matricial nehari problems j inner matrix functions and the muckenhoupt condition abstract the classes of regular and strongly regular generating matrices and j inner matrix valued functions arose in the investigation of the matricial nehari problem bitangential interpolation problems and inverse problems for canonical systems as well as the theory of characteristic functions of operators and operator nodes in this paper new characterizations of these classes are developed in particular the property of strong regularity is characterized in terms of a matricial muckenhoupt a2 condition in the treil volberg form these results are based on parametrizations that are intimately connected with darlington representations of matrix valued functions in the schur and caratheodory classes as a byproduct of this analysis examples of strongly regular generating matrices and entire j inner matrix valued functions that are unbounded on the circle and the real line respectively are presented

on continuous dependence for an inverse initial boundary value problem for the wave equation abstract we show that the bounded coefficient q x of the wave equation u tt u qu x t 0 t u x 0 u t x 0 0 x u v x x x t 0 t can be continuously determined by its neumann to dirichlet map in certain function spaces

stability estimate for the aligned magnetic field in a periodic quantum waveguide from dirichlet to neumann map in this article we study the boundary inverse problem of determining the aligned magnetic field appearing in the magnetic schrodinger equation in a periodic quantum cylindrical waveguide by knowledge of the dirichlet to neumann map we prove a holder stability estimate with respect to the dirichlet to neumann map by means of the geometrical optics solutions of the magnetic schrodinger equation

on the zeros of some continuous analogues of matrix orthogonal polynomials and a related extension problem with negative squares a new proof of a recent theorem of ellis gohberg and lay which identifies the number of roots of a continuous matrix orthogonal polynomial in the open upper halfplane with the number of negative eigenvalues of a related integral operator is presented a related extension problem is then formulated and solved in assorted classes of functions which are analytic in the open upper half plane apart from a finite number of poles a discrete analogue of this extension problem is also formulated and solved 1994 john wiley sons inc

the sharp bound for the hilbert transform on weighted lebesgue spaces in terms of the classical ap characteristic we show that the norm of the hilbert transform as an operator in the weighted space l p r for 2 p oo is bounded by a constant multiple of the first power of the classical ap characteristic of this result is sharp we also prove a bilinear imbedding theorem with simple conditions

on the inverse scattering problem for jacobi matrices with the spectrum on an interval a finite system of intervals or a cantor set of positive length when solving the inverse scattering problem for a discrete sturm liouville operator with a rapidly decreasing potential one gets reflection coefficients s and invertible operators where is the hankel operator related to the symbol s the marchenko faddeev theorem 8 in the continuous case for the discrete case see 4 6 guarantees the uniqueness of the solution of the inverse scattering problem in this article we ask the following natural question can one find a precise condition guaranteeing that the inverse scattering problem is uniquely solvable and that operators are invertible can one claim that uniqueness implies invertibility or vise versa moreover we are interested here not only in the case of decreasing potential but also in the case of asymptotically almost periodic potentials so we merge here two mostly developed cases of the inverse problem for sturm liouville operators the inverse problem with almost periodic potential and the inverse problem with the fast decreasing potential

convex body domination and weighted estimates with matrix weights abstract we introduce the so called convex body valued sparse operators which generalize the notion of sparse operators to the case of spaces of vector valued functions we prove that calderon zygmund operators as well as haar shifts and paraproducts can be dominated by such operators by estimating sparse operators we obtain weighted estimates with matrix weights we get two weight a 2 a estimates that in the one weight case give us the estimate t l 2 w l 2 w c w a 2 1 2 w a c w a 2 3 2 where t is either calderon zygmund operator with modulus of continuity satisfying the dini condition or a haar shift or a paraproduct

on a new class of structured reproducing kernel spaces abstract a class of reproducing kernel spaces with reproducing kernels of the form k j j with p a a is characterized in terms of invariance under a pair of generalized shift operators and a structural identity this incorporates a characterization of de branges for the line case and a later analogue due to ball for the circle case as well as many other possibilities by specializing the choice of these results also permit the extension of some earlier characterizations by the authors of finite dimensional spaces with reproducing kernels of the form given above to the infinite dimensional case the non hermitian case is also considered

observability and stabilization of magnetic schrodinger equations abstract we are mainly interested in extending the known results on observability inequalities and stabilization for the schrodinger equation to the magnetic schrodinger equation that is in presence of a magnetic potential we establish observability inequalities and exponential stabilization by extending the usual multiplier method under the same geometric condition that needed for the schrodinger equation we also prove with the help of elliptic carleman inequalities logarithmic stabilization results through a resolvent estimate although the approach is classical these results on logarithmic stabilization seem to be new even for the schrodinger equation

