bases for some reciprocity algebras ii abstract we construct bases for the stable branching algebras for the symmetric pairs gl n o n o n m o n o m and sp 2 n gl n

a positive mass theorem in three dimensional cauchy riemann geometry in this note we summarize the results from 6 on the positive mass problem in 3 dimensional cr cauchy riemann geometry

on the positivity of the effective action in a theory of random surfaces it is shown that the functional s eta frac 1 24 pi int left frac 1 2 left nabla eta right 2 2 eta right d mu 0 defined onc functions on the two dimensional sphere satisfies the inequalitys 0 if is subject to the constraint int e eta 1 d mu 0 0 the minimums 0 is attained at the solutions of the euler lagrange equations the proof is based on a sharper version of moser trudinger s inequality due to aubin which holds under the additional constraint int e eta xd mu 0 0 this condition can always be satisfied by exploiting the invariance ofs under the conformal transformations ofs2 the result is relevant for a recently proposed formulation of a theory of random surfaces

sharp constants for moser trudinger inequalities on spheres in complex space ℂn the main results of this paper concern sharp constants for the moser trudinger inequalities on spheres in complex space c n we derive moser trudinger inequalities for smooth functions and holomorphic functions with different sharp constants see theorem 1 1 the sharp moser trudinger inequalities under consideration involve the complex tangential gradients for the functions and thus we have shown here such inequalities in the cr setting though there is a close connection in spirit between inequalities proven here on complex spheres and those on the heisenberg group for functions with compact support in any finite domain proven earlier by the same authors 17 derivation of the sharp constants for moser trudinger inequalities on complex spheres are more complicated and difficult to obtain than on the heisenberg group variants of moser onofri type inequalities are also given on complex spheres as applications of our sharp inequalities see theorems 1 2 and 1 3 one of the key ingredients in deriving the main theorems is a sharp representation formula for functions on the complex spheres in terms of complex tangential gradients see theorem 1 4 c 2004 wiley periodicals inc

compact isospectral sets of surfaces on etudie des ensembles de surfaces qui sont isospectrales par rapport a l operateur de laplace beltrami

semi linear liouville theorems in the heisenberg group via vector field methods we consider a global liouville type theorem for semi linear elliptic equation on heisenberg group our main idea is the method of integration by parts for a vector field analogous to the quantity in gidas and spruck 1981 5 c 2009 elsevier inc all rights reserved

hausdorff dimension of limit sets for spherical cr manifolds abstract let m 2 n 1 n 1 be a compact spherical cr manifold suppose m 2 n 1 is its universal cover and m 2 n 1 s 2 n 1 is on injective cr developing map where s 2 n 1 is the standard unit sphere in the complex n 1 space c n 1 then m 2 n 1 is of the quotient form where is a simply connected open set in s 2 n 1 and is a complex klein group acting on properly discontinuously in this paper we show that if the cr yamabe invariant of m 2 n 1 is positive then the carnot hausdorff dimension of the limit set of is bounded above by n s m 2 n 1 where s m 2 n 1 1 and is a cr invariant the method that we adopt is analysis of the cr invariant laplacian we also explain the geometric origin of this question

improved moser trudinger type inequalities in the hyperbolic space hn we establish an improved version of the moser trudinger inequality in the hyperbolic space hnhn n 2n 2 namely we prove the following result for any 0 n 1nn then we have supu c0 hn hn gu gndvolg hn u ndvolg 1 hn n n u nn 1 dvolg where n n n 11n 1 n 1 n 1 denotes the surface area of the unit sphere in rnrn and n t et j 0n 2tjj this improves the moser trudinger inequality in hyperbolic spaces obtained recently by mancini and sandeep 2010 by mancini et al 2013 and by adimurthi and tintarev 2010 for 0 0 in the limiting case n 1n n we prove a moser trudinger inequality with exact growth in hnhn supu c0 hn hn gu gndvolg n 1n n hn u ndvolg 11 hn u ndvolg hn n n u nn 1 1 u nn 1dvolg this improves the moser trudinger inequality with exact growth in hnhn established by lu and tang 2016 these inequalities are achieved from the comparison of the symmetric non increasing rearrangement of a function both in the hyperbolic and the euclidean spaces and the same inequalities in the euclidean space this approach seems to be new comparing with the previous ones

