sub hermitian geometry and the quantitative newlander nirenberg theorem abstract given a finite collection of c 1 complex vector fields on a c 2 manifold m such that they and their complex conjugates span the complexified tangent space at every point the classical newlander nirenberg theorem gives conditions on the vector fields so that there is a complex structure on m with respect to which the vector fields are t 0 1 in this paper we give intrinsic diffeomorphic invariant necessary and sufficient conditions on the vector fields so that they have a desired level of regularity with respect to this complex structure i e smooth real analytic or have zygmund regularity of some finite order by addressing this in a quantitative way we obtain a holomorphic analog of the quantitative theory of sub riemannian geometry initiated by nagel stein and wainger we call this sub hermitian geometry moreover we proceed more generally and obtain similar results for manifolds which have an associated formally integrable elliptic structure this allows us to introduce a setting which generalizes both the real and complex theories

tangential lipschitz gain for holomorphic functions on domains of finite type functions that are holomorphic and lipschitz in a smoothly bounded domain enjoy a gain in the order of lipschitz regularity in the complex tangential directions near the boundary we describe this gain explicitly in terms of the defining function near points of finite type in the boundary

estimates for the bar partial neumann problem in pseudoconvex domains of finite type in c2problem in pseudoconvex domains of finite type in c2

the aronsson euler equation for absolutely minimizing lipschitz extensions with respect to carnot caratheodory metrics we derive the euler lagrange equation also known in this setting as the aronsson euler equation for absolute minimizers of the l variational problem formula math where c g is an open subset of a carnot group 0 u denotes the horizontal gradient of u r and the lipschitz class is defined in relation to the carnot caratheodory metric in particular we show that absolute minimizers are infinite harmonic in the viscosity sense as a corollary we obtain the uniqueness of absolute minimizers in a large class of groups this result extends previous work of jensen and of crandall evans and gariepy we also derive the aronsson euler equation for more regular absolutely minimizing lipschitz extensions corresponding to those carnot caratheodory metrics which are associated to free systems of vector fields

closed finitely generated ideals in algebras of holomorphic functions and smooth to the boundary in strictly pseudoconvex domains define in a m a a natural estructure of banach resp fr6chet algebra in this paper we are interested in the closed finitely generated ideals of the algebras am as much as we know there are very few papers on the ideal structure of these algebras in dimension n 1 in 22 nagel proved that if d is piecewise real analytic and g ok are holomorphic in a neighbourhood o f the ideal in a generated by g gk is closed our first main result establishes a sort of reciprocal of nagel s theorem

regularity of a class of quasilinear degenerate elliptic equations which is in a form of eq 1 in general convex solution u of 2 is at most in c 1 if k is only assumed to be smooth and nonnegative e g see ckns and g1 in g1 a sharp sufficient condition was introduced to establish c 1 regularity of 2 a basic question left is when is a solution u of 2 smooth or better than c we remark that if k is positive and smooth by elliptic theory u is smooth one may expect u to be smooth if the decay of k near its null set is under control say of finite type unfortunately this is not true the function u x x 2 2 n provides an example that the solution u of eq 2 article no ai971677

a characterization of hajlasz sobolev and triebel lizorkin spaces via grand littlewood paley functions abstract in this paper we establish the equivalence between the hajlasz sobolev spaces or classical triebel lizorkin spaces and a class of grand triebel lizorkin spaces on euclidean spaces and also on metric spaces that are both doubling and reverse doubling in particular when p n n 1 we give a new characterization of the hajlasz sobolev spaces m 1 p r n via a grand littlewood paley function

multi parameter singular radon transforms ii the lp theory the purpose of this paper is to study the l2 boundedness of operators of the form f x f t x k t dt where t x is a c function defined on a neighborhood of the origin in t x n n satisfying 0 x x is a c cut off function supported on a small neighborhood of 0 n and k is a multi parameter singular kernel supported on a small neighborhood of 0 n the goal is given an appropriate class of kernels k to give conditions on such that every operator of the above form is bounded on l2 the case when k is a calderon zygmund kernel was studied by christ nagel stein and wainger we generalize their conditions to the case when k has a multi parameter structure for example when k is given by a product kernel even when k is a calderon zygmund kernel our methods yield some new results this is the first paper in a three part series the later two of which are joint with e m stein the second paper deals with the related question of lp boundedness while the third paper deals with the special case when is real analytic

