\ell }_p$-geodesics. We also prove the equivalence of ${\mathrm{TCD}}_p^{⁎}(K,N)$ and ${\mathrm{TMCP}}^{⁎}(K,N)$ to their respective entropic counterparts in the sense of Cavalletti and Mondino.

Some of these results are obtained under timelike p-essential nonbranching, a concept which is a priori weaker than timelike nonbranching.

For any pseudo-Riemannian hyperbolic space X over $R,C,H$ or $O$, we show that the resolvent $R\left(z\right)={(\square -z\mathrm{Id})}^{-1}$ of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators $C_c^{\infty }\left(X\right)\to D^{\prime }\left(X\right)$. Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in $D^{\prime }\left(X\right)$ forms a representation of the isometry group of X, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces.

For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which $R\left(z\right)$ extends, and the residue representations can be infinite-dimensional.

We obtain sharp maximal vanishing order at a given time level for solutions to parabolic equations with a $C^1$ potential V. Our main result Theorem 1.1 is a parabolic generalization of a well known result of Donnelly-Fefferman and Bakri. It also sharpens a previous result of Zhu that establishes similar vanishing order estimates which are instead averaged over time. The principal tool in our analysis is a new quantitative version of the well-known Escauriaza-Fernandez-Vessella type Carleman estimate that we establish in our setting.

In a general $L^2$ extension theorem of Demailly for log canonical pairs, the $L^2$ criterion with respect to a measure called the Ohsawa measure determines when a given holomorphic function can be extended. Despite the analytic nature of the Ohsawa measure, we establish a geometric characterization of this analytic criterion using the theory of log canonical centers from algebraic geometry. Along the way, we characterize when the Ohsawa measure fails to have generically smooth positive density, which answers an essential question arising from Demailly's work.

We describe the variation of the Minkowski, packing and Hausdorff dimensions of a set moving under a holomorphic motion, as well as the variation of its area. Our method provides a new, unified approach to various celebrated theorems about quasiconformal mappings, including the work of Astala on the distortion of area and dimension under quasiconformal mappings and the work of Smirnov on the dimension of quasicircles.

In this paper we are concerned with global maximal regularity estimates for elliptic equations with degenerate weights. We consider both the linear case and the non-linear case. We show that higher integrability of the gradients can be obtained by imposing a local small oscillation condition on the weight and a local small Lipschitz condition on the boundary of the domain. Our results are new in the linear and non-linear case. We show by example that the relation between the exponent of higher integrability and the smallness parameters is sharp even in the linear or the unweighted case.

We present a different symplectic point of view in the definition of weighted modulation spaces $M_m^{p,q}\left(R^d\right)$ and weighted Wiener amalgam spaces $W(F{L}_{m_1}^p,L_{m_2}^q)\left(R^d\right)$. All the classical time-frequency representations, such as the short-time Fourier transform (STFT), the τ-Wigner distributions and the ambiguity function, can be written as metaplectic Wigner distributions $\mu \left(A\right)(f\otimes \overline{g})$, where $\mu \left(A\right)$ is the metaplectic operator and $A$ is the associated symplectic matrix. Namely, time-frequency representations can be represented as images of metaplectic operators, which become the real protagonists of time-frequency analysis. In [13], the authors suggest that any metaplectic Wigner distribution that satisfies the so-called shift-invertibility condition can replace the STFT in the definition of modulation spaces. In this work, we prove that shift-invertibility alone is not sufficient, but it has to be complemented by an upper-triangularity condition for this characterization to hold, whereas a lower-triangularity property comes into play for Wiener amalgam spaces. The shift-invertibility property is necessary: Rihaczek and conjugate Rihaczek distributions are not shift-invertible and they fail the characterization of the above spaces. We also exhibit examples of shift-invertible distributions without upper-triangularity condition which do not define modulation spaces. Finally, we provide new families of time-frequency representations that characterize modulation spaces, with the purpose of replacing the time-frequency shifts with other atoms that allow to decompose signals differently, with possible new outcomes in applications.

Obstruction flatness of a strongly pseudoconvex hypersurface Σ in a complex manifold refers to the property that any (local) Kähler-Einstein metric on the pseudoconvex side of Σ, complete up to Σ, has a potential $-\log u$ such that u is $C^{\infty }$-smooth up to Σ. In general, u has only a finite degree of smoothness up to Σ. In this paper, we study obstruction flatness of hypersurfaces Σ that arise as unit circle bundles $S\left(L\right)$ of negative Hermitian line bundles $(L,h)$ over Kähler manifolds $(M,g)$. We prove that if $(M,g)$ has constant Ricci eigenvalues, then $S\left(L\right)$ is obstruction flat. If, in addition, all these eigenvalues are strictly less than one and $(M,g)$ is complete, then we show that the corresponding disk bundle admits a complete Kähler-Einstein metric. Finally, we give a necessary and sufficient condition for obstruction flatness of $S\left(L\right)$ when $(M,g)$ is a Kähler surface $(\dim M=2$) with constant scalar curvature.

We study the focusing energy-critical NLS(NLS)$i{\partial }_{t}u+{\Delta }_{x,y}u=-|u{|}^{\frac{4}{d-1}}u$ on the waveguide manifold $R_x^d\times T_y$ with $d\ge 2$. We reveal the somewhat counterintuitive phenomenon that despite the energy-criticality of the nonlinear potential, the long time dynamics of (NLS) are purely determined by the semivirial-vanishing geometry which possesses an energy-subcritical characteristic. As a starting point, we consider a minimization problem $m_c$ defined on the semivirial-vanishing manifold with prescribed mass c. We prove that for all sufficiently large mass the variational problem $m_c$ has a unique optimizer $u_c$ satisfying ${\partial }_y{u}_c=0$, while for all sufficiently small mass, any optimizer of $m_c$ must have non-trivial y-dependence. Afterwards, we prove that $m_c$ characterizes a sharp threshold for the bifurcation of finite time blow-up ($d=2,3$) and globally scattering ($d=3$) solutions of (NLS) in dependence of the sign of the semivirial. To the author's knowledge, the paper also gives the first large data scattering result for focusing NLS on product spaces in the energy-critical setting.

Hermitian, pluriclosed metrics with vanishing Bismut-Ricci form give a natural extension of Calabi-Yau metrics to the setting of complex, non-Kähler manifolds, and arise independently in mathematical physics. We reinterpret this condition in terms of the Hermitian-Einstein equation on an associated holomorphic Courant algebroid, and thus refer to solutions as Bismut Hermitian-Einstein. This implies Mumford-Takemoto slope stability obstructions, and using these we exhibit infinitely many topologically distinct complex manifolds in every dimension with vanishing first Chern class which do not admit Bismut Hermitian-Einstein metrics. This reformulation also leads to a new description of pluriclosed flow in terms of Hermitian metrics on holomorphic Courant algebroids, implying new global existence results, in particular on all complex non-Kähler surfaces of Kodaira dimension $\kappa \ge 0$. On complex manifolds which admit Bismut-flat metrics we show global existence and convergence of pluriclosed flow to a Bismut-flat metric, which in turn gives a classification of generalized Kähler structures on these spaces.