Let be a classical pseudodifferential operator of order on an -dimensional manifold . For the truncation to a smooth subset there is a well-known theory of boundary value problems when has the transmission property (preserves ) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian with , are not covered. They have instead the -transmission property defined in Hörmander's books, mapping into . In an unpublished lecture note from 1965, Hörmander described an -solvability theory for -transmission operators, departing from Vishik and Eskin's results. We here develop the theory in Sobolev spaces ( ) in a modern setting. It leads to not only Fredholm solvability statements but also regularity results in full scales of Sobolev spaces ( ). The solution spaces have a singularity at the boundary that we describe in detail. We moreover obtain results in Hölder spaces, which radically improve recent regularity results for fractional Laplacians.
Let be the upper half Euclidean space and let be any real number between 0 and 2. Consider the following Dirichlet problem involving the fractional Laplacian: Instead of using the conventional extension method of Caffarelli and Silvestre , we employ a new and direct approach by studying an equivalent integral equation Applying the , we prove the non-existence of positive solutions in the critical and subcritical cases under no restrictions on the growth of the solutions.
Let P be a classical pseudodifferential operator of order m epsilon C on an n-dimensional C-infinity manifold Omega(1). For the truncation P-Omega to a smooth subset Omega there is a well known theory of boundary value problems when P-Omega has the transmission property (preserves C-infinity ((Omega) over bar)) and is of integer order; the calculus of Boutet de Monvel. Many interesting operators, such as for example complex powers of the Laplacian (-Delta)(mu) with mu is not an element of Z, are not covered. They have instead the mu-transmission property defined in Hormander's books, mapping x(n)(mu) C-infinity ((Omega) over bar) into C-infinity((Omega) over bar). In an unpublished lecture note from 1965, Hormander described an L-2-solvability theory for mu-transmission operators, departing from Vishik and Eskin's results. We here develop the theory in Lp Sobolev spaces (1 infinity). The solution spaces have a singularity at the boundary that we describe in detail. We moreover obtain results in Holder spaces, which radically improve recent regularity results for fractional Laplacians. (C) 2014 Elsevier Inc. All rights reserved.
We introduce the notion of parallel Ricci tensor for real hypersurfaces in the complex quadric . According to the -principal or the -isotropic unit normal vector field , we give a complete classification of real hypersurfaces in with parallel Ricci tensor.
In this paper we prove that the focusing, -dimensional mass critical nonlinear Schrödinger initial value problem is globally well-posed and scattering for , , where is the ground state, and . We first establish an interaction Morawetz estimate that is positive definite when , and has the appropriate scaling. Next, we will prove an interaction Morawetz estimate localized to low frequencies, similar to the estimates made in . See also for localization to high frequencies in the energy critical case.
A long-standing conjecture of Stanley states that every Cohen–Macaulay simplicial complex is partitionable. We disprove the conjecture by constructing an explicit counterexample. Due to a result of Herzog, Jahan and Yassemi, our construction also disproves the conjecture that the Stanley depth of a monomial ideal is always at least its depth.
We prove, through a KAM algorithm, the existence of large families of stable and unstable quasi-periodic solutions for the NLS in any number of independent frequencies. The main tools are the existence of a non-degenerate integrable normal form proved in and a generalization of the quasi-Töplitz functions defined in .
We study the C 4 smooth convex bodies K⊂Rn+1 satisfying K(x)=u(x) 1-p , where x∈Sn, K is the Gauss curvature of ∂K, u is the support function of K, and p is a constant. In the case of n=2, either when p∈[-1, 0] or when p∈(0, 1) in addition to a pinching condition, we show that K must be the unit ball. This partially answers a conjecture of Lutwak, Yang, and Zhang about the uniqueness of the L p -Minkowski problem in R3. Moreover, we give an explicit pinching constant depending only on p when p∈(0, 1).
For a finite set and an integer , we say that is compatible with if is a Hadamard matrix. Let denote the uniformly discrete probability measure on . We prove that the class of infinite convolution (Moran measure) is a spectral measure provided that there is a common compatible to all the and . We also give some examples to illustrate the hypotheses and results, in particular, the last condition on is essential.
We initiate a new study of differential operators with symmetries and combine this with the study of branching laws for Verma modules of reductive Lie algebras. By the criterion for discretely decomposable and multiplicity-free restrictions of generalized Verma modules (T. Kobayashi (2012) ), we are brought to natural settings of parabolic geometries for which there exist unique equivariant differential operators to submanifolds. Then we apply a new method (F-method) relying on the Fourier transform to find singular vectors in generalized Verma modules, which significantly simplifies and generalizes many preceding works. In certain cases, it also determines the Jordan–Hölder series of the restriction for singular parameters. The F-method yields an explicit formula of such unique operators, for example, giving an intrinsic and new proof of Juhl's conformally invariant differential operators (Juhl (2009) ) and its generalizations to spinor bundles. This article is the first in the series, and the next ones include their extension to curved cases together with more applications of the F-method to various settings in parabolic geometries.
In this paper we consider the following of equations on a compact surface: which is motivated by the study of models in non-abelian Chern–Simons theory. Here are smooth positive functions, two positive parameters, points of the surface and non-negative numbers. We prove a general existence result using variational methods. The same analysis applies to the following mean field equation which arises in fluid dynamics.