the szegő kernel as a singular integral kernel on a family of weakly pseudoconvex domains the szego kernels on the weakly pseudoconvex domains im z2 izl12k k e z have been computed by greiner and stein after constructing a global nonisotropic pseudometric suitable for calderon zygmund singular integral theory on the boundaries of the domains we study principal value operators associated to these szego kernels we prove that the principal value operators are bounded on lp for 1 p oo and that they preserve certain nonisotropic lipschitz classes we then derive a plemelj formula that relates the principal value operators to the szego projections from this formula we deduce that the szego projections are also bounded on lp for 1 p oo and that they preserve the same nonisotropic lipschitz classes

large deviations for stochastic reaction diffusion systems with multiplicative noise and non lipshitz reaction term following classical work by freidlin trans amer math soc 1988 305 665 657 and subsequent works by sowers ann probab 1992 20 504 537 and peszat probab theory related fields 1994 98 113 136 we prove large deviation estimates for the small noise limit of systems of stochastic reaction diffusion equations with globally lipschitz but unbounded diffusion coefficients however assuming the reaction terms to be only locally lipschitz with polynomial growth this generalizes results of the above mentioned authors our results apply in particular to systems of stochastic ginzburg landau equations with multiplicative noise

gaussian upper bounds for the heat kernels of some second order operators on riemannian manifolds abstract we describe a method of obtaining gaussian upper bounds on heat kernels which unifies and improves recent results for hypoelliptic operators in divergence form and for laplace beltrami operators on complete riemannian manifolds

dynamique des applications holomorphes propres de domaines reguliers et probleme de l injectivite this paper deals with proper holomorphic self maps of smoothly bounded pseudoconvex domains in open image in new window we study the dynamical properties of their extension to the boundary and show that their non wandering sets are always contained in the weakly pseudoconvex part of the boundary in the case of complete circular domains we combine this fact with an entropy degree argument to show that the maps are automorphisms some of our results remain true in open image in new window

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

the square_ b heat equation and multipliers via the wave equation recently nagel and stein studied the square b heat equation where square b is the kohn laplacian on the boundary of a weakly pseudoconvex domain of finite type in mathbb c 2 they showed that the schwartz kernel of e t square b satisfies good off diagonal estimates while that of e t square b pi satisfies good on diagonal estimates where denotes the szego projection we offer a simple proof of these results which easily generalizes to other similar situations our methods involve adapting the well known relationship between the heat equation and the finite propagation speed of the wave equation to this situation in addition we apply these methods to study multipliers of the form m square b in particular we show that m square b is an nis operator where m satisfies an appropriate mihlin hormander condition

a variational approach to stochastic nonlinear parabolic problems abstract this work is concerned with an optimal control approach to stochastic nonlinear parabolic diffusion equations with monotonically increasing nonlinearity this approach leads to sharper existence and uniqueness results under minimal growth conditions on nonlinear diffusion coefficients

large deviations for infinite dimensional stochastic dynamical systems the large deviations analysis of solutions to stochastic differential equations and related processes is often based on approximation the construction and justification of the approximations can be onerous especially in the case where the process state is infinite dimensional in this paper we show how such approximations can be avoided for a variety of infinite dimensional models driven by some form of brownian noise the approach is based on a variational representation for functionals of brownian motion proofs of large deviations properties are reduced to demonstrating basic qualitative properties existence uniqueness and tightness of certain perturbations of the original process

reduced heat kernels on nilpotent lie groups letu be a basis representation of an irreducible unitary representation of a nilpotent lie groupg inl2 rk and letdu denote the representation of the lie algebrag obtained by differentiation ifb1 b d is a basis ofg andb i du b i we consider the operators h sum limits i j 1 d c ij b i b j sum limits i 1 d c i b i wherec c ij is a real symmetric strictly positive matrix andc i c thenh generates a continuous semigroups holomorphic in the open right half plane with a reduced kernek defined by s z varphi x int limits r k dy kappa z x y varphi y we prove gaussian off diagonal bounds and exponential on diagonal bounds for for example ifc i 0 we establish that left kappa t x y right leqq a 1 wedge varepsilon mu t k mathord left vphantom k 2 right kern nulldelimiterspace 2 e lambda 1 t e d x y 2 4 1 varepsilon t 1 for allt 0 and 0 1 where is the smallest eigenvalue ofc 1 is the smallest eigenvalue ofh andd is a natural distance associated with the coefficientsc and the representationu bounds are also obtained forc i and complext alternatively ifh is self adjoint then left kappa z x y right leqq ae lambda 1 operatorname re z e b left x right alpha left y right alpha for allz c with rez 1 for some 0 2

