gaussian bounds for degenerate parabolic equations abstract let a be a real symmetric degenerate elliptic matrix whose degeneracy is controlled by a weight w in the a 2 or qc class we show that there is a heat kernel w t x y associated to the parabolic equation w u t div a u and w t satisfies classic gaussian bounds w t x y c 1 t n 2 exp c 2 x y 2 t we then use this bound to derive a number of other properties of the kernel

structure of basic lie superalgebras and of their affine extensions we generalize to the case of superalgebras several properties of simple lie algebras involving the use of dynkin diagrams if to a simple lie algebra can be associated one dynkin diagram it is a finite set of nonequivalent ones which can be constructed for a basic superalgebra or b s a the knowledge of these diagrams which can be obtained for each b s a in a systematic way allows us to deduce the regular subsuperalgebras of a b s a the symmetries of the dynkin diagrams are related to outer automorphisms of b s a and lead to some singular subsuperalgebras finally we consider the extended dynkin diagrams in order to classify the affine b s a and use their symmetries to construct the twisted basic superalgebras

riemannian symmetric superspaces and their origin in random matrix theory gaussian random matrix ensembles defined over the tangent spaces of the large families of cartan s symmetric spaces are considered such ensembles play a central role in mesoscopic physics as they describe the universal ergodic limit of disordered and chaotic single particle systems the generating function for the spectral correlations of each ensemble is reduced to an integral over a riemannian symmetric superspace in the limit of large matrix dimension such a space is defined as a pair g h m r where g h is a complex analytic graded manifold homogeneous with respect to the action of a complex lie supergroup g and m r is a maximal riemannian submanifold of the support of g h

involutive automorphisms and iwasawa decomposition of some hyperbolic kac moody algebras the involutive automorphisms of hyperbolic kac moody algebras are computed from satake diagrams of these algebras which are then used to furnish a general procedure of their iwasawa decomposition in particular 1 1 and 2 1 are considered as representative examples to illustrate the efficacy of the underlying treatment

involutive automorphisms of the affine kac moody algebra a1 1 the matrix formulation of automorphisms of an affine kac moody algebra that was developed in a previous paper is here applied to the determination of all the conjugacy classes of involutive automorphisms of the affine kac moody algebra a1 1 where it is compared with the application of more traditional structural techniques

weil petersson teichmuller space iii dependence of riemann mappings for weil petersson curves the classical riemann mapping theorem implies that there exists a so called riemann mapping which takes the upper half plane onto the left domain bounded by a jordan curve in the extended complex plane the primary purpose of the paper is to study the basic problem how does a riemann mapping depend on the corresponding jordan curve we are mainly concerned with those jordan curves in the weil petersson class namely the corresponding riemann mappings can be quasiconformally extended to the whole plane with beltrami coefficients being square integrable under the poincare metric after giving a geometric characterization of a weil petersson curve we endow the space of all normalized weil petersson curves with a new real hilbert manifold structure in a geometric manner and show that this new structure is topologically equivalent to the standard complex hilbert manifold structure which implies that an appropriately chosen riemann mapping depends continuously on a weil petersson curve and vice versa this can be considered as the first result about the continuous dependence of riemann mappings on non smooth jordan curves

a classification of involutive automorphisms of an affine kac moody lie algebra abstract in this thesis we classify the conjugacy classes of involutions in aut g where g is an affine kac moody lie algebra we distinguish between two kinds of involutions those which preserve the conjugacy class of a borel subalgebra and those which don t we give a complete and non redundant list of representatives of involutions of the first kind and we compute their fixed points sets we prove that any involution of the first kind has a conjugate which leaves invariant the components of the gauss decomposition g n h n we also give a complete list of representatives of the conjugacy classes of involutions of the second kind

satake superdiagrams real forms and iwasawa decomposition of classical lie superalgebras satake superdiagrams corresponding to the lie superalgebras c n together with the exceptional ones are constructed from their dynkin diagrams with a view to determine the real forms of these superalgebras the involutive automorphisms are computed from a modified formula to account for all pertinent cases in a consistent way this mechanism is also used to furnish a general treatment of the iwasawa and langlands decompositions of these superalgebras in particular we compute these decompositions for for illustrational purposes and obtain the corresponding induced representations using the method of schmidt construction

chord arc curves and the beurling transform we study the relation between the geometric properties of a quasicircle gamma and the complex dilatation mu of a quasiconformal mapping that maps the real line onto gamma denoting by s the beurling transform we characterize bishop jones quasicircles in terms of the boundedness of the operator i mu s on a particular weighted l 2 space and chord arc curves in terms of its invertibility as an application we recover the l 2 boundedness of the cauchy integral on chord arc curves

general theory of the matrix formulation of the automorphisms of affine kac moody algebras the determination of conjugacy classes within the group of all automorphisms of an affine kac moody algebra is discussed in detail and the limited role of cartan preserving automorphisms is emphasized a comprehensive method of dealing with all the automorphisms of an untwisted affine kac moody algebra is developed this is based on a matrix formulation of the untwisted affine kac moody algebras four different types of automorphism called type 1a type 1b type 2a and type 2b are identified and analysed within this matrix formulation the development of the detailed properties of automorphisms in the matrix formulation includes formulae for the products and inverses of automorphisms together with the conjugacy and involutive conditions

