the effect of dissipation on solutions of the generalized korteweg de vries equation recent numerical simulations of the generalized korteweg de vries equation u l u p u x u xxx 0 indicate that for p 4 smooth solutions of the initial value problem may form singularities in finite time it is the purpose of this paper to ascertain what effect dissipation has on the instability of solitary waves and the associated blow up phenomena that are related to this singularity formation two different dissipative mechanisms are appended to in our study a burgers type term u xx and a simple zeroth order term u for both of these types of dissipation it is found that for small values of the positive parameters and solutions continue to form singularities in finite time however for given initial data u o it appears there are critical values c and c such that if c or c the solution associated with u o is globally defined and decays as t in the case wherein the singularity formation is averted by larger values of or a simple analysis shows the solution to approach its mean value exponentially fast theoretical analysis in the case when u o is a perturbed solitary wave leads to a conjecture about how c and c depend on the amplitude and spread of u o the numerical simulations indicate the analysis to be surprisingly sharp in predicting the qualitative dependence of c and c on u o

analysis on besov spaces ii embedding and duality theorems abstract this paper gives several results on besov spaces of holomorphic functions on a very large class of domains d in c n they include duality theorem embedding theorem best growth estimate and boundedness of multiplication operators on besov spaces

on the boundedness and compactness of operators of hankel type abstract boundedness and compactness results are obtained for commutators of multiplication operators with a general class of integral operators j of a critical homogeneity which are modeled on the bergman projection it is shown that the commutator m f j is bounded or compact on l p whenever the function is in an appropriately defined bmo or vmo space respectively special cases of the general results include commutators with the bergman projection in strictly pseudoconvex domains in c n and finite type domains in c 2 in addition it is shown that in the case of the bergman projection in a strictly pseudoconvex domain the sufficient condition is also necessary

composition operators on qk spaces abstract suppose that is a nonconstant analytic self map of the unit disk d compactness of composition operator c between two mobius spaces q k 1 and q k 2 is studied moreover necessary and sufficient condition for c from the dirichlet space d to a general class of analytic functions f p q s to be compact is given in terms of the map

spectra of certain jacobi operators we discuss the spectra and in particular the essential spectra of some bounded self adjoint jacobi operators associated with orthogonal polynomial sequences obtained via polynomial mappings

morrey boundedness and compactness characterizations of integral commutators with singular kernel on strictly pseudoconvex domains in cn abstract we prove the morrey boundedness and compactness characterization of integral commutators with singular kernel of calderon zygmund type on pseudo convex domains in c n our results extend some of the ones of krantz li in 16 17

asymptotic decomposition of nonlinear dispersive wave equations with dissipation abstract provided 0 solutions of the generalized regularized long wave burgers equation u t u x p u x u xx u xxt 0 that begin with finite energy decay to zero as t becomes unboundedly large consideration is given here to the case where p vanishes at least cubically at the origin in this case solutions of may be decomposed exactly as the sum of a solution of the corresponding linear equation and a higher order correction term an explicit asymptotic form for the l 2 norm of the higher order correction is presented here the effect of the nonlinearity is felt only in the higher order term a similar decomposition is given for the generalized korteweg de vries burgers equation u t u x p u x u xx u xxx 0

commutators of singular integrals and fractional integral operators the commutators of integral and differential operators are very important tools in harmonic analysis and pde recently the third author of this paper gave a survey front math china 2011 6 5 821 833 on some late results and put forward a few problems on commutators of many important operators in harmonic analysis in this paper we present some results about the commutators of singular integrals and fractional integrals with rough or variable kernels which can be seen as a supplement of the aforementioned survey

hypercyclic operators that commute with the bergman backward shift the backward shift b on the bergman space of the unit disc is known to be hypercyclic meaning it has a dense orbit here we ask which operators that commute with b inherit its hypercyclicity we show that the problem reduces to the study of operators of the form 9p b where so is a holomorphic self map of the unit disc that multiplies the dirichlet space into itself and that the question of hypercyclicity for such an operator depends on how freely 9p z is allowed to approach the unit circle as lz s 1

explicit inverse of a tridiagonal k toeplitz matrix we obtain explicit formulas for the entries of the inverse of a nonsingular and irreducible tridiagonal k toeplitz matrix a the proof is based on results from the theory of orthogonal polynomials and it is shown that the entries of the inverse of such a matrix are given in terms of chebyshev polynomials of the second kind we also compute the characteristic polynomial of a which enables us to state some conditions for the existence of a 1 our results also extend known results for the case when the residue mod k of the order of a is equal to 0 or k 1 numer math 10 1967 pp 153 161