proof that pyramid networks are 1 hamiltonian connected with high probability wu revealed in 2001 that pyramid networks are hamiltonian connected this investigation demonstrates that a pyramid network with one faulty node or one faulty edge is hamiltonian connected excluding some special faulty cases by building a hamiltonian path between any two distinct nodes in it although a pyramid network with one fault is not hamiltonian connected this study indicates that a pyramid network is 1 hamiltonian connected with a very high probability

sharp sobolev inequalities on the sphere and the moser trudinger inequality where de denotes normalized surface measure v is the conformal gradient and q 2n n 2 a modern folklore theorem is that by taking the infinitedimensional limit of this inequality one obtains the gross logarithmic sobolev inequality for gaussian measure which determines nelson s hypercontractive estimates for the hermite semigroup see 8 one observes using conformal invariance that the above inequality is equivalent to the sharp sobolev inequality on rn for which boundedness and extremal functions can be easily calculated using dilation invariance and geometric symmetrization the roots here go back to hardy and littlewood the advantage of casting the problem on the sphere is that the role of the constants is evident and one is led immediately to the conjecture that this inequality should hold whenever possible for example 2 q 0o if n 2 this is in fact true and will be demonstrated in section 2 a clear question at this point is what is the situation in dimension 2 two important arguments 25 26 27 dealt with this issue both motivated by geometric variational problems because q goes to infinity for dimension 2 the appropriate function space is the exponential class responding in part

on a conformally invariant variational problem on differential forms abstract we consider a conformally invariant constrained variational problem on p differential forms for p dim m 2 2 coming from conformal geometry and the associated yamabe type system of partial differential equations it is shown that the infimum p m of the functional is not bigger than p r n and if p m p r n then p m is achieved by a c 2 p form for some 0

sharp moser trudinger inequality on the heisenberg group at the critical case and applications let h cn r be the n dimensional heisenberg group q 2n 2 be the homogeneous dimension of h q qq 1 and z 4 t2 14 be the homogeneous norm of z t h then we prove the following sharp moser trudinger inequality on h theorem 1 6 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q 0 q 1 q there holds sup u 1 1 h1 exp u q q 1 k 0q 2 kk u kq q 1 c q the constant q 1 q is best possible in the sense that the supremum is infinite if q 1 q here is any positive number and u 1 h hu q h u q 1 q our result extends the sharp moser trudinger inequality by cohn and lu 2001 19 on domains of finite measure on h and sharpens the recent result of cohn et al 2012 18 where such an inequality was studied for the subcritical case q 1 q we carry out a completely different and much simpler argument than that in cohn et al 2012 18 to conclude the critical case our method avoids using the rearrangement argument which is not available in an optimal way on the heisenberg group and can be used in more general settings such as riemanian manifolds appropriate metric spaces etc as applications we establish the existence and multiplicity of nontrivial nonnegative solutions to certain nonuniformly subelliptic equations of q laplacian type on the heisenberg group theorems 1 8 1 9 1 10 and 1 11 divh hu q 2 hu v u q 2u f u eh with nonlinear terms f of maximal exponential growth exp u qq 1 as u in particular when e 0 the existence of a nontrivial solution is also given

concentration compactness principle for the sharp adams inequalities in bounded domains and whole space rn abstract we prove an improvement of the sharp adams inequality in w 0 m n m where is a bounded domain in r n inspired by lions concentration compactness principle for the sharp moser trudinger inequality our method gives an alternative approach to a concentration compactness principle in w 0 m n m recently established by do o and macedo furthermore we obtain a sharp threshold for m odd improving the one of do o and macedo our approach is also successfully applied to whole space r n to establish the improvements of the sharp adams inequalities in w m n m r n this type of improvement is still unknown in general except the case m 1 due to do o de souza de medeiros and severo our method is a further development of the one of cerny cianchi and hencl combining with some estimates for the decreasing rearrangement of a function in terms of the one of its higher order derivatives

