Regularity, monotonicity and symmetry of positive solutions of m-Laplace equations

We consider the Dirichlet problem for positive solutions of the equation −Δ_m(u)=f(u) in a bounded smooth domain Ω, with f locally Lipschitz continuous, and prove some regularity results for weak solutions. In particular when f(s)>0 for s>0 we prove summability properties of , and Sobolev's and Poincaré type inequalities in weighted Sobolev spaces with weight |Du|^m−2. The point of view of considering |Du|^m−2 as a weight is particularly useful when studying qualitative properties of a fixed solution. In particular, exploiting these new regularity results we can prove a weak comparison principle for the solutions and, using the well known Alexandrov-Serrin moving plane method, we then prove a general monotonicity (and symmetry) theorem for positive solutions u of the Dirichlet problem in bounded (and symmetric in one direction) domains when f(s)>0 for s>0 and m>2. Previously, results of this type in general bounded (and symmetric) domains had been proved only in the case 1<m<2.     

Estimate of the L-Fourier transform norm on nilpotent Lie groups

Let 1<p?2 and q be such that . It is well known that the norm of the L^p-Fourier transform of the additive group is , where . For a nilpotent Lie group G, we obtain the estimate , where m is the maximal dimension of the coadjoint orbits. Such a result was known only for some particular cases.     

Improved Sobolev inequalities and Muckenhoupt weights on stratified Lie groups

We study in this article the improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms corresponding to Sobolev spaces and Besov spaces . When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in Rn and is classically obtained by a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When p=1, these two techniques are not available anymore and following M. Ledoux (2003) [12], in Rn, we will treat here the critical case p=1 for general stratified Lie groups in a weighted functional space setting. Finally, we will go a step further with a new generalization of improved Sobolev inequalities using weak-type Sobolev spaces.     

On duality between étale groupoids and Hopf algebroids

For any étale Lie groupoid G over a smooth manifold M, the groupoid convolution algebra of smooth functions with compact support on G has a natural coalgebra structure over the commutative algebra which makes it into a Hopf algebroid. Conversely, for any Hopf algebroid A over we construct the associated spectral étale Lie groupoid over M such that is naturally isomorphic to G. Both these constructions are functorial, and is fully faithful left adjoint to . We give explicit conditions under which a Hopf algebroid is isomorphic to the Hopf algebroid of an étale Lie groupoid G.     

Minimum number of ideal generators for a linear center perturbed by homogeneous polynomials

Using the algorithm presented in [J. Giné, X. Santallusia, On the Poincaré-Liapunov constants and the Poincaré series, Appl. Math. (Warsaw) 28 (1) (2001) 17-30] the Poincaré-Liapunov constants are calculated for polynomial systems of the form , , where P_n and Q_n are homogeneous polynomials of degree n. The objective of this work is to calculate the minimum number of ideal generators i.e., the number of functionally independent Poincaré-Liapunov constants, through the study of the highest fine focus order for n=4 and n=5 and compare it with the results that give the conjecture presented in [J. Giné, On the number of algebraically independent Poincaré-Liapunov constants, Appl. Math. Comput. 188 (2) (2007) 1870-1877]. Moreover, the computational problems which appear in the computation of the Poincaré-Liapunov constants and the determination of the number of functionally independent ones are also discussed.     

Hardy space estimates for the wave equation on compact Lie groups

Let −L be the Laplacian. In this paper, we prove that on a compact Lie group G of dimension n, the multiplier operator , s∈(0,1], extends to a bounded operator on the Hardy space Hp(G), 0<p<∞, if and only if . The result is an analogue of a well-known theorem in Euclidean space.     

A simple proof of inequalities of integrals of composite functions

In this paper we give a simple proof of inequalities of integrals of functions which are the composition of nonnegative continuous convex functions on a vector space Rm and vector-valued functions in a weakly compact subset of a Banach vector space generated by m-spaces for 1?p<+∞. Also, the same inequalities hold if these vector-valued functions are in a weakly* compact subset of a Banach vector space generated by m-spaces instead.     

