the range of the tangential cauchy riemann system to a cr embedded manifold we prove that every compact pseudoconvex orientable cr manifold of bbb c n bounds a complex manifold in the c sense in particular bar partial b has closed range

morse inequalities for fourier components of kohn rossi cohomology of cr manifolds with s 1 action let x be a compact connected cr manifold of dimension 2n 1 n ge 2 with a transversal cr s 1 action on x we study the fourier components of the kohn rossi cohomology with respect to the s 1 action by studying the szego kernel of the fourier components we establish the morse inequalities on x using the morse inequalities we have established on x we prove that there are abundant cr functions on x when x is weakly pseudoconvex and strongly pseudoconvex at a point

a general framework for population based distributed optimization over networks abstract this paper presents a population based solution for the distributed optimization problem where the overall objective function is defined as an average of local cost functions corresponding to the nodes of a network populations are introduced for the nodes to cooperatively find the global optimum of the overall objective function the main challenge is that each population cannot know the overall objective function and thus cannot directly evaluate the quality of their individuals in each iteration to overcome this difficulty we introduce consensus methods to design a message passing protocol under which the local estimates can converge to the same value we present a general framework that consists of consensus search consensus evaluation population evolution and local stopping steps compared with mathematical methods this framework solves in network distributed optimization problems without requiring a convexity assumption on the objective functions the standard pso and one of its variants called mco are introduced for testing and comparison two simulated examples under different network topologies illustrate the feasibility of our approach

holder estimates on cr manifolds with a diagonalizable levi form a for all e 0 the operator cl maps lip s po into lip s 2 m e po here lip s p denotes the space of square integrable forms of degree p q with 1 g q otheoperators o l o l q and0 map lip s p into lip s l m e po c for all e o the operators 8b8zo 8z8ho 1 q a aa cl 8 8 8 q 8 and 8 0 ab map lip s p into lip s p hebre aiag web asm acts on forms of type p q with 1 q 0 e zf f is a square integrable function on a with f e lip p then s f e lip s e p for all e 0 where s is the orthogonal projection on square integrable cr functions

subelliptic boundary conditions for spinc dirac operators gluing relative indices and tame fredholm pairs abstract let x be a spin manifold with boundary such that the spin structure is defined near the boundary by an almost complex structure which is either strictly pseudoconvex or pseudoconcave and hence contact using generalized szeg projectors we define modified neumann boundary conditions eo for spinors which lead to subelliptic fredholm boundary value problems for the spin dirac operator eo to study the index of these boundary value problems we introduce a generalization of fredholm pairs to the tame category in this context we show that the index of the graph closure of eo eo equals the tame relative index on the boundary between eo and the calderon projector let x0 and x1 be strictly pseudoconvex spin manifolds as above let bx1 bx0 be a contact diffeomorphism s0 s1 denote generalized szeg projectors on bx0 bx1 respectively and 0eo 1eo the subelliptic boundary conditions they define if x1 is the manifold x1 with its orientation reversed then the glued manifold x x0 x1 has a canonical spin structure and dirac operator xeo applying these results we obtain a formula for the relative index r ind s0 s1 as a special case this formula verifies a conjecture of atiyah and weinstein 1997 rims kokyuroku 1014 1 14 for the index of the quantization of a contact transformation between cosphere bundles

embedding compact three dimensional cr manifolds of finite type in mathbf c n an abstract three dimensional cr manifold x is a smooth manifold without boundary of real dimension three equipped with a smooth one dimensional complex sub bundle t0 of its complexified tangent bundle which for each x e x satisfies txol n txo 1 o canonically associated to the cr structure is a first order differential operator db which maps functions on x to sections of the bundle b 1 dual to to and is defined by