estimates and extremals for zeta function determinants on four manifolds leta be a positive integral power of a natural conformally covariant differential operator on tensor spinors in a riemannian manifold suppose thata is formally self adjoint and has positive definite leading symbol for example a could be the conformal laplacian yamabe operator l or the square of the dirac operator open image in new window within the conformal class left g e 2w g 0 w in c infty m right of an einstein locally symmetric background metricgo on a compact four manifoldm we use an exponential sobolev inequality of adams to show that bounds on the functional determinant ofa and the volume ofg imply bounds on thew2 2 norm of the conformal factorw provided that a certain conformally invariant geometric constantk k m g o a is strictly less than 32 2 we show for the operatorsl and open image in new window that indeedk 32 2 except when m g o is the standard sphere or a hyperbolic space form on the sphere a centering argument allows us to obtain a bound of the same type despite the fact thatk is exactly equal to 32 2 in this case finally we use an inequality of beckner to show that in the conformal class of the standard four sphere the determinant ofl or of open image in new window is extremized exactly at the standard metric and its images under the conformal transformation groupo 5 1

extremal functions for moser s inequality let q be a bounded smooth domain in ri and u x a c1 function with compact support in q moser s inequality states that there is a constant co depending only on the dimension n such that 1 wn1 n j ee n u dx co where iqi is the lebesgue measure of q and wn 1 the surface area of the unit ball in rn we prove in this paper that there are extremal functions for this inequality in other words we show that the sup j nw u dx u e wj 11vuh1n 1 is attained earlier results include carleson chang 1986 q is a ball in any dimension and flucher 1992 q is any domain in 2 dimensions

optimal path cover problem on block graphs abstract let g v e be a block graph first we show that an algorithm for finding the path partition number p g by j h yan and g j chang gives wrong answers to some block graphs then we present an efficient algorithm for finding a minimum path partition of g not just the path partition number p g the complexity of this algorithm is o m where m e

fundamental solutions for non divergence form operators on stratified groups we construct the fundamental solutions and for the non divergence form operators i j a i j x t x i x j t and i j a i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we also provide gaussian estimates of r and its derivatives and some results for the relevant cauchy problem suitable long time estimates of allow us to construct using both t saturation and approximation arguments

compactness of pseudohermitian structures with integral bounds on curvature in this paper we show a compactness criterion for contact forms in a fixed cr structure i e conformal pseudohermitian structures assuming a volume bound and lp bounds p 2 on the tanaka webster curvature the pseudohermitian torsion and its covariant derivative we also need the l2 bound on the derivative of the tanaka webster curvature along the characteristic vector field as an application we can show that the cr automorphism group is compact if m is not cr spherical or the cr yamabe constant is negative

an extension problem for the cr fractional laplacian we show that the conformally invariant fractional powers of the sub laplacian on the heisenberg group are given in terms of the scattering operator for an extension problem to the siegel upper halfspace remarkably this extension problem is different from the one studied among others by caffarelli and silvestre we also prove an energy identity that yields a sharp trace sobolev embedding

noncommutative residue for heisenberg manifolds applications in cr and contact geometry abstract this paper has four main parts in the first part we construct a noncommutative residue for the hypoelliptic calculus on heisenberg manifolds that is for the class of h do operators introduced by beals greiner and taylor this noncommutative residue appears as the residual trace on integer order h dos induced by the analytic extension of the usual trace to non integer order h dos moreover it agrees with the integral of the density defined by the logarithmic singularity of the schwartz kernel of the corresponding h do in addition we show that this noncommutative residue provides us with the unique trace up to constant multiple on the algebra of integer order h dos in the second part we give some analytic applications of this construction concerning zeta functions of hypoelliptic operators logarithmic metric estimates for green kernels of hypoelliptic operators and the extension of the dixmier trace to the whole algebra of integer order h dos in the third part we present examples of computations of noncommutative residues of some powers of the horizontal sublaplacian and the contact laplacian on contact manifolds in the fourth part we present two applications in cr geometry first we give some examples of geometric computations of noncommutative residues of some powers of the horizontal sublaplacian and of the kohn laplacian second we make use of the framework of noncommutative geometry and of our noncommutative residue to define lower dimensional volumes in pseudohermitian geometry e g we can give sense to the area of any 3 dimensional cr manifold endowed with a pseudohermitian structure on the way we obtain a spectral interpretation of the einstein hilbert action in pseudohermitian geometry