a bound for the degree of nonholonomy in the plane let v1 vs be a system made with vector fields v1 vs in rn whose coordinates are polynomials of degree d to such a system is associated the control system x u1v1 it is proven that in the case n 2 the degree of nonholonomy of such a system is bounded by a function 2 d 6d2 2d 2

stochastic generalized porous media and fast diffusion equations abstract we present a generalization of krylov rozovskii s result on the existence and uniqueness of solutions to monotone stochastic differential equations as an application the stochastic generalized porous media and fast diffusion equations are studied for finite reference measures where the drift term is given by a negative definite operator acting on a time dependent function which belongs to a large class of functions comparable with the so called n functions in the theory of orlicz spaces

maximal hypoellipticity for the neumann problem abstract we establish maximal hypoellipticity for the neumann problem on smooth bounded pseudoconvex domains in c n under the weakest possible condition on the levi form in particular maximal hypoellipticity holds on the level of n 1 forms for pseudoconvex domains of finite commutator type

estimates for the bergman and szego kernels in mathbf c 2 the purpose of this paper some of whose conclusions were announced in nrsw is to study the bergman and szegd projection operators on pseudoconvex domains q of finite type in c2 the results we obtain are of three kinds i the size estimates of the bergman and szeg6 kernels and their derivatives ii the cancellation properties of those kernels expressed in terms of the actions of these operators on suitable bump functions these properties have an interest in their own right but are also crucial in iii below iii the sharp mapping properties of these operators on functions spaces such as lp the nonisotropic sobolev spaces and the nonisotropic lipschitz spaces which are naturally attached to the geometry of the boundary this also leads ultimately to sharp results in the regularity properties of solutions of the equation du f in q and 9bu f on the boundary let us describe these matters in more detail our starting points are certain basic geometric constructs associated to q a metric p defined in terms of the vector fields x1 and x2 which are the real and imaginary parts of the tangential levi vector field on d 2 and a function a p 8 a polynomial in 8 for p e d 2 which represents the higher levi invariant attached to d 9 the significance of these to the geometry of d q can be understood from the following facts the ball b p 8 c d q centered at p of radius 8 in the metric p when viewed in the appropriate coordinates has width 8 in the complex tangential directions but its width is a p 8 in the complementary direction thus the volume of the ball is 82a p 8 among other crucial properties

holder regularity for weak solutions to divergence form degenerate quasilinear parabolic systems abstract in this paper we establish holder regularity of weak solutions to the diagonal divergence form degenerate quasilinear parabolic system related to hormander type vector fields by deriving a parabolic poincare inequality and the higher integrability of gradients of weak solutions

a partial differential equation connected to option pricing with stochastic volatility regularity results and discretization this paper completes a previous work on a black and scholes equation with stochastic volatility this is a degenerate parabolic equation which gives the price of a european option as a function of the time of the price of the underlying asset and of the volatility when the volatility is a function of a mean reverting orstein uhlenbeck process possibly correlated with the underlying asset the analysis involves weighted sobolev spaces we give a characterization of the domain of the operator which permits us to use results from the theory of semigroups we then study a related model elliptic problem and propose a finite element method with a regular mesh with respect to the intrinsic metric associated with the degenerate operator for the error estimate we need to prove an approximation result

convex hypersurfaces and fourier transforms hilbert transforms and maximal functions along curves and surfaces spectral synthesis problems and the study of certain operators related to hyperbolic partial differential and pseudodifferential operators the problem of estimating such fourier transforms has a long history see for example hlawka 3 herz 2 littman 4 randol 9 10 svensson 15 sogge and stein 12 13 stein 14 marshall 5 etc the results of these investigations suggest a strong relation between the local curvature properties of the surface s and the decay of the function a we are mainly interested in how the geometry of s near a fixed point xo affects the decay of a6 for by using a partition of unity on s it is clearly enough to study integrals of the type

estimation on invariant distances on pseudoconvex domains of finite type in dimension two we study the class of smooth bounded weakly pseudoconvex domains open image in new window that are of finite type in the sense of j kohn and prove effective estimates on the invariant distances of bergman and kobayashi and also for the inner distance that is associated to the caratheodory distance

existence of isoperimetric regions in contact sub riemannian manifolds abstract we prove existence of regions minimizing perimeter under a volume constraint in contact sub riemannian manifolds whose quotient by the group of contact transformations preserving the sub riemannian metric is compact

boundary behavior of non negative solutions to degenerate sub elliptic equations let x x 1 x m be a system of c infinity vector fields in r n satisfying hormander s finite rank condition and let omega be a non tangentially accessible domain with respect to the carnot