decay of solutions of some nonlinear wave equations abstract we study the large time behaviour of solutions to the initial value problem for the korteweg de vries equation and for the regularized long wave equation with a dissipative term appended using energy estimates a maximum principle and a transformation of cole hopf type sharp rates of temporal decay of certain norms of the solution are obtained

compactness for the commutators of singular integral operators with rough variable kernels abstract let t be the singular integral operator with variable kernel defined by t f x p v r n x x y x y n f y d y where x y is homogeneous of degree zero in the second variable y and s n 1 x z d z 0 for any x r n in this paper the authors prove that if l r n l q s n 1 for some q 2 n 1 n then the commutator generated by a cmo r n function and t and the associated lacunary maximal operator are compact on l 2 r n the associated continuous maximal commutator is also considered

relative growth of linear iterations and orthogonal polynomials on several intervals assume a c2 2 and y0 c2 let yn yn a y0 be defined by thelinear iteration yn ayn 1 n 1 2 then for every p 0 and for every norm in c2 there is a constant c depending only on p and such that sup a 1 y0 0 yl a y0 p lk 0 yk a y0 p cl l 1 2 where a denotes the spectral radius of a applications to orthogonal polynomials pn with respect to a positive borel measure on several intervals are studied in particular the growth of orthogonal polynomials relative to their sums satisfies limn supx supp pn x p n 1k 0 pk x p 0 provided that the three term recurrence coefficients of pn are asymptotically periodic

orthogonal polynomials and cubic transformations abstract starting from a sequence pn of orthogonal polynomials with respect to a quasi definite or regular linear functional we construct a new sequence qn of orthogonal polynomials such that q3n x pn x3 the basic tool is the three term recurrence relation which such a family satisfies as an application we present some examples concerning chebyshev and hermite polynomials

characterizations of generalized hermite and sieved ultraspherical polynomials a new characterization of the generalized hermite polynomials and of the orthogonal polynomials with respect to the measure ixy 1 x2 1 2dx is derived which is based on a reversing property of the coefficients in the corresponding recurrence formulas and does not use the representation in terms of laguerre and jacobi polynomials a similar characterization can be obtained for a generalization of the sieved ultraspherical polynomials of the first and second kind these results are applied in order to determine the asymptotic limit distribution for the zeros when the degree and the parameters tend to infinity with the same order

compact composition operators on the bloch space necessary and sufficient conditions are given for a composition operator cof f o q to be compact on the bloch space w and on the little bloch space o weakly compact composition operators on o are shown to be compact if q e o is a conformal mapping of the unit disk d into itself whose image q d approaches the unit circle t only in a finite number of nontangential cusps then co is compact on 7o on the other hand if there is a point of t n q d at which q d does not have a cusp then co is not compact

asymptotics in the critical case for whitham type equations abstract consider the cauchy problem for nonlinear dissipative evolution equations u t n u u l u 0 x r t 0 u 0 x u 0 x x r where l is the linear pseudodifferential operator l u f x l u and the nonlinearity is a quadratic pseudodifferential operator n u u f x r a t y u t y u t y d y u f x u is direct fourier transformation let the initial data u 0 h 0 h 0 1 2 are sufficiently small and have a non zero total mass m u 0 x d x 0 here h n m l 2 x m i x n x l 2 is the weighted sobolev space then we prove that the main term of the large time asymptotics of solutions in the critical case is given by the self similar solution defined uniquely by the total mass m of the initial data

sieved ultraspherical polynomials the continuous q ultraspherical polynomials contain a number of important examples as limiting or special cases one of these arose in allaway s ph d thesis in a previous paper we solved a characterization problem essentially equivalent to allaway s and showed that these polynomials arose from the q ultraspherical polynomials when q approached a root of unity a second class of such polynomials is found and the recurrence relation and orthogonality relation are found for each of these polynomials the orthogonality is interesting because the weight function has a finite number of zeros in 1 1 generating functions and other formulas are also found

compactness of hankel operators and analytic discs in the boundary of pseudoconvex domains abstract using several complex variables techniques we investigate the interplay between the geometry of the boundary and compactness of hankel operators let be a function smooth up to the boundary on a smooth bounded pseudoconvex domain c n we show that if is convex or the levi form of the boundary of is of rank at least n 2 then compactness of the hankel operator h implies that is holomorphic along analytic discs in the boundary furthermore when is convex in c 2 we show that the condition on is necessary and sufficient for compactness of h

composition operators induced by symbols defined on a polydisk abstract suppose is a holomorphic mapping from the polydisk d m into the polydisk d n or from the polydisk d m into the unit ball b n we consider the action of the associated composition operator c on hardy and weighted bergman spaces of d n or b n we first find the optimal range spaces and then characterize compactness as a special case we show that if z 1 z n z z z 1 z n is a holomorphic self map of the polydisk d n then c maps a p d n boundedly into a p d n the weight n 2 2 is best possible and the operator c a p d n a p d n is compact if and only if the function k 1 n 1 z k 2 n k 1 n 1 k z 2 tends to 0 as z approaches the full boundary of d n this settles an outstanding problem concerning composition operators on the polydisk

