The spectral projections and the resolvent for scattering metrics

In this paper, we consider a compact manifold with boundaryX equipped with a scattering metricg as defined by Melrose [9]. That is,g is a Riemannian metric in the interior ofX that can be brought to the formg=x ⁻⁴ dx²+x^{−2 h’} near the boundary, wherex is a boundary defining function andh’ is a smooth symmetric 2-cotensor which restricts to a metrich on ϖX. LetH=Δ+V, whereV∈x ²C^∞ (X) is real, soV is a ‘short-range’ perturbation of Δ. Melrose and Zworski started a detailed analysis of various operators associated toH in [11] and showed that the scattering matrix ofH is a Fourier integral operator associated to the geodesic flow ofh on ϖX at distance π and that the kernel of the Poisson operator is a Legendre distribution onX×ϖX associated to an intersecting pair with conic points. In this paper, we describe the kernel of the spectral projections and the resolvent,R(σ±i0), on the positive real axis. We define a class of Legendre distributions on certain types of manifolds with corners and show that the kernel of the spectral projection is a Legendre distribution associated to a conic pair on the b-stretched productX _b ² (the blowup ofX ² about the corner, (ϖX)²). The structure of the resolvent is only slightly more complicated. As applications of our results, we show that there are ‘distorted Fourier transforms’ forH, i.e., unitary operators which intertwineH with a multiplication operator and determine the scattering matrix; we also give a scattering wavefront set estimate for the resolventR(σ±i0) applied to a distributionf.     

Renormalization group, operator expansion, and anomalous scaling in a simple model of turbulent diffusion

Using the renormalization group method and the operator expansion in the Obukhov-Kraichnan model that describes the intermixing of a passive scalar admixture by a random Gaussian field of velocities with the correlator 〈v(t,x)v(t′,x)〉−〈v(t,x)v(t′,x′)〉∝δ(t−t′)|x−x′|^ε, we prove that the anomalous scaling in the inertial interval is caused by the presence of “dangerous” composite operators (powers of the local dissipation rate) whose negative critical dimensions determine the anomalous exponents. These exponents are calculated up to the second order of the ε expansion. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 2, pp. 309–314, August, 1999.     

A chart preserving the normal vector and extensions of normal derivatives in weighted function spaces

Given a domain Ω of class C ^k,1, k ∈ ℕ, we construct a chart that maps normals to the boundary of the half space to normals to the boundary of in the sense that (∂/∂x _n )α(x′, 0) = − N(x′) and that still is of class C ^k,1. As an application we prove the existence of a continuous extension operator for all normal derivatives of order 0 to k on domains of class C ^k,1. The construction of this operator is performed in weighted function spaces where the weight function is taken from the class of Muckenhoupt weights.     

Nonlinear Boundary Value Problems for Second Order Differential Inclusions

In this paper we study two boundary value problems for second order strongly nonlinear differential inclusions involving a maximal monotone term. The first is a vector problem with Dirichlet boundary conditions and a nonlinear differential operator of the form x ↦ a(x, x′)′. In this problem the maximal monotone term is required to be defined everywhere in the state space ℝ^N. The second problem is a scalar problem with periodic boundary conditions and a differential operator of the form x ↦ (a(x)x′)′. In this case the maximal monotone term need not be defined everywhere, incorporating into our framework differential variational inequalities. Using techniques from multivalued analysis and from nonlinear analysis, we prove the existence of solutions for both problems under convexity and nonconvexity conditions on the multivalued right-hand side.     

Perturbed boundary eigenvalue problem for the Schrödinger operator on an interval

A perturbed two-parameter boundary value problem is considered for a second-order differential operator on an interval with Dirichlet conditions. The perturbation is described by the potential μ⁻¹ V((x − x ₀)ɛ⁻¹), where 0 < ɛ ≪ 1 and μ is an arbitrary parameter such that there exists δ > 0 for which ɛ/μ = o(ɛ^δ). It is shown that the eigenvalues of this operator converge, as ɛ → 0, to the eigenvalues of the operator with no potential. Complete asymptotic expansions of the eigenvalues and eigenfunctions of the perturbed operator are constructed.     

A class of oscillatory singular integrals on triebel-lizorkin spaces

The boundedness on Triebel-Lizorkin spaces of oscillatory singular integral operator T in the form e^i｜x｜^aΩ（x）｜x｜^-n is studied,where a∈R,a≠0,1 and Ω∈L^1（S^n-1） is homogeneous of degree zero and satisfies certain cancellation condition. When kernel Ω（x＇ ）∈Llog＋L（S^n-1 ）, the Fp^a,q（R^n） boundedness of the above operator is obtained. Meanwhile ,when Ω（x） satisfies L^1- Dini condition,the above operator T is bounded on F1^0,1 （R^n）.     