attractors met x elliptic operators abstract we consider degenerate parabolic and damped hyperbolic equations involving an operator l that is x elliptic with respect to a family of locally lipschitz continuous vector fields x x 1 x m the local well posedness is established under subcritical growth restrictions on the nonlinearity f which are determined by the geometry and functional setting naturally associated with the family of vector fields x assuming additionally that f is dissipative the global existence of solutions follows and we can characterize their longtime behavior using methods from the theory of infinite dimensional dynamical systems

smoothness of subriemannian isometries we show that the group of isometries i e distance preserving homeomorphisms of an equiregular subriemannian manifold is a finite dimensional lie group of smooth transformations the proof is based on a new pde argument in the spirit of harmonic coordinates establishing that in an arbitrary subriemannian manifold there exists an open dense subset where all isometries are smooth

the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type we show that the hull of holomorphy of a nonisotropic ball in a real hypersurface of finite type in cn contains an open set in cn which emanates from the hypersurface a distance which is proportional to the length of the minor axis of the nonisotropic ball in addition we prove a maximal function estimate for plurisubharmonic functions which is important in the study of boundary values of holomorphic functions

the lindelof principle in several complex variables we formulate and prove a new lindelof principle in the function theory of several complex variables inspired by the classical result as improved later by lehto and virtanen this new result meshes closely with the well established fatou theorems of koranyi and stein in particular this is a lindelof principle for admissible approach regions we further adapt the new principle to the levi geometry of various domains the results in this paper improve on earlier results of cirka cima krantz abate and abate tauraso

small time gaussian estimates of heat diffusion kernels ii the theory of large deviations la deuxieme partie d un article etudiant le noyau de chaleur d un operateur differentiel d ordre 2 sur une variete c

existence of strong solutions for stochastic porous media equation under general monotonicity conditions this paper addresses existence and uniqueness of strong solutions to stochastic porous media equations dx x dt b x dw t in bounded domains of r d with dirichlet boundary conditions here is a maximal monotone graph in r r possibly multivalued with

regularity in the obstacle problem for parabolic non divergence operators of hormander type in this paper we continue the study initiated in 15 concerning the obstacle problem for a class of parabolic non divergence operators structured on a set of vector fields x x1 xq in rn with c coefficients satisfying hormander s finite rank condition i e the rank of lie x1 xq equals n at every point in rn in 15 we proved under appropriate assumptions on the operator and the obstacle the existence and uniqueness of strong solutions to a general obstacle problem the main result of this paper is that we establish further regularity in the interior as well as at the initial state of strong solutions compared to 15 we in this paper assume in addition that there exists a homogeneous lie group g rn ring operator such that x1 xq are left translation invariant on g and such that x1 xq are homogeneous of degree one

stein weiss inequalities for the fractional integral operators in carnot groups and applications in this article we consider the fractional integral operator i on any carnot group x1d53e i e nilpotent stratified lie group in the weighted lebesgue spaces l p x x1d53e we establish stein weiss inequalities for i and obtain necessary and sufficient conditions on the parameters for the boundedness of the fractional integral operator i from the spaces l p x x1d53e to l q x x1d53e and from the spaces l 1 x x1d53e to the weak spaces wl q x x1d53e by using the stein weiss inequalities in the limiting case we prove that the modified fractional integral operator is bounded from the space l p x x1d53e to the weighted bounded mean oscillation bmo space bmo x x1d53e where q is the homogeneous dimension of x1d53e as applications of the properties of the fundamental solution of sub laplacian on x1d53e we prove two sobolev stein embedding theorems on weighted lebesgue and weighted besov spaces in the carnot group setting as another application we prove the boundedness of i from the weighted besov

sub riemannian calculus on hypersurfaces in carnot groups we develop a sub riemannian calculus for hypersurfaces in graded nilpotent lie groups we introduce an appropriate geometric framework such as horizontal levi civita connection second fundamental form and horizontal laplace beltrami operator we analyze the relevant minimal surfaces and prove some basic integration by parts formulas using the latter we establish general first and second variation formulas for the horizontal perimeter in the heisenberg group such formulas play a fundamental role in the sub riemannian bernstein problem

schilder theorem for the brownian motion on the diffeomorphism group of the circle abstract we prove a large deviation principle for flows associated to stochastic differential equations with non lipschitz coefficients as an application we establish a schilder theorem for the brownian motion on the group of diffeomorphisms of the circle