the szegő projection sobolev estimates in regular domains the szego projection preserves global smoothness in weakly pseudoconvex domains that are regular in the sense of diederich fornxess and catlin it preserves local smoothness near boundary points of finite type the bergman projection being linked to the 8 problem plays an important role in several complex variables as demonstrated by recent applications to the boundary behavior of holomorphic maps 7 16 the szego projection is interesting both by analogy and by its connection with the ab problem and hence with unsolvable equations of lewy type while in strictly pseudoconvex domains the bergman and szego projections are both well understood in general much more is known about the bergman projection this is no surprise for the theory of the 8 problem is further advanced than the theory of the 8b problem nonetheless i believe that the szego projection ought not be any more mysterious than the bergman projection this paper is intended to support this contention the most important domains for which the bergman projection is known to be regular are domains with a good a neumann operator here it is shown that in such domains the szego projection also is regular more precisely in smooth bounded pseudoconvex domains satisfying catlin s property p the most general condition presently known that guarantees global regularity of the neumann operator the szego projection is shown to be globally regular theorem 4 1 moreover at boundary points of finite type the szego projection is locally regular theorem 5 1 in both cases the szego projection exactly preserves the degree of differentiability measured in sobolev norms the method of proof is not the obvious one of directly generalizing from the strictly peudoconvex case for there are serious obstacles to such a generalization for instance the approach of analysis on the heisenberg group used by phong and stein 29 to derive estimates for both the bergman and szego projections cannot yet be extended because the theory of singular integrals on weakly pseudoconvex surfaces is still in its early stages 28 15 similarly one does not know how to carry over the fourier integral operator approach of boutet de monvel and sjostrand 10 to the weakly pseudoconvex case explicit construction of cauchy fantappie forms received by the editors march 11 1986 paper presented april 11 1986 to the 826th meeting of the american mathematical society indianapolis special session on several complex variables 1980 mathematics subject classification 1985 revision primary 32a25 32h10 partially supported by national science foundation grants mcs 8201063 and dms 8501758 1987 american mathematical society 0002 9947 87 1 00 25 per page

optimal hoelder and l p estimates for b on the boundaries of real ellipsoids in c n let d be a real ellipsoid in cn n 3 with defining function p z en x k yk2k 1 zk xk iyk where nk mk e n in this paper we study the sharp holder and lp estimates for the solutions of the tangential cauchy riemann equations 0b on the boundary od of d using the integral kernel method in particular we proved that if a e l o j od such that oba 0 on ad in the distribution sense then there exists a u e a112m od satisfying obu a and iiuiia 2 9d 0 independent of a where al 2m od is the lipschitz space with exponent 21 and 2m maxl k f min 2nk 2mk is the type of the m k domain d

the range of the tangential cauchy riemann operator over a small ball in this paper we study the closed range property of tangential cauchy riemann equations of an abstract cr structure over a small ball let d be the set x e r 1 1x1 4 kuranishi 141 and akahori 11 have studied the l theory

hearing pseudoconvexity with the kohn laplacian a bounded domain in open image in new window with connected lipschitz boundary is pseudoconvex if the bottom of the essential spectrum of the kohn laplacian on the space of 0 q forms 1 q n 1 with l2 coefficients is positive

spectral theory for tensor products of hermitian holomorphic line bundles let l be a quasipositive line bundle over a compact complex hermitian manifold it has been conjectured that the smallest positive eigenvalue of the laplacian acting on the k th tensor power of l is uniformly bounded independent of k various special cases of the conjecture are proved but a counterexample is given to the general conjecture

a relative index on the space of embeddable cr structures i we study the problem of embeddability for three dimensional cr manifolds let m m0b denote a compact embeddable strictly pseudoconvex cr manifold and os the orthogonal projection on ker 0 b if 16b denotes a deformation of this cr structure then 1 b is embeddable if and only if

the cr paneitz operator and the stability of cr pluriharmonic functions abstract we give a condition which ensures that the paneitz operator of an embedded three dimensional cr manifold is nonnegative and has kernel consisting only of the cr pluriharmonic functions our condition requires uniform positivity of the webster scalar curvature and the stability of the cr pluriharmonic functions for a real analytic deformation as an application we show that the real ellipsoids in c 2 are such that the cr paneitz operator is nonnegative with kernel consisting only of the cr pluriharmonic functions

equation de lewy resolubilite globale de l equation partial_b u f sur la frontiere de domaines faiblement pseudo convexes de mathbf c 2 mathrm ou mathbf c n

the complex green operator on cr submanifolds of ℂⁿ of hypersurface type compactness we establish compactness estimates for b on a compact pseudoconvex cr submanifold of c of hypersurface type that satisfies property p when the submanifold is orientable these estimates were proved by a raich in 32 using microlocal methods our proof deduces the estimates from a slight extension when q 1 of those known on hypersurfaces via the fact that locally cr submanifolds of hypersurface type are cr equivalent to a hypersurface the relationship between two potential theoretic conditions is also clarified