global uniqueness in the inverse scattering problem for the schrodinger operator with external yang mills potentials we consider the schrodinger equation in r n n 3 with external yang mills potentials having compact supports we prove the uniqueness modulo a gauge transformation of the solution of the inverse boundary value problem in a bounded convex domain a similar uniqueness result holds for the inverse scattering problem at a fixed energy

criteria for the strong regularity of j inner functions and γ generating matrices abstract the class of left and right strongly regular j inner mvf s plays an important role in bitangential interpolation problems and in bitangential direct and inverse problems for canonical systems of integral and differential equations a new criterion for membership in this class is presented in terms of the matricial muckenhoupt condition a 2 that was introduced for other purposes by treil and volberg analogous results are also obtained for the class of generating functions that intervene in the nehari problem the new criterion is simpler than the criterion that we presented earlier a determinental criterion is also presented

bounds for the hilbert transform with matrix a2 weights we establish a new estimate for the hilbert transform in l 2 space endowed with a matrix weight

rationality of the moduli spaces of plane curves of sufficiently large degree we prove that the moduli space of plane curves of degree d is rational for all sufficiently large d

strichartz estimates for the magnetic schrodinger equation abstract we prove global scale invariant strichartz estimates for the linear magnetic schrodinger equation with small time dependent magnetic field this is done by constructing an appropriate parametrix as an application we show a global regularity type result for schrodinger maps in dimensions n 6

stability estimates for the hyperbolic dirichlet to neumann map in anisotropic media abstract in this paper we prove stability estimates for the hyperbolic dirichlet to neumann map associated to the wave equation in anisotropic media

embeddings of vector valued bergman spaces abstract we show that a dyadic version of the carleson embedding theorem for the bergman space extends to vector valued functions and operator valued measures this is in contrast to a result by nazarov treil volberg in the context of the hardy space we also discuss some embeddings for analytic vector valued functions

the tb theorem on non homogeneous spaces 0 introduction main objects and results 3 0 1 main results 4 0 2 an application of t1 heorem electric intensity capacity 7 0 3 how to interpret calderon zygmund operator t 9 0 3 1 bilinear form is defined on lipschitz functions 10 0 3 2 bilinear form is defined for smooth functions 11 0 3 3 apriori boundedness 12 0 4 plan of the paper 15

stability estimate for an inverse problem for the magnetic schrodinger equation from the dirichlet to neumann map abstract we consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic schrodinger equation in a bounded smooth domain of r n with input dirichlet data from measured neumann boundary observations this information is enclosed in the dynamical dirichlet to neumann map associated to the solutions of the magnetic schrodinger equation we prove in dimension n 2 that the knowledge of the dirichlet to neumann map for the magnetic schrodinger equation measured on the boundary determines uniquely the magnetic field and we prove a holder type stability in determining the magnetic field induced by the magnetic potential

sampling and interpolation problems for vector valued signals in the paley wiener spaces a new approach to sampling interpolation problem for vector valued signals is proposed for a function signal f from the paley wiener space pw pi bbc n of entire vector valued functions the sampling data are defined as the inner products langf lambdan alphan rang of f lambdan and vectors alphan it is proved that alphan lambdan is a sampling and interpolating sequence for pw pi bbc n if and only if the family alphanei lambdan t forms a riesz basis in l 2 pi pi bbc n this relation allows to characterize all sampling and interpolating sequences in terms of a generating matrix function to produce a wide class of matrix functions generating sampling and interpolating sequences we consider the matrix sturm liouville problems and apply the boundary control theory for hyperbolic dynamical systems this approach allows to construct both asymptotically uniform and nonuniform sampling and interpolating sequences

wavelets and the angle between past and future the main result the riesz projectionp or equivalently hilbert transformt is bounded in the weighted spacel2 w wherewis a matrix valued weight if and only ifsupi 1 i iw 1 2 1 i iw 1 1 2 wheresupremumis taken over all intervalsi motivation for this problem comes from stationary processes riesz projection is bounded means the angle between past and future of a stationary process with spectral measurewis nonzero in the scalar case the result is the well known hunt muckenhoupt wheeden theorem the main step in our proof is to show that a vector haar system forms an unconditional basis inl2 w as a byproduct of our approach we get some new results about bases of wavelets in weighted spaces in both scalar and vector valued cases