multiparameter hardy space theory on carnot caratheodory spaces and product spaces of homogeneous type this paper is inspired by the work of nagel and stein in which the lp 1 p theory has been developed in the setting of the product carnot caratheodory spaces m m1 mn formed by vector fields satisfying hormander s finite rank condition the main purpose of this paper is to provide a unified approach to develop the multiparameter hardy space theory on product spaces of homogeneous type this theory includes the product hardy space its dual the product bmo space the boundedness of singular integral operators and calderon zygmund decomposition and interpolation of operators as a consequence we obtain the endpoint estimates for those singular integral operators considered by nagel and stein 2004 in fact we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric all of our results are established by introducing certain banach spaces of test functions and distributions developing discrete calderon identity and discrete littlewood paley stein theory our methods do not rely on the journe type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product hardy spaces

a property of the functions and distributions annihilated by a locally integrable system of complex vector fields page introduction 387 1 basic notation and ingredients 389 2 approximation and constancy on the fibers of the classical solutions of the homogeneous equations 393 3 first consequences and interpretation of the fundamental theorem 398 4 distribution solutions of the homogeneous equations 402 5 regularity of the solutions of the homogeneous equations 409 6 example of a system that is analytic hypo elliptic but not co hypo elliptic 413 references 420

large deviations for stochastic differential delay equations abstract in this work a freidlin wentzell type large deviation principle is established for stochastic differential delay equations the result in mohammed and zhang 2006 6 is improved

nonsmooth solutions for a class of fully nonlinear pde s on lie groups abstract in this paper we prove the existence of nonsmooth viscosity solutions for dirichlet problems involving a class a fully non linear operators on lie groups in particular we consider the elementary symmetric functions of the eigenvalues of the hessian built with left invariant vector fields

ck α regularity of solutions to quasilinear equations structured on hormander s vector fields abstract for a linear nonvariational operator structured on smooth hormander s vector fields with holder continuous coefficients we prove a regularity result in the scale of c x k spaces we deduce an analogous regularity result for nonvariational degenerate quasilinear equations

spectral theory for commutative algebras of differential operators on lie groups the joint spectral theory of a system of pairwise commuting self adjoint left invariant differential operators l1 ln on a connected lie group g is studied under the hypothesis that the algebra generated by them contains a weighted subcoercive operator of ter elst and robinson 1998 52 the joint spectrum of l1 ln in every unitary representation of g is characterized as the set of the eigenvalues corresponding to a particular class of generalized joint eigenfunctions of positive type of l1 ln connections with the theory of gelfand pairs are established in the case l1 ln generate the algebra of k invariant left invariant differential operators on g for some compact subgroup k of aut g

green functions for weighted subelliptic p laplace operator constructed by hormander s vector fields abstract this paper deals with the weighted subelliptic p laplace operator constructed by hormander s vector fields l p u d i v x a x x u x x u x p 2 2 a x x u x where u w 1 p u w 1 p q and a x is a bounded measurable and m m symmetric matrix satisfying 1 w x 2 p 2 a x w x 2 p 2 r m w x a p we first prove existence of the modified green function for l p by virtue of minty browder theorem and then existence of the green function for l p by checking the convergence of sequence of modified green functions next we derive upper bounds of the modified green function for l p by establishing the interpolation inequality in the weighted weak l p spaces finally the bounds of the green function for l p are also obtained by virtue of results for the modified green function and the compact weighted embedding theorem

harnack inequalities and gaussian estimates for a class of hypoelliptic operators we prove a global harnack inequality for a class of degenerate evolution operators by using repeatedly an invariant local harnack inequality as a consequence we obtain an accurate gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators

the two phase stochastic stefan problem this work is concerned with existence for a stochastic free boundary problem arising in phase transition the stefan two phase problem the existence of an invariant ergodic measure associated with the corresponding transition semigroup p t is also proved together with an integration by parts formula for the generator of p t

boundary behavior of functions holomorphic in domains of finite type abstract we study several notions of distance and ball on the boundary of domains of finite type in cn we use these notions to develop a theory of maximal functions and area integrals for functions holomorphic in such domains

a variational representation for certain functionals of brownian motion in this paper we show that the variational representation log ee f w inf v e 1 2 int 0 1 parallel v s parallel 2 ds f w int 0 cdot v s ds holds where w is a standard d dimensional brownian motion f is any bounded measurable function that maps c 0 1 mathbb r d into mathbb r and the infimum is over all processes v that are progressively measurable with respect to the augmentation of the filtration generated by w an application is made to a problem concerned with large deviations and an extension to unbounded functions is given