compactness of the complex green operator on cr manifolds of hypersurface type the purpose of this article is to study compactness of the complex green operator on cr manifolds of hypersurface type we introduce cr pq a potential theoretic condition on 0 q forms that generalizes catlin s property pq to cr manifolds of arbitrary codimension we prove that if an embedded cr manifold of hypersurface type of real dimension at least five satisfies cr pq and cr pn 1 q then the complex green operator is a compact operator on the sobolev spaces h s 0 q m and h s 0 n 1 q m if 1 q n 2 and s 0 we use cr plurisubharmonic functions to build a microlocal norm that controls the totally real direction of the tangent bundle 1 introduction and results in this article we introduce property cr pq a potential theoretic condition on 0 q forms we show that if an embedded cr manifold of hypersurface type satisfies cr pq and cr pn 1 q then the complex green operator is a compact operator on the sobolev spaces h s 0 q m and h s 0 n 1 q m if 1 q n 2 we use cr plurisubharmonic functions to build a microlocal norm that controls the bad direction of the tangent bundle we first prove the closed range and compactness results on l 2 q m and use an elliptic regularization

nonexistence of homotopy formula for 0 1 forms on hypersurfaces in mathbb c 3 soit g une 0 1 forme definie sur une hypersurface strictement convexe dans c 3 et on suppose que b g 0 mais b g est petit on considere la resolution locale approchee de l equation b u g

local existence theorems with estimates for b on weakly pseudo convex cr manifolds in particular we prove the solvability for 0 1 on co for 1 q n 2 when m is a pseudo convex cr hypersurface of finite type in the sense defined in d angelo 11 and c co is the intersection of m with a flat hypersurface m0 see theorem 1 in sect i near a point zo of finite type we also show that there exists a neighborhood basis co of z0 such that 0 1 is solvable in co for any ct satisfying 0 2 see corol lary 1 1 we also obtain the interior regularity of the solutions for 0 1 in the sobolev spaces and show that the range of b is closed in the frechet norm l p q co loc see theorem 2 and corol lary 2 1 in sect 1 since the existence of the solution u of 0 1 was proved on co for any c satisfying 0 2 in co theorems 1 and 2 are semi global results when we assume that mo is only levi flat and with a stein ne ighborhood basis then we prove that for any relatively compact subset co c co there exists a u satisfying 0 2 in co see theorem 3 we note that the local solvability for 0 1 was not known previously even on a small subset in co on general weakly pseudo convex cr manifolds in this paper we shall prove the

sobolev estimates for the bar partial neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary neumann operator on domains inc n admitting a defining function that is plurisubharmonic on the boundary

higher order bergman functions and explicit construction of moduli space for complete reinhardt domains in this article we introduce higher order bergman functions for bounded complete reinhardt domains in a variety with possibly isolated singularities these bergman functions are invariant under biholomorhic maps we use bergman functions to determine all the biholomorhic maps between two such domains as a result we can construct an infinite family of numerical invariants from the bergman functions for such domains in an variety x y z c xy z these infinite family of numerical invariants are actually a complete set of invariants for either the set of all bounded strictly pseudoconvex complete reinhardt domain in an variety or the set of all bounded pseudoconvex complete reinhardt domains with real analytic boundaries in an variety in particular the moduli space of these domains in an variety is constructed explicitly as the image of this complete family of numerical invariants it is well known that an variety is the quotient of cyclic group of order n 1 on c we prove that the moduli space of bounded complete reinhardt domains in an variety coincides with the moduli space of the corresponding bounded complete reinhardt domains in c since our complete family of numerical invariants are computable we have solved the biholomorphically equivalent problem for large family of domains in c

extension of c r structures on three dimensional compact pseudoconvex c r manifolds let m be a smooth compact orientable pseudoconvex c r manifold of real dimension three and assume that there is a smooth function which is strictly subharmonic in any direction where the levi form vanishes on m then we extend the given c r structure on m to an integrable almost complex structure on the concave side of m as an application if m is a non compact pseudoconvex c r manifold of real dimension three we prove that the given c r structure on m can be locally extended to an integrable almost complex structure on the concave side of m