carleman estimate for the schrodinger equation and application to magnetic inverse problems abstract in this paper we prove lipschitz stable determination of the complex valued electric potential and the direction of the magnetic field appearing in the dynamic schrodinger equation with static coefficients by finitely many partial boundary measurements of the solution this is by means of the bukhgeim klibanov method based on an appropriate carleman estimate since the time symmetrization of the static magnetic schrodinger equation around t 0 is not possible we preliminarily establish a carleman inequality specifically designed for this problem

trigonometric bases for matrix weighted lp spaces abstract we give a complete characterization of 2 periodic matrix weights w for which the vector valued trigonometric system forms a schauder basis for the matrix weighted space l p t w then trigonometric quasi greedy bases for l p t w are considered quasi greedy bases are systems for which the simple thresholding approximation algorithm converges in norm it is proved that such a trigonometric basis can be quasi greedy only for p 2 and whenever the system forms a quasi greedy basis the basis must actually be a riesz basis

an inverse problem of the wave equation this paper is organized as follows in sect 1 the statement of the inverse problem is formulated the basic results are formulated and proved in sect 2 some auxiliary material on the x ray transform is given in a self contained way for convenience of the reader this material is used in section one

failure of the matrix weighted bilinear carleson embedding theorem abstract we prove failure of the natural formulation of a matrix weighted bilinear carleson embedding theorem featuring a matrix valued carleson sequence as well as products of norms for the embedding we show that assuming an a 2 weight is also not sufficient indeed a uniform bound on the conditioning number of the matrix weight is necessary and sufficient to get the bilinear embedding we prove the optimal dependence of the embedding on this quantity we show that any improvement of a recent matrix weighted bilinear embedding featuring a scalar carleson sequence and inner products instead of norms must fail in particular replacing the scalar sequence by a matrix sequence results in failure even when maintaining the formulation using inner products any formulation using norms even in the presence of a scalar carleson sequence must fail as a positive result we prove the so called matrix weighted redundancy condition in full generality

the theory of weights and the dirichlet problem for elliptic equations in this work we consider questions from the area of partial differential equations and the related questions from harmonic analysis specifically we will study the dirichlet problem for second order divergence form elliptic operators with bounded measurable coefficients roughly speaking we consider the following question suppose that we are given two operators lo and li and that we have good estimates for the dirichlet problem for lo good in this context could mean several different things for instance it might mean that there exists p 1 p oo so that the dirichlet problem is solvable in a suitable sense with lp data what are the optimal conditions on the difference between the coefficients of lo and l1 which guarantee that the dirichlet problem for li also has good estimates this question has been extensively studied recently cf f j k r f and d4 our work uses in a crucial way the results in r f the main results in d4 and r f are proved using an ingenious differential inequality for a family of harmonic measures introduced in d4 another of our goals in this article is to give new direct proofs of the results in d4 and r f without the use of this differential inequality theorems 2 4 2 5 and 2 18 these direct approaches have already had important applications k p2 an interesting class of examples for the theorems quoted above arises when one pulls back the laplacian via a quasiconformal mapping of the plane into itself specializing our theorem to this case suggests a new characterization of the class of am weights introduced in connection with several problems in harmonic analysis by muckenhoupt mul and coifman c fefferman co f we shall present in theorem 2 3 a criterion in terms of carleson measures which is very close to log w e bmo and is necessary and sufficient for w e ad

inverse problems for the schrodinger equations with time dependent electromagnetic potentials and the aharonov bohm effect we consider the inverse boundary value problem for the schrodinger operator with time dependent electromagnetic potentials in domains with obstacles we extend the resuls of the author s works inverse probl 19 49 2003 19 985 2003 20 1497 2004 to the case of time dependent potentials we relate our results to the aharonov bohm effect caused by magnetic and electric fluxes

cmv matrices with asymptotically constant coefficients szego blaschke class scattering theory abstract we develop a modern extended scattering theory for cmv matrices with asymptotically constant verblunsky coefficients we show that the traditional faddeev marchenko condition is too restrictive to define the class of cmv matrices for which there exists a unique scattering representation the main results are 1 the class of twosided cmv matrices acting in l 2 whose spectral density satisfies the szego condition and whose point spectrum the blaschke condition corresponds precisely to the class where the scattering problem can be posed and solved that is to a given cmv matrix of this class one can associate the scattering data and the fm space the cmv matrix corresponds to the multiplication operator in this space and the orthonormal basis in it corresponding to the standard basis in l 2 behaves asymptotically as the basis associated with the free system 2 from the point of view of the scattering problem the most natural class of cmv matrices is that one in which a the scattering data determine the matrix uniquely and b the associated gelfand levitan marchenko transformation operators are bounded necessary and sufficient conditions for this class can be given in terms of an a 2 kind condition for the density of the absolutely continuous spectrum and a carleson kind condition for the discrete spectrum similar conditions close to the optimal ones are given directly in terms of the scattering data