on sieved orthogonal polynomials iii orthogonality on several intervals on introduit deux generalisations des polynomes de chelryshev dont le spectre continu est x 2 c 1 c t k x 2 c 1 c ou c est un parametre positif on definit les fonctions poids on obtient les fonctions generatrices et les formules explicites

weakly compact composition operators on vector valued bmoa abstract weak compactness of the analytic composition operator f f is studied on bmoa x the space of x valued analytic functions of bounded mean oscillation and its subspace vmoa x where x is a complex banach space it is shown that the composition operator is weakly compact on bmoa x if x is reflexive and the corresponding composition operator is compact on the scalar valued bmoa a concrete example is given which shows that bmoa x differs from the weak vector valued bmoa for infinite dimensional banach spaces x

trace ideal criteria for toeplitz operators for a complex measure on the open unit disk u define an operator t on a hilbert space h of analytic functions with reproducing kernel k z w by u for a certain scale of hilbert spaces h 1 which includes the hardy space h2 and weighted bergman spaces a2 conditions are obtained which imply t belongs to a schatten ideal sp if is a positive measure then these conditions are necessary and sufficient application to composition operators and restriction operators are indicated

an integrable system connected with the uvarov chihara problem for orthogonal polynomials a new non local integrable chain with continuous time and discrete space variable is considered in contrast to the case of toda and volterra chains the general solution can be presented in explicit form in terms of two arbitrary sequences it is shown that this solution is connected with the so called uvarov chihara problem inserting a discrete mass at the centre of the spectral interval of symmetric orthogonal polynomials the asymptotic behaviour of the recurrence coefficients of such polynomials is considered

on semiclassical orthogonal polynomials via polynomial mappings abstract we consider orthogonal polynomials via polynomial mappings in the framework of the semiclassical class we prove that this class is stable under polynomial transformations several consequences of this fact are deduced as an application we analyze in detail cubic transformations for semiclassical orthogonal polynomials of class at most 2 recovering and extending some results proved recently for class 1 and producing new examples of semiclassical orthogonal polynomials of class 2 in particular we show how to obtain integral representations for the regular functionals with respect to which these new semiclassical families are orthogonal

essential norm estimates for composition operators on bmoa abstract we provide two function theoretic estimates for the essential norm of a composition operator c acting on the space bmoa one in terms of the n th power n of the symbol and one which involves the nevanlinna counting function we also show that if the symbol is univalent then the essential norm of c is comparable to its essential norm on the bloch space

decay estimates for the solutions of some nonlinear evolution equations abstract we study decay estimates for the solutions and their derivatives to the initial value problems for some generalized nonlinear evolution equations which have lower order diffusion terms

bmo on strongly pseudoconvex domains hankel operators duality and overline partial estimates we study the condition that characterizes the symbols of bounded hankel operators on the bergman space of a strongly pseudoconvex domain and show that it is equivalent to bmo plus analytic here we mean the bergman metric bmo of berger coburn and zhu in the course of the proof we obtain new 0 estimates that may be of independent interest some applications include a decomposition of bmo similar to the classical li l and two characterizations of the dual of vmo which is also a predual of bmo in addition we obtain some partial results on the boundedness of hankel operators in l1 norm

characterizations for besov spaces and applications part i the main theorem of this paper gives a characterization for holomorphic besov space bp d over a large class of bounded domains d in cn which states that there is a bounded linear operator vd bp d lp d d so that pvd i on bp d where p is the bergman projection and d z k z z dv is the biholomorphic invariant measure with k z z being bergman kernel function for d moreover some application for characterizing schatter von neumann p class small hankel operation is given as a direct consequence of this theorem

global existence of solution of cauchy problem for nonlinear pseudo parabolic equation abstract in this paper we prove that the cauchy problem for the nonlinear pseudo parabolic equation v t v x x t v x x v x f v x v x x g v g v x x admits a unique global generalized solution in c 1 0 h s r a unique global classical solution and asymptotic behavior of the solution we also prove that the cauchy problem for the equation v t v x x t v x x g v g v x x has a unique global generalized solution in c 1 0 w m p r l r a unique global classical solution and asymptotic behavior of the solution