embeddability for three dimensional cr manifolds a question of principal interest in the theory of compact three dimensional cr manifolds is to understand when a given strictly pseudoconvex cr structure can be realized by an embedding in c we call a structure for which this is possible embeddable and nonembeddable otherwise this question is only of interest in three dimensions as a theorem of boutet de monvel states that any strictly pseudoconvex cr structure on a compact 2d 1 manifold is realizable as an embedding in some c provided d 1 on the other hand there is a classical example of rossi which shows that an arbitrarily small real analytic perturbation of the standard structure on the three sphere may fail to be embeddable ros later nirenberg constructed w perturbations for which the associated ob equation fails to have even local solutions n in fact this situation is generic in the f topology it is now understood principally from the work of eliashberg that cr structures with

on three dimensional cauchy riemann manifolds of the almost complex tensor of c2 multiplication by i the collection of these subspaces hp m forms a bundle hm c tm called the horizontal or contact bundle indeed strict pseudoconvexity of m implies that the plane field hpm is nondegenerate i e defines a contact structure the almost complex tensor of c2 then restricts to a bundle endomorphism j hm hm such that j2 id the bundle hm c tm together with this endomorphism j defines the cr structure of m slightly more generally let d d u m be a four dimensional compact manifold with boundary m interior d endowed with a smooth almost complex tensor which is integrable on d if d is strictly pseudoconvex i e in a neighborhood of any p e m there is a smooth strictly plurisubharmonic function u negative on d 0 on m but du 0 on m then m inherits a strictly pseudoconvex cr structure from d in this case we shall say that m bounds a strictly pseudoconvex surface this leads us to the abstract definition a strictly pseudoconvex threedimensional cr manifold is a compact manifold m without boundary dim m 3 endowed with a contact structure hm hpm p e m c tm and an endomorphism j of hm such that j2 id throughout this paper unless otherwise stated we shall be working with infinitely differentiable objects thus m hm j are assumed to be smooth in this sense furthermore a differentiable function f m c is a cr function if for any p e m the restriction df ihm hpm tf p c

global regularity for b on weakly pseudoconvex cr manifolds abstract let m 2 n 1 be a compact orientable weakly pseudoconvex manifold of dimension at least five embedded in c n n n of codimension one or more and endowed with the induced cr structure we show the tangential cauchy riemann operator b has closed range on such a manifold m hence we get global existence and regularity results for the b problem we also show the middle i e 1 q n 2 b cohomology groups of m h 0 p q m b h s p q m b and h p q m b with respect to l 2 sobolev s norm and c coefficients respectively are finite and isomorphic to each other the results are obtained by microlocalization using a new type of weight function called strongly cr plurisubharmonic

estimates for the parametrix of the kohn laplacian on certain domains soit le demi espace superieur degenere generalise dans c 3 d p w 1 w 2 w imw p w 1 w 2 2 ou p c p 0 0 p est reel p 0 p 0 0 k p 0 0 pour k 2 on etudie la taille de la solution fondamentale de b sur des formes 0 1 sur la frontiere de d p

maximum principle for singular stochastic control problems in this paper an optimal singular stochastic control problem is considered for this model a general stochastic maximum principle is obtained by using a time transformation this is the first version of the stochastic maximum principle that covers nonlinear cases

on the singularities of the szegő projections on lower energy forms let x be an abstract not necessarily compact orientable cr manifold of dimension 2n 1 n geqslant 2 let box q b be the gaffney extension of kohn laplacian on 0 q forms we show that the spectral function of box q b admits a full asymptotic expansion on the non degenerate part of the levi form as a corollary we deduce that if x is compact and the levi form is non degenerate of constant signature on x then the spectrum of box q b in 0 infty consists of point eigenvalues of finite multiplicity moreover we show that a certain microlocal conjugation of the associated szeg kernel admits an asymptotic expansion under a local closed range condition as applications we establish the szeg kernel asymptotic expansions on some weakly pseudoconvex cr manifolds and on cr manifolds with transversal cr s 1 actions by using these asymptotics we establish some local embedding theorems on cr manifolds and we give an analytic proof of a theorem of lempert asserting that a compact strictly pseudoconvex cr manifold of dimension three with a transversal cr s 1 action can be cr embedded into mathbb c n for some n in mathbb n