exact controllability for the schrodinger equation the exact controllability of schrodinger equation in bounded domains with dirichlet boundary condition is studied both the boundary controllability and the internal controllability problems are considered concerning the boundary controllability the paper proves the exact controllability in h 1 omega with l 2 boundary control on the other hand the exact controllability in l 2 omega is proved with l 2 controls supported in a neighborhood of the boundary both results hold for an arbitrarily small time the method of proof combines both the hum hilbert uniqueness method and multiplier techniques

stability estimate for the hyperbolic inverse boundary value problem by local dirichlet to neumann map abstract in this paper we consider the stability of the inverse problem of determining a function q x in a wave equation t 2 u u q x u 0 in a bounded smooth domain in r n from boundary observations this information is enclosed in the hyperbolic dynamic dirichlet to neumann map associated to the solutions to the wave equation we prove in the case of n 2 that q x is uniquely determined by the range restricted to a subboundary of the dirichlet to neumann map whose stability is a type of double logarithm

determining a magnetic schrodinger operator from partial cauchy data in this paper we show in dimension n 3 that knowledge of the cauchy data for the schrodinger equation in the presence of a magnetic potential measured on possibly very small subsets of the boundary determines uniquely the magnetic field and the electric potential we follow the general strategy of 7 using a richer set of solutions to the dirichlet problem that has been used in previous works on this problem

discreteness of the spectrum of schrodinger operators with non negative matrix valued potentials abstract we prove three results giving sufficient and or necessary conditions for discreteness of the spectrum of schrodinger operators with non negative matrix valued potentials i e operators acting on l 2 r n c d by the formula h v v where the potential v takes values in the set of non negative hermitian d d matrices the first theorem provides a characterization of discreteness of the spectrum when the potential v is in a matrix valued a class thus extending a known result in the scalar case d 1 we also discuss a subtlety in the definition of the appropriate matrix valued a class the second result is a sufficient condition for discreteness of the spectrum which allows certain degenerate potentials i e such that det v 0 to formulate the condition we introduce a notion of oscillation for subspace valued mappings our third and last result shows that if v is a 2 2 real polynomial potential then v has discrete spectrum if and only if the scalar operator has discrete spectrum where x is the minimal eigenvalue of v x

determination of coefficients for a dissipative wave equation via boundary measurements in this paper we consider the inverse problem of recovering the viscosity coefficient in a dissipative wave equation via boundary measurements we obtain stability estimates by considering all possible measurements implemented on the boundary we also prove that the viscosity coefficient is uniquely determined by a finite number of measurements on the boundary provided that it belongs to a given finite dimensional vector space

a study of the matrix carleson embedding theorem with applications to sparse operators abstract in this paper we study the dyadic carleson embedding theorem in the matrix weighted setting we provide two new proofs of this theorem which highlight connections between the matrix carleson embedding theorem and both maximal functions and h 1 bmo duality along the way we establish boundedness results about maximal functions associated to matrix a 2 weights and duality results concerning h 1 and bmo sequence spaces in the matrix setting as an application we then use this carleson embedding theorem to show that if s is a sparse operator then the operator norm of s on l 2 w satisfies s l 2 w l 2 w w a 2 3 2 for every matrix a 2 weight w

the bitangential inverse spectral problem for canonical systems the theory of the direct and bitangential inverse input impedance problem is used to solve the direct and bitangential inverse spectral problem the analysis of the direct spectral problem uses and extends a number of results that appear in the literature special attention is paid to the class of canonical integral systems with matrizants that are strongly regular j inner matrix valued functions in the sense introduced in 7 the bitangential inverse spectral problem is solved in this class in our considerations the data for this inverse problem is a given nondecreasing p p matrix valued function on r and a normalized monotonic continuous chain of pairs b3t b4t 0 t d of entire inner p p matrix valued functions each such chain defines a class of canonical integral systems in which we find a solution of the inverse problem for the given spectral function a detailed comparison of our investigations of inverse problems with those of sakhnovich is presented