a new approach to sharp moser trudinger and adams type inequalities a rearrangement free argument abstract the main purpose of this paper is two fold on the one hand we will develop a new approach to establish sharp singular moser trudinger and adams type inequalities in unbounded domains of euclidean spaces without using the standard symmetrization on the other hand we will prove the sharp singular adams type inequality on high order sobolev spaces w m n m r n of arbitrary integer order m theorem 1 1 which improves the results of ruf and sani 2013 48 where sharp adams inequalities were established for even m and those of the authors lam and lu 2012 28 29 for odd m but with different and more restricted norms we first establish the sharp local singular adams inequality on domains in r n of finite measure theorem 1 4 we take a perspective that any function in the high order sobolev spaces w m n m r n can be represented as a bessel potential thus we can fully use the tools from harmonic analysis and the kernel properties of the polyharmonic operators i m 2 once we have established this sharp local adams inequality then we can adapt the rearrangement free method we will develop in this paper to derive a global sharp adams inequality from a local one our argument substantially simplifies those in ruf and sani 2013 48 and lam and lu 2012 28 29 and avoids the use of rather deep and complicated comparison principle of solutions to polyharmonic operators used in ruf and sani 2013 48 lam and lu 2012 28 29 moreover our theorem holds on sobolev spaces w n r n of any positive fractional order n theorems 1 6 and 1 7 as consequences we can also establish sharp adams inequalities with less restrictions on the sobolev norms see theorems 1 2 1 3 and 1 5 our approach is surprisingly simple and general and can be easily applied to scenarios such as riemannian and sub riemannian manifolds where symmetrization argument does not work see e g on the heisenberg group lam and lu 2012 30

on the moser trudinger inequality in complex space abstract in this paper we prove the pluricomplex counterpart of the moser trudinger and sobolev inequalities in complex space we consider these inequalities for plurisubharmonic functions with finite pluricomplex energy and we estimate the concerned constants

prescribing webster scalar curvature on cr manifolds of negative conformal invariants abstract in this paper we are interested in solving the following partial differential equation u r u f u 1 2 n on a compact strictly pseudo convex cr manifold m of dimension 2 n 1 with n 1 this problem naturally arises when solving the prescribing webster scalar curvature problem on m with the prescribed function f using variational techniques we prove several non existence existence and multiplicity results when the function f is sign changing

isocapacity estimates for hessian operators abstract through a new powerful potential theoretic analysis this paper is devoted to discovering the geometrically equivalent isocapacity forms of chou wang s sobolev type inequality and tian wang s moser trudinger type inequality for the fully nonlinear 1 k n 2 hessian operators

cr yamabe constant cr yamabe flow and its soliton abstract on a compact strictly pseudoconvex cr manifold m we consider the cr yamabe constant of its infinite conformal covering by using the maximum principles we then prove a uniqueness theorem for the cr yamabe flow on a complete noncompact cr manifold finally we obtain some properties of the cr yamabe soliton on complete noncompact cr manifolds

multiplicity result for a nonhomogeneous yamabe type equation involving the kohn laplacian abstract we consider the nonhomogeneous yamabe equation on a bounded set of the heisenberg group h u u q 2 u f where f is a small perturbation in the c 0 sense under suitable hypotheses we will state a multiplicity existence result for positive solutions with zero dirichlet boundary conditions

fundamental solution for the q laplacian and sharp moser trudinger inequality in carnot groups abstract for a general carnot group g with homogeneous dimension q we prove the existence of a fundamental solution of the q laplacian u q and a constant a q 0 such that exp a q u q is a homogeneous norm on g this implies a representation formula for smooth functions on g which is used to prove the sharp carnot group version of the celebrated moser trudinger inequality

moser trudinger and beckner onofri s inequalities on the cr sphere we derive sharp moser trudinger inequalities on the cr sphere the rst type is in the adams form for powers of the sublaplacian and for general spectrally dened operators on the space of cr pluriharmonic functions we will then obtain the sharp beckner onofri inequality for crpluriharmonic functions on the sphere and as a consequence a sharp logarithmic hardy littlewood sobolev inequality in the form given by carlen and loss