satake superdiagrams and iwasawa decomposition of some hyperbolic kac moody superalgebras the involutive automorphisms of hyperbolic kac moody superalgebras are computed from the satake superdiagrams corresponding to these algebras these are then used to furnish a general treatment of the iwasawa decomposition of these algebras in particular we consider 1 2 1 and 1 0 1 as representative examples for the purpose of illustration

corrigendum to gaussian bounds for degenerate parabolic equations j funct anal 255 2 2008 283 312 abstract we rectify an error in the proof of the gaussian estimates for the heat kernel associated to certain weighted elliptic equations

involutive automorphisms and iwasawa decomposition of affine kac moody superalgebras the involutive automorphisms of affine kac moody superalgebras are computed from satake superdiagrams corresponding to these algebras these are then used to furnish a general treatment of the iwasawa decomposition of these algebras in particular we consider a 1 0 1 and b 1 1 1 as representative examples for the purpose of illustration

quasiconformal mappings and chord arc curves given a quasiconformal mapping p on the plane what conditions on its dilatation guarantee that p r is rectifiable and pir is locally abso lutely continuous we show in this paper that if satisfies certain quadratic carleson measure conditions with small norm then p r is a chord arc curve with small constant and p x p 0 g0 ea t dt for x e r with a e bmo having small norm conversely given any such map from r c we show that it has an extension to c with the right kind of dilatation similar results hold with r replaced by a chord arc curve examples are given that show that it would be hard to improve these results applications are given to conformal welding and the theorem of coifman and meyer on the real analyticity of the riemann mapping on the manifold of chord arc curves let p be a quasiconformal map of the plane onto itself thus p is a homeomorphism with locally integrable distributional derivatives and ps upz where u e l c il ul100 1 here we use the notations fz af azf 2 z id f fz 2 aa ida f this function u is called the complex dilatation of p the mapping theorem for q c maps states that for each u e l c il ul100 1 there is a q c map p on c with dilatation u and p is unique up to normalization a basic problem is to understand how geometric properties of p are reflected in u for example one would like to have natural conditions on u which imply that p r is rectifiable and p ir is absolutely continuous this question arises naturally when considering problems in conformal mappings and conformal welding in this paper we obtain such estimates for the mapping theorem and we also give some applications our results involve bmo aoo weights chord arc curves and carleson measures and so we first review the appropriate definitions a locally integrable function f on r lies in bmo if llfil sup lil if x fil dx received by the editors june 1 1986 1980 mathematics subject classification 1985 revision primary 30g60 secondary 42b20 30e20

dynkin diagrams of hyperbolic kac moody superalgebras hyperbolic kac moody superalgebras are classified in terms of their dynkin diagrams these types of kac moody superalgebras are those whose diagrams revert to either that of simple or affine superalgebras upon deletion of one of the vertices it is found that the maximum rank of this type of algebra is 6

the conjugacy classes of the involutive automorphisms of affine kac moody algebra a2 1 all the conjugacy classes of involutive automorphisms of the affine kac moody algebra a2 1 are obtained using the matrix formulation of automorphisms of an affine kac moody algebra that was developed in a previous paper

iwasawa decomposition of affine kac moody algebras using satake diagrams we carry out the iwasawa decomposition of the affine kac moody algebras within the general framework of involutive automorphisms which are determined with the help of the satake diagrams corresponding to these algebras the parabolic subalgebras essential for the construction of induced representations are computed from the langlands decomposition in particular we consider the untwisted versions viz a1 1 and a2 1 as representative examples for the purpose of illustration

beltrami operators in the plane we determine optimal lp properties for the solutions of the general nonlinear elliptic system in the plane of the form fz h z fz h l c where h is a measurable function satisfying h z w1 h z w2 k w1 w2 and k is a constant k 1 we will also establish the precise invertibility and spectral properties in lp c for the operators i t i t and t where t is the beurling transform these operators are basic in the theory of quasiconformal mappings and in linear and nonlinear elliptic partial differential equations in two dimensions in particular we prove invertibility in lp c whenever 1 p 1 1 we also prove related results with applications to the regularity of weakly quasiconformal mappings

quasiconformal mappings and ahlfors david curves we show that if p is a quasiconformal mapping of the plane whose dilatation satisfies a certain quadratic carleson measure condition relative to an ahlfors david curve f then p is differentiable almost everywhere on f and log ip l e bmo when f is chord arc we show that its image is a bishopjones curve if the carleson norm is small then we show that p is absolutely continuous on f the image of f is an ahlfors david curve and p ea where a e bmo with a small norm

satake diagrams of affine kac moody algebras satake diagrams of affine kac moody algebras untwisted and twisted are obtained from their dynkin diagrams these diagrams give a classification of restricted root systems associated with these algebras in the case of simple lie algebras these root systems and satake diagrams correspond to symmetric spaces which have recently found many physical applications in quantum integrable systems quantum transport problems random matrix theories etc we hope these types of root systems may have similar applications in theoretical physics in future and may correspond to symmetric spaces analogue of affine kac moody algebras if they exist