Metric aspects of noncommutative homogeneous spaces

For a closed cocompact subgroup Γ of a locally compact group G, given a compact abelian subgroup K of G and a homomorphism satisfying certain conditions, Landstad and Raeburn constructed equivariant noncommutative deformations of the homogeneous space G/Γ, generalizing Rieffel's construction of quantum Heisenberg manifolds. We show that when G is a Lie group and G/Γ is connected, given any norm on the Lie algebra of G, the seminorm on induced by the derivation map of the canonical G-action defines a compact quantum metric. Furthermore, it is shown that this compact quantum metric space depends on ρ continuously, with respect to quantum Gromov-Hausdorff distances.     

Compactification and desingularization of spaces of polynomial Liénard equations

The paper deals with polynomial Liénard equations of type (m,n), i.e. planar vector fields associated to a scalar second order differential equation , with f and g polynomials of respective degree m and n. It is shown that, besides compactifying the phase plane, or the Liénard plane, one can also compactify and desingularize the space of Liénard equations of type (m,n) for each (m,n) separately, by adding both singular perturbation problems and Hamiltonian perturbation problems.     

Transportation cost inequalities on path and loop groups

Let G be a connected Lie group with the Lie algebra . The action of Cameron-Martin space on the path space P_e(G) introduced by L. Gross (Illinois J. Math. 36 (1992) 447) is free. Using this fact, we define the H-distance on P_e(G), which enables us to establish a transportation cost inequality on P_e(G). This method will be generalized to the path space over the loop group , so that we obtain a transportation cost inequality for heat measures on .     

On the topology of components of some Springer fibers and their relation to Kazhdan-Lusztig theory

We describe the irreducible components of Springer fibers for hook and two-row nilpotent elements of as iterated bundles of flag manifolds and Grassmannians. We then relate the topology (in particular, the intersection homology Poincaré polynomials) of the pairwise intersections of these components with the inner products of the Kazhdan-Lusztig basis elements of irreducible representations of the rational Iwahori-Hecke algebra of type A corresponding to the hook and two-row Young shapes.     

On cohomology rings of infinite groups

Let R be any ring (with 1), G a torsion free group and RG the corresponding group ring. Let be the cohomology ring associated with the RG-module M. Let H be a subgroup of finite index of G. The following is a special version of our main Theorem: Assume the profinite completion of G is torsion free. Then an element is nilpotent (under Yoneda’s product) if and only if its restriction to is nilpotent. In particular this holds for the Thompson group.There are torsion free groups for which the analogous statement is false.     

Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories

Let G be a Lie group which is the union of an ascending sequence G₁⊆G₂⊆? of Lie groups (all of which may be infinite-dimensional). We study the question when in the category of Lie groups, topological groups, smooth manifolds, respectively, topological spaces. Full answers are obtained for G the group Diffc(M) of compactly supported C^∞-diffeomorphisms of a σ-compact smooth manifold M; and for test function groups of compactly supported smooth maps with values in a finite-dimensional Lie group H. We also discuss the cases where G is a direct limit of unit groups of Banach algebras, a Lie group of germs of Lie group-valued analytic maps, or a weak direct product of Lie groups.     

On a conjecture concerning the orientation number of a graph

For a connected graph G containing no bridges, let D(G) be the family of strong orientations of G; and for any D∈D(G), we denote by d(D) the diameter of D. The orientation number of G is defined by . Let G(p,q;m) denote the family of simple graphs obtained from the disjoint union of two complete graphs K_p and K_q by adding m edges linking them in an arbitrary manner. The study of the orientation numbers of graphs in G(p,q;m) was introduced by Koh and Ng [K.M. Koh, K.L. Ng, The orientation number of two complete graphs with linkages, Discrete Math. 295 (2005) 91-106]. Define and . In this paper we prove a conjecture on α proposed by K.M. Koh and K.L. Ng in the above mentioned paper, for q≥p+4.