adams inequalities on measure spaces abstract in 1988 adams obtained sharp moser trudinger inequalities on bounded domains of r n the main step was a sharp exponential integral inequality for convolutions with the riesz potential in this paper we extend and improve adams results to functions defined on arbitrary measure spaces with finite measure the riesz fractional integral is replaced by general integral operators whose kernels satisfy suitable and explicit growth conditions given in terms of their distribution functions natural conditions for sharpness are also given most of the known results about moser trudinger inequalities can be easily adapted to our unified scheme we give some new applications of our theorems including sharp higher order moser trudinger trace inequalities sharp adams moser trudinger inequalities for general elliptic differential operators scalar and vector valued for sums of weighted potentials and for operators in the cr setting

concentration compactness principle for trudinger moser inequalities with logarithmic weights and their applications abstract in this paper we establish a sharp concentration compactness principle associated with the trudinger moser inequality on sobolev spaces with logarithmic weights as applications we establish the existence of ground state solutions to the following equation with critical double exponential nonlinearity d i v u n 2 u x f x u in b 1 0 u 0 in b 1 0 u 0 on b 1 0 where x log e x n 1 is the logarithmic weight the nonlinear term f x u is continuous radial in x b 1 0 and has critical double exponential growth which behaves like exp e e n u n

extremals of determinants of laplacians on etudie le determinant associe au laplacien en fonction de la metrique sur une surface donnee et en particulier ses valeurs extremes quand la metrique est bien restreinte

existence of extremal functions for the stein weiss inequalities on the heisenberg group abstract in this paper we establish the existence of extremals for two kinds of stein weiss inequalities on the heisenberg group more precisely we prove the existence of extremals for the stein weiss inequalities with full weights in theorem 1 1 and the stein weiss inequalities with horizontal weights in theorem 1 4 different from the proof of the analogous inequality in euclidean spaces given by lieb 26 using riesz rearrangement inequality which is not available on the heisenberg group we employ the concentration compactness principle to obtain the existence of the maximizers on the heisenberg group our result is also new even in the euclidean case because we don t assume that the exponents of the double weights in the stein weiss inequality 1 1 are both nonnegative see theorem 1 3 and more generally theorem 1 5 therefore we extend lieb s celebrated result of the existence of extremal functions of the stein weiss inequality in the euclidean space to the case where the exponents are not necessarily both nonnegative see theorem 1 3 furthermore since the absence of translation invariance of the stein weiss inequalities additional difficulty presents and one cannot simply follow the same line of lions idea to obtain our desired result our methods can also be used to obtain the existence of optimizers for several other weighted integral inequalities theorem 1 5

a sharp inequality of j moser for higher order derivatives in fact go nwl 1 where wnis the area of the surface of the unit n ball also he notes that if 3 exceeds go then 1 can no longer hold with some co independent of the u satisfying 2 the fact that this result holds for some 3 0 independent of u has been observed now by several authors with a variety of proofs see 24 20 23 22 10 4 1 2 13 14 for example the earliest of these arguments expands the exponential function in a power series then observing that the norm in the imbedding

convergence of the cr yamabe flow we consider the cr yamabe flow on a compact strictly pseudoconvex cr manifold m of real dimension 2n 1 we prove convergence of the cr yamabe flow when n 1 or m is spherical

the asymptotic expansion of a cr invariant and grauert tubes cr manifolds the abstract models of real hypersurfaces in complex manifolds are 2n 1 dimensional manifolds m with a codimension one subbundle h of the tangent bundle which carries a complex structure the cr refers to cauchy riemann because for m c c n l the subbundle h consists of induced cauchy riemann operators there is a wealth of geometry and analysis associated with these structures especially when the cr manifolds are strictly pseudoconvex for example two strictly pseudoconvex domains are biholomorphically equivalent if and only if their boundaries are cr equivalent a fundamental problem in cr geometry is to find computable invariants associated with the cr structures the global cr invariant we will consider in this paper is the chem simons type invariant discovered by bums and epstein b e 1 it is a real valued global cr invariant of a compact 3dimensional strictly pseudo convex cr manifold whose holomorphic tangent bundle is trivial cheng and lee independently found this invariant and extend the definition of b e invariant to a relative invariant on an arbitrary compact 3 dimensional cr manifold cf c l we will evaluate this g asymptotically on the boundary of small grauert tubes before posing the question in a more precise form we will first say a few workds about grauert tubes let x be a real analytic manifold then every coordinate patch u c r n can be thickened to obtain an open set r c c n since the coordinate changes of x are real analytic maps by taking power series expansions and by shrinking c u to get convergence they can be extended holomorphically to such enlarged domains and thus they can be used as holomorphic transition