compact and weakly compact composition operators from the bloch space into mobius invariant spaces we obtain exhaustive results and treat in a unified way the question of boundedness compactness and weak compactness of composition operators from the bloch space into any space from a large family of conformally invariant spaces that includes the classical spaces like bmoa q and analytic besov spaces bp in particular by combining techniques from both complex and functional analysis we prove that in this setting weak compactness is equivalent to compactness for the operators into the corresponding small spaces we also characterize the boundedness and show that it is equivalent to compactness

orthogonal polynomials on several intervals via a polynomial mapping starting from a sequence pn x no of orthogonal polynomials with an orthogonality measure yurj supported on eo c 1 1 we construct a new sequence p x fi of orthogonal polynomials on t 1 eq t is a polynomial of degree tv with an orthogonality measure i that is related to no if eo 1 1 then e t 1 l l will in general consist of tv intervals we give explicit formulas relating pn x fi and pn x fio and show how the recurrence coefficients in the three term recurrence formulas for these orthogonal polynomials are related if one chooses t to be a chebyshev polynomial of the first kind then one gets sieved orthogonal polynomials

sieved orthogonal polynomials vii generalized polynomial mappings systems of symmetric orthogonal polynomials whose recurrence relations are given by compatible blocks of second order difference equations are studied in detail applications are given to the theory of the recently discovered sieved orthogonal polynomials the connection with polynomial mappings is examined an example of a family of orthogonal polynomials having discrete masses in the interior of the spectrum is included

on co recursive orthogonal polynomials and their application to potential scattering abstract let pn x n 0 be a system of polynomials orthogonal with respect to a positive definite moment functional and satisfying the recurrence relation pn x x cn pn 1 x npn 2 x n 1 2 where p0 x 1 and p 1 x 0 the corresponding co recursive orthogonal polynomials p n x n 0 satisfy the same recurrence relations except for n 1 where now p 1 x x c 1 0 and p 0 x 1 the p n are orthogonal with respect to a moment functional which is positive definite for 0 and quasi definite for p n x separation theorems true interval of orthogonality etc can be determined from those of the pn x these polynomials occur in the l2 solution of the radial schrodinger equation for a separable potential where pn x is the tchebichef polynomial of the second kind in the case of s waves

analytic discs in the boundary and compactness of hankel operators with essentially bounded symbols abstract we derive conditions for compactness of hankel operators h f a 2 l 2 h f g i p f g with bounded holomorphic symbols f for a large class of convex and bounded domains with d k

orthogonal polynomials on the unit circle via a polynomial mapping on the real line let f n n 0 be a sequence of monic orthogonal polynomials on the unit circle opuc with respect to a symmetric and finite positive borel measure email protected on 0 email protected and let 1 a 0 a 1 a 2 be the associated sequence of verblunsky coefficients in this paper we study the sequence email protected n n 0 of monic opuc whose sequence of verblunsky coefficients is 1 b 1 b 2 b n 1 a 0 b n 1 b 2 b 1 a 1 b 1 b 2 b n 1 a 2 b n 1 b 2 b 1 a 3 where b 1 b 2 b n 1 are n 1 fixed real numbers such that b email protected 1 1 for all j 1 2 n 1 so that email protected n n 0 is also orthogonal with respect to a symmetric and finite positive borel measure email protected on the unit circle we show that the sequences of monic orthogonal polynomials on the real line oprl corresponding to f n n 0 and email protected n n 0 by szego s transformation are related by some polynomial mapping giving rise to a one to one correspondence between the monic opuc email protected n n 0 on the unit circle and a pair of monic oprl on a subset of the interval 1 1 in particular we prove email protected q email protected n 1 q email protected q n q email protected q n q q n q supported on a subset of the union of 2n intervals contained in 0 email protected such that any two of these intervals have at most one common point and where up to an affine change in the variable z n 1 and email protected n are algebraic polynomials in email protected of degrees n 1 and n respectively defined only in terms of a 0 b 1 b n 1 this measure induces a measure on the unit circle supported on the union of 2n arcs pairwise symmetric with respect to the real axis the restriction to symmetric measures or real verblunsky coefficients is needed in order that szego s transformation may be applicable

rigidity of composition operators on the hardy space h p abstract let be an analytic map taking the unit disk d into itself we establish that the class of composition operators f c f f exhibits a rather strong rigidity of non compact behaviour on the hardy space h p for 1 p and p 2 our main result is the following trichotomy which states that exactly one of the following alternatives holds i c is a compact operator h p h p ii c fixes a linearly isomorphic copy of l p in h p but c does not fix any copies of l 2 in h p iii c fixes a copy of l 2 in h p moreover in case iii the operator c actually fixes a copy of l p 0 1 in h p provided p 1 we reinterpret these results in terms of norm closed ideals of the bounded linear operators on h p which contain the compact operators k h p in particular the class of composition operators on h p does not reflect the quite complicated lattice structure of such ideals