Maximal average along variable lines

We prove the L ^p boundedness of the maximal operator associated with a family of lines l _x = {(x ₁, x ₂) − t(1, a(x ₁)): t ∈ [0, ∞)} when a′ is a positive increasing function. This work was supported by the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MOST) (Grant No. R01-2007-000-10527-0)     

Gibbsianness of fermion random point fields

We consider fermion (or determinantal) random point fields on Euclidean space ℝ^d. Given a bounded, translation invariant, and positive definite integral operator J on L²(ℝ^d), we introduce a determinantal interaction for a system of particles moving on ℝ^d as follows: the n points located at x₁,· · ·,x_n ∈ ℝ^d have the potential energy given by where j(x−y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d≥2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I+J)⁻¹ is a Gibbs measure admitted to the specification.     

Special functions associated with a certain fourth-order differential equation

We develop a theory of “special functions” associated with a certain fourth-order differential operator D_m,n\mathcal{D}_{\mu,\nu} on ℝ depending on two parameters μ,ν. For integers μ,ν≥−1 with μ+ν∈2ℕ₀, this operator extends to a self-adjoint operator on L ²(ℝ₊,x ^μ+ν+1 dx) with discrete spectrum. We find a closed formula for the generating functions of the eigenfunctions, from which we derive basic properties of the eigenfunctions such as orthogonality, completeness, L ²-norms, integral representations, and various recurrence relations.     

Affine symplectic geometry I: Applications to geometric inequalities

We use the integral geometric formulas in the symplectic space of geodesics of a Riemannian manifold to derive various inequalities of isoperimetric type. We give a sharp lower bound for the area of the minimal bubble spanning a spherical curve in ℝ³. We also present an “inverse Croke inequality” relating the area of the boundary of a complex domain in a Riemannian manifold to the injectivity radius and the volume of the domain. We prove a sharp lower bound for the ground state of the harmonic oscillator operator inL ²(M), whereM is a Hadamard manifold. This article is dedicated to my dear friend Julia Rashba     

Universality Limits for Exponential Weights

We establish universality in the bulk for fixed exponential weights on the whole real line. Our methods involve first-order asymptotics for orthogonal polynomials and localization techniques. In particular, we allow exponential weights such as | x | ^2β g ²(x)exp (−2Q(x)), where β>−1/2, Q is convex and Q ^′′ satisfies some regularity conditions, while g is positive, and has a uniformly continuous and slowly growing or decaying logarithm.      

On the Ricci curvature of a Randers metric of isotropic <Emphasis Type="Italic">S</Emphasis>-curvature

We derive the integral inequality of a Randers metric with isotropic S-curvature in terms of its navigation representation. Using the obtained inequality we give some rigidity results under the condition of Ricci curvature. In particular, we show the following result： Assume that an n-dimensional compact Randers manifold （M, F） has constant S-curvature c. Then （M, F） must be Riemannian if its Ricci curvature satisfies that Ric 〈 -（n - 1）c^2.     

Lower bounds at infinity of solutions of partial differential equations in the exterior of a proper cone

An extension of a classical theorem of Rellich to the exterior of a closed proper convex cone is proved: Let Γ be a closed convex proper cone inR ⁿ and −Γ′ be the antipodes of the dual cone of Γ. Let be a partial differential operator with constant coefficients inR ⁿ, whereQ(ζ)≠0 onR ⁿ−iΓ′ andP _i is an irreducible polynomial with real coefficients. Assume that the closure of each connected component of the set {ζ∈R ⁿ−iΓ′;P _j(ζ)=0, gradP _j(ζ)≠0} contains some real point on which gradP _j≠0 and gradP _j∉Γ∪(−Γ). LetC be an open cone inR ⁿ−Γ containing both normal directions at some such point, and intersecting each normal plane of every manifold contained in {ξ∈R ⁿ;P(ξ)=0}. Ifu∈ℒ′∩L _loc ² (R ⁿ−Γ) and the support ofP(−i∂/∂x)u is contained in Γ, then the condition implies that the support ofu is contained in Γ.     

Fundamental solution of the cauchy problem corresponding to the one-speed linear Boltzman equation for anisotropic media

We consider the fundamental solution E (t,x,s;s ₀) of the Cauchy problem for the one-speed linear Boltzman equation (∂/∂t+c(s,grad _x)+γ)E(t,x,s;s ₀)=γν∫ f((s, s′))E(t,x,s′; s₀)ds′+Ωδ(t)δ(x)δ (s−s ₀) that is assumed to be valid for any (t,x)∈Rⁿ⁺¹; morevoer, for t<0 the condition E(t,x,s; s₀)=0 holds. By using the Fourier-laplace transform in space-time arguments, the problem reduces to the study of an integral equation in the variables. For 0<ν≤1, the uniqueness and existence of the solution of the original problem are proved for any fixeds in the space of tempered distributions with supports in the front space-time cone. If the scattering media are of isotropic type (f(.)=1), the solution of the integral equation is given in explicit form. In the approximation of “small mean-free paths,” various weak limits of the solution are obtained with the help of a Tauberian-type theorem, for distributions. Bibliography: 4 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 250, 1998, pp. 319–332. Translated by Yu. B. Yanushanets.     