harnack inequality for non divergence form operators on stratified groups we prove lower bounds for the fundamental solutions of the non divergence form operators j i j x t x i x j t and i j i j x x i x j where the x i s are hormander vector fields generating a stratified group g and a i j i j is a positive definite matrix with holder continuous entries we then prove an invariant harnack inequality for such operators as a byproduct we also study some relevant properties of the green functions on bounded domains

sharp subcritical moser trudinger inequalities on heisenberg groups and subelliptic pdes the aim of this paper is to prove a sharp subcritical moser trudinger inequality on the whole heisenberg group let h cn rh cn r be the n n dimensional heisenberg group q 2n 2q 2n 2 be the homogeneous dimension of hh q qq 1 and z 4 t2 14 be the homogeneous norm of z t h z t h then we establish the following inequality on hh theorem 1 1 there exists a positive constant q q 2 n 12 q 12 q2 1 n 1 q 1 such that for any pair satisfying 0 q0 q 0 q 1 q there exists a constant 0 c c 0 c c such that the following inequality holds sup hu lq h 11 u lq h q h1 exp u q q 1 k 0q 2 kk u kq q 1 c the above result is the best possible in the sense when q 1 q the integral is still finite for any u w1 q h u w1 q h but the supremum is infinite in contrast to the analogous inequality in euclidean spaces proved in adachi and tanaka 1999 6 using symmetrization our argument is completely different and avoids the symmetrization method which is not available on the heisenberg group in an optimal way moreover our restriction on the norm hu lq h 1 hu lq h 1 of the function uu is much weaker than hu lq h u lq h 1 which was assumed in lam and lu 2012 16 as a consequence our inequality fails at q 1 q in contrast to the one in 16 as an application of this inequality we will prove that the following nonlinear subelliptic equation of qq laplacian type without perturbation equation 0 1 qu v u q 2u f u in h has a nontrivial weak solution where the nonlinear term ff has the critical exponential growth e u qq 1 as u u but does not satisfy the ambrosetti rabinowitz condition

an equation of monge ampere type in conformal geometry and four manifolds of positive ricci curvature we formulate natural conformally invariant conditions on a 4 manifold for the existence of a metric whose schouten tensor satisfies a quadratic inequality this inequality implies that the eigenvalues of the ricci tensor are positively pinched

the approximability of the weighted hamiltonian path completion problem on a tree given a graph the hamiltonian path completion problem is to find an augmenting edge set such that the augmented graph has a hamiltonian path in this paper we show that the hamiltonian path completion problem will unlikely have any constant ratio approximation algorithm unless np p this problem remains hard to approximate even when the given subgraph is a tree moreover if the edge weights are restricted to be either 1 or 2 the hamiltonian path completion problem on a tree is still np hard then it is observed that this problem is strongly np hard so it does not have any fully polynomial time approximation scheme fptas unless np p when the given tree is a k tree we give an approximation algorithm with performance ratio 1 5

lq regularity for p laplace type baouendi grushin equations abstract in this work we prove the interior l q estimates for the weak solution of a class of p laplace type baouendi grushin equations by the compactness method and its intrinsic metric associated with the geometry of the baouendi grushin vector fields

extremal metrics of zeta function determinants on 4 manifolds in conformal geometry the sobolev inequality at a critical exponent has received much attention in particular the determination of the best constants has played a crucial role in the yamabe problem in dimension two the analogous problem deals with the moser trudinger inequality on a compact riemann surface m2 there exists a constant c c m so that f e47w2 c m if f ivw12 1 and f w 0 the connection of this inequality with geometry comes through the zeta functional determinant of the laplacian as defined by ray singer for a riemannian metric g let 0 a1 a2 be the spectrum