Let be a symmetrizable Kac–Moody algebra, and the corresponding quantum group. We showed in that the braided Coxeter structure on integrable, category representations of which underlies the -matrix actions arising from the Levi subalgebras of and the quantum Weyl group action of the generalized braid group can be transferred to integrable, category representations of . We prove in this paper that, up to unique equivalence, there is a unique such structure on the latter category with prescribed restriction functors, -matrices, and local monodromies. This extends, simplifies and strengthens a similar result of the second author valid when is semisimple, and is used in to describe the monodromy of the rational Casimir connection of in terms of the quantum Weyl group operators of . Our main tool is a refinement of Enriquez's universal algebras, which is adapted to the describing a Lie bialgebra graded by the non-negative roots of .
We introduce different Finsler metrics on the space of smooth Kähler potentials that will induce a natural geometry on various finite energy classes . Motivated by questions raised by R. Berman, V. Guedj and Y. Rubinstein, we characterize the underlying topology of these spaces in terms of convergence in energy and give applications of our results to existence of Kähler–Einstein metrics on Fano manifolds.
We study the C-4 smooth convex bodies K subset of Rn+1 satisfying K (x) = u(x)(1-P), where x is an element of S-n K is the Gauss curvature of partial derivative K, u is the support function of K, and p is a constant. In the case of n = 2, either when p is an element of [-1,0] or when p is an element of (0,1) in addition to a pinching condition, we show that K must be the unit ball. This partially answers a conjecture of Lutwak, Yang, and Zhang about the uniqueness of the L-p-Minkowski problem in R-3. Moreover, we give an explicit pinching constant depending only on p when p is an element of (0,1). (C) 2015 Elsevier Inc. All rights reserved.
The first goal of this paper is to provide an abstract framework in which to formulate and study local duality in various algebraic and topological contexts. For any stable ∞-category together with a collection of compact objects we construct local cohomology and local homology functors satisfying an abstract version of local duality. When specialized to the derived category of a commutative ring and a suitable ideal in , we recover the classical local duality due to Grothendieck as well as generalizations by Greenlees and May. More generally, applying our result to the derived category of quasi-coherent sheaves on a quasi-compact and separated scheme implies the local duality theorem of Alonso Tarrío, Jeremías López, and Lipman. As a second objective, we establish local duality for quasi-coherent sheaves over many algebraic stacks, in particular those arising naturally in stable homotopy theory. After constructing an appropriate model of the derived category in terms of comodules over a Hopf algebroid, we show that, in familiar cases, the resulting local cohomology and local homology theories coincide with functors previously studied by Hovey and Strickland. Furthermore, our framework applies to global and local stable homotopy theory, in a way which is compatible with the algebraic avatars of these theories. In order to aid computability, we provide spectral sequences relating the algebraic and topological local duality contexts.
We show that whenever and are nonamenable factors in a large class of von Neumann algebras that we call and which contains all free Araki–Woods factors, the tensor product factor retains the integer and each factor up to stable isomorphism, after permutation of the indices. Our approach unifies the Unique Prime Factorization (UPF) results from and moreover provides new UPF results in the case when are free Araki–Woods factors. In order to obtain the aforementioned UPF results, we show that Connes's bicentralizer problem has a positive solution for all type factors in the class .
This paper deals with left non-degenerate set-theoretic solutions to the Yang–Baxter equation (= LND solutions), a vast class of algebraic structures encompassing groups, racks, and cycle sets. To each such solution there is associated a shelf (i.e., a self-distributive structure) which captures its major properties. We consider two (co)homology theories for LND solutions, one of which was previously known, in a reduced form, for biracks only. An explicit isomorphism between these theories is described. For groups and racks we recover their classical (co)homology, whereas for cycle sets we get new constructions. For a certain type of LND solutions, including quandles and non-degenerate cycle sets, the (co)homologies split into the degenerate and the normalized parts. We express 2-cocycles of our theories in terms of group cohomology, and, in the case of cycle sets, establish connexions with extensions. This leads to a construction of cycle sets with interesting properties.
With the intent of laying the groundwork for a program that aims at explicitly describing the space of Cartan (i.e. multiplicative) connections on a general proper Lie groupoid, we begin to investigate the space of such connections in the regular case. We point out that there is a close relationship between Cartan connections on a proper regular groupoid and representations of the groupoid on its own longitudinal bundle (i.e. on the vector distribution tangent to its orbits). This observation enables us to reduce the original problem to a simpler one. We carry out a prospective study of the latter problem, and apply the resulting analysis to produce a number of examples in rank two which serve to illustrate the diversity of the possible obstructions to the existence of multiplicative connections.
We prove that the notion of Drinfeld center defines a functor from the category of indecomposable multi-tensor categories with morphisms given by bimodules to that of braided tensor categories with morphisms given by monoidal bimodules. Moreover, we apply some ideas from the physics of topological orders to prove that the center functor restricted to indecom- posable multi-fusion categories (with additional conditions on the target category) is fully faithful. As byproducts, we provide new proofs to some important known results in fusion categories. In physics, this fully faithful functor gives the precise mathematical description of the boundary-bulk relation for 2+1D anomaly-free topological orders with gapped boundaries.