Phase portraits of the quadratic vector fields with a polynomial first integral

In this work we classify the phase portraits of all quadratic polynomial differential systems having a polynomial first integral. IfH(x, y) is a polynomial of degreen+1 then the differential systemx′=−∂H/∂y,y′=∂H/∂x is called a Hamiltonian system of degreen. We also prove that all the phase portraits that we obtain in this paper are realizable by Hamiltonian systems of degree 2.     

Estimates for the wave kernel near focal points on compact manifolds

This article deals with analogue statements of the so-called basic Strichartz inequality for certain values of the time variable t on a smooth compact manifold; that is we prove L^q′ → L^q bounds for the modified half-wave operator e^itP P⁻ (n+1)(1/2− 1/q) where for a set of times t which depends on the global behavior of the geodesic flow. Then we give estimates for the blow-up of the bounds as approaching the limit points of this set. In doing this we use facts from differential geometry and the calculus of variations.     

A counterexample to the dominating set conjecture

The metric polytope met _n is the polyhedron associated with all semimetrics on n nodes and defined by the triangle inequalities x _ij − x _ik − x _jk ≤ 0 and x _ij + x _ik + x _jk ≤ 2 for all triples i, j, k of {1,..., n}. In 1992 Monique Laurent and Svatopluk Poljak conjectured that every fractional vertex of the metric polytope is adjacent to some integral vertex. The conjecture holds for n ≤ 8 and, in particular, for the 1,550,825,600 vertices of met₈. While the overwhelming majority of the known vertices of met₉ satisfy the conjecture, we exhibit a fractional vertex not adjacent to any integral vertex.     

On the Wave Equation Associated to the Hermite and the Twisted Laplacian

The dispersive properties of the wave equation u _tt +Au=0 are considered, where A is either the Hermite operator −Δ+|x|² or the twisted Laplacian −(∇ _x −iy)²/2−(∇ _y +ix)²/2. In both cases we prove optimal L ¹−L ^∞ dispersive estimates. More generally, we give some partial results concerning the flows exp (itL ^ν ) associated to fractional powers of the twisted Laplacian for 0<ν<1.     

Extremals for Logarithmic Hardy-Littlewood-Sobelov Inequalities on Compact Manifolds

Let M be a closed, connected surface and let Γ be a conformal class of metrics on M with each metric normalized to have area V. For a metric g Γ, denote the area element by dV and the Laplace–Beltrami operator by Δ _g . We define the Robin mass m(x) at the point x M to be the value of the Green’s function G(x, y) at y = x after the logarithmic singularity has been subtracted off. The regularized trace of Δ _g ⁻¹ is then defined by trace Δ⁻¹ = ∫ _M m dV. (This essentially agrees with the zeta functional regularization and is thus a spectral invariant.) Let be the Laplace–Beltrami operator on the round sphere of volume V. We show that if there exists g Γ with trace Δ _g ⁻¹ < trace then the minimum of trace Δ⁻¹ over Γ is attained by a metric in Γ for which the Robin mass is constant. Otherwise, the minimum of trace Δ⁻¹ over Γ is equal to trace . In fact we prove these results in the general setting where M is an n-dimensional closed, connected manifold and the Laplace–Beltrami operator is replaced by any non-negative elliptic operator A of degree n which is conformally covariant in the sense that for the metric g we have . In this case the role of is assumed by the Paneitz or GJMS operator on the round n-sphere of volume V. Explicitly these results are logarithmic HLS inequalities for (M, g). By duality we obtain analogs of the Onofri–Beckner theorem. Received: February 2006, Accepted: March 2006     

Tubes and eigenvalues for negatively curved manifolds

We investigate the structure of the spectrum near zero for the Laplace operator on a complete negatively curved Riemannian manifoldM. If the manifold is compact and its sectional curvaturesK satisfy 1 ≤K < 0, we show that the smallest positive eigenvalue of the Laplacian is bounded below by a constant depending only on the volume ofM. Our result for a complete manifold of finite volume with sectional curvatures pinched between −a² and −1 asserts that the number of eigenvalues of the Laplacian between 0 and (n− 1)²/4 is bounded by a constant multiple of the volume of the manifold with the constant depending ona and the dimension only. Research supported in part by the Swiss National Science Foundation, the US National Science Foundation, and the PSC-CUNY Research Award Program.