the fefferman metric and pseudo hermitian invariants c fefferman has shown that a real strictly pseudoconvex hypersurface in complex n space carries a natural conformai lorentz metric on a circle bundle over the manifold this paper presents two intrinsic constructions of the metric valid on an abstract cr manifold one is in terms of tautologous differential forms on a natural circle bundle the other is in terms of webster s pseudohermitian invariants these results are applied to compute the connection and curvature forms of the fefferman metric explicitly

optimal control in prescribing webster scalar curvatures on 3 dimensional pseudo hermitian manifolds abstract in this work we give new existence and multiplicity results for the solutions of the prescription problem for the webster scalar curvature on a 3 dimensional pseudo hermitian manifold the critical points of prescribed functions verify mixed conditions we establish some morse inequalities at infinity and a poincare hopf type formula to give a lower bound on the number of solutions as well as an upper bound for the morse index of such solutions

a cr poincare inequality on the complex sphere abstract the main purpose of this paper is to prove a cr poincare inequality with sharp exponent on the sphere in complex space we use the complex tangential gradient on the sphere instead of the usual laplace beltrami gradient on the sphere

the dirichlet problem for the kohn laplacian on the heisenberg group ii for x y t rn rn r denote xj xj 2yj t yj yj 2xj t and l 14 j 1nxj2 yj2 t when n 2q la represents the action of the kohn laplacian b on q forms on the heisenberg group for n n we construct a parametrix for the dirichlet problem in smooth domains d near non characteristic points of d a point w of d is non characteristic if one of x1 xn y1 yn is transverse to d at w this yields sharp local estimates in the dirichlet problem in the appropriate non isotropic lipschitz classes the main new tool is a convolution calculus of pseudo differential operators that can be applied to the relevant layer potentials for which the usual asymptotic composition formula is false characteristic points are treated in part ii

an integral type brezis nirenberg problem on the heisenberg group abstract this paper is devoted to a class of integral type brezis nirenbreg problems on the heisenberg group they are a class of nonlinear integral equations on the bounded domains of heisenberg group and related to the cr yamabe problems on cr manifold based on the sharp hardy littlewood sobolev inequalities on the heisenberg group the nonexistence and existence results are obtained

sharp hardy littlewood sobolev inequalities on quaternionic heisenberg groups this paper is the second one following christ et al nonlinear anal 130 361 395 2016 in a series considering sharp hardy littlewood sobolev inequalities on groups of heisenberg type the first important breakthrough was made in frank et al ann math 176 349 381 2012 in this paper analogous results are obtained for the octonionic heisenberg group

on certain rings of highest weight vectors abstract let r m n be the ring of highest weight vectors of the action of o m gl n on the polynomial algebra of m n matrices we determine r m 2 and find generators for r m 3 in particular the results about r m 2 give branching rules and information about the structure of holomorphic representations of sp 4

positive solutions of yamabe type equations with function coefficients on graphs abstract in this paper we study the yamabe type semi linear equation u a x u b x u 1 on both finite and locally finite graphs we determine certain conditions on the coefficients of the equation that guarantee existence of positive solutions in both cases some uniqueness results are also established

adams inequalities for bi laplacian and extremal functions in dimension four let be a smooth oriented bounded domain be the sobolev space and be the first eigenvalue of the bi laplacian operator 2 then for any a 0a o we have and the above supremum is infinity when a o this strengthens adams inequality in dimension 4 d adams a sharp inequality of j moser for high order derivatives ann of math 128 1988 365 398 where he proved the above inequality holds for a 0 moreover we prove that for sufficiently small a an extremal function for the above inequality exists as a special case of our results we thus show that there exists with such that this establishes the existence of an extremal function of the original adams inequality in dimension 4

sub elliptic global high order poincare inequalities in stratified lie groups and applications sharp poincare inequalities on balls or chain type bounded domains have been extensively studied both in classical euclidean space and carnot caratheodory spaces associated with sub elliptic vector fields e g vector fields satisfying hormander s condition in this paper we investigate the validity of sharp global poincare inequalities of both first order and higher order on the entire nilpotent stratified lie groups or on unbounded extension domains in such groups we will show that simultaneous sharp global poincare inequalities also hold and weighted versions of such results remain to be true more precisely let g be a nilpotent stratified lie group and f be in the localized non isotropic sobolev space w m p loc g w here 1 p q mand q is the homogeneous dimension of the lie group g suppose that the mth sub elliptic derivatives of f is globally l p integrable i e g x m f x p dx is finite but assume that lower order sub elliptic derivatives are only locally l p integrable we denote the space of such functions as b m p g we prove a high order poincare inequality for f minus a polynomial of order m 1 over the entire space g or unbounded extension domains as applications we will prove a density theorem stating that smooth functions with compact support are dense in b m p g modulus a finite dimensional subspace

geometric inequalities on locally conformally flat manifolds through the study of some elliptic and parabolic fully nonlinear pdes we establish conformal versions of the quermassintegral inequality the sobolev inequality and the moser trudinger inequality for the geometric quantities associated to the schouten tensor on locally conformally flat manifolds

sharp hardy adams inequalities for bi laplacian on hyperbolic space of dimension four abstract we establish sharp hardy adams inequalities on hyperbolic space b 4 of dimension four namely we will show that for any 0 there exists a constant c 0 such that b 4 e 32 2 u 2 1 32 2 u 2 d v 16 b 4 e 32 2 u 2 1 32 2 u 2 1 x 2 4 d x c for any u c 0 b 4 with b 4 h 9 4 h u u d v 1 as applications we obtain a sharpened adams inequality on hyperbolic space b 4 and an inequality which improves the classical adams inequality and the hardy inequality simultaneously the later inequality is in the spirit of the hardy trudinger moser inequality on a disk in dimension two given by wang and ye in 46 and on any convex planar domain by lu and yang in 33 the fourier analysis techniques on hyperbolic and symmetric spaces play an important role in our work

existence and nonexistence of extremals for critical adams inequalities in r4 and trudinger moser inequalities in r2 abstract though much progress has been made with respect to the existence of extremals of the critical first order trudinger moser inequalities in w 1 n r n and higher order adams inequalities on finite domain r n whether there exists an extremal function for the critical higher order adams inequalities on the entire space r n still remains open the current paper represents the first attempt in this direction by considering the critical second order adams inequality in the entire space r 4 the classical blow up procedure cannot apply to solving the existence of critical adams type inequality because of the absence of the polya szego type inequality in this paper we develop some new ideas and approaches based on a sharp fourier rearrangement principle see 31 sharp constants of the higher order gagliardo nirenberg inequalities and optimal poly harmonic truncations to study the existence and nonexistence of the maximizers for the adams inequalities in r 4 of the form s sup u h 2 1 r 4 exp 32 2 u 2 1 u 2 d x where 32 2 we establish the existence of the threshold where 32 2 2 b 2 2 and b 2 1 24 2 such that s is attained if 32 2 and is not attained if 32 2 this phenomenon has not been observed before even in the case of first order trudinger moser inequality therefore we also establish the existence and non existence of an extremal function for the trudinger moser inequality on r 2 furthermore the symmetry of the extremal functions can also be deduced through the fourier rearrangement principle

nodal solutions for the fractional yamabe problem on heisenberg groups we prove that the fractional yamabe equation rm cal l gamma u vert u vert 4 gamma q 2 gamma u on the heisenberg group n has n 1 2 sequences of nodal sign changing weak solutions whose elements have mutually different nodal properties where rm cal l gamma denotes the cr fractional sub laplacian operator on n q 2n 2 is the homogeneous dimension of n and gamma in bigcup nolimits k 1 n k kq q 1 our argument is variational based on a ding type conformal pulling back transformation of the original problem into a problem on the cr sphere s2n 1 combined with a suitable hebey vaugon type compactness result and group theoretical constructions for special subgroups of the unitary group u n 1

optimal path cover problem on block graphs and bipartite permutation graphs abstract the optimal path cover problem is to find a minimum number of vertex disjoint paths which together cover all the vertices of the graph in this paper we present linear time algorithms for the optimal path cover problem for the class of block graphs and bipartite permutation graphs

degenerate principal series representations of gln c and gln r let f be the field c or r for 1 k n 2 let pk be the maximal parabolic subgroup of gln f with its levi factor isomorphic to gln k f glk f in this paper we explicitly calculate the action of the enveloping algebra of gln f on the degenerate principal series representations induced from characters of pk we address composition series k types and unitarity