new abstract hardy spaces the aim of this paper is to propose an abstract construction of spaces which keep the main properties of the already known hardy spaces h1 we construct spaces through an atomic or molecular decomposition we prove some results about continuity from these spaces into l1 and some results about interpolation between these spaces and the lebesgue spaces we also obtain some results on weighted norm inequalities finally we present partial results in order to understand a characterization of the duals of hardy spaces

semigroup kernels poisson bounds and holomorphic functional calculus letlbe the generator of a continuous holomorphic semigroupswhose action is determined by an integral kernelkon a scale of spaceslp x under mild geometric assumptions on x we prove that iflhas a boundedh functional calculus onl2 x andksatisfies bounds typical for the poisson kernel thenlhas a boundedh functional calculus onlp x for eachp 1 moreover if x is of polynomial type andksatisfies second order gaussian bounds we establish criteria forlto have a bounded hormander functional calculus or a bounded davies helffer sjostrand functional calculus

gaussian heat kernel upper bounds via the phragmen lindelof theorem we prove that in presence of l gaussian estimates so called davies gaffney estimates on diagonal upper bounds imply precise off diagonal gaussian upper bounds for the kernels of analytic families of operators on metric measure spaces

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

an operator valued tb theorem we establish conditions similar to the tb theorem of david journe and semmes which guarantee the boundedness of an integral transformation t with l x valued kernel on lxp rn where 1 p and x is a banach space with the unconditionality property of martingale differences umd

a t 1 theorem for non integral operators let x be a space of homogeneous type and let l be a sectorial operator with bounded holomorphic functional calculus on l 2 x we assume that the semigroup e tl t 0 satisfies davies gaffney estimates associated with l are certain approximations of the identity we call an operator t a non integral operator if compositions involving t and these approximations satisfy certain weighted norm estimates the davies gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on t in calderon zygmund theory in this paper we show under the additional assumption that a vertical littlewood paley stein square function associated with l is bounded on l 2 x that a non integral operator t is bounded on l 2 x if and only if t 1 in bmo l x and t 1 in bmo l x here bmo l x and bmo l x denote the recently defined bmo x spaces associated with l that generalize the space bmo x of john and nirenberg generalizing a recent result due to f bernicot we show a second version of a t 1 theorem under weaker off diagonal estimates which gives a positive answer to a question raised by him as an application we prove l 2 x boundedness of a paraproduct operator associated with l we moreover study criterions for a t b theorem to be valid

hardy spaces associated to operators satisfying davies gaffney estimates and bounded holomorphic functional calculus abstract let x be a space of homogeneous type assume that an operator l has a bounded holomorphic functional calculus on l 2 x and the kernel of the heat semigroup e t l t 0 satisfies the davies gaffney estimates without the assumption that l is self adjoint we develop a theory of hardy spaces h l p x 0 p 1 which includes a molecular decomposition an atomic decomposition a square function characterization duality of hardy and lipschitz spaces and a marcinkiewicz type interpolation theorem as applications we show that l has a bounded holomorphic functional calculus on h l p x for all p 0 and certain riesz transforms associated to l are bounded from h l p x to l p x for all 0 p 2

weak type p p estimates for riesz transforms we consider the problem of l p boundedness of higher order riesz transforms associated to elliptic operators l of order 2m on as an application of the recently solved kato conjecture we show for all this generalizes the result of auscher and tchamitchian restricted to the case d 2m

quadratic estimates and functional calculi of perturbed dirac operators we prove quadratic estimates for complex perturbations of dirac type operators and thereby show that such operators have a bounded functional calculus as an application we show that spectral projections of the hodge dirac operator on compact manifolds depend analytically on l changes in the metric we also recover a unified proof of many results in the calderon program including the kato square root problem and the boundedness of the cauchy operator on lipschitz curves and surfaces

the poincare inequality is an open ended condition let p 1 and let x d be a complete metric measure space with borel and doubling that admits a 1 p poincare inequality then there exists e 0 such that x d admits a 1 q poincare inequality for every q p e quantitatively

a numerical method for solving systems of linear and nonlinear integral equations of the second kind by hat basis functions abstract in this paper we use hat basis functions to solve the system of fredholm integral equations sfies and the system of volterra integral equations svies of the second kind this method converts the system of integral equations into a linear or nonlinear system of algebraic equations also we consider the order of convergence of the method and show that it is o h 2 application of the method on some examples show its accuracy and efficiency

chebyshev polynomial solutions of systems of linear integral equations a chebyshev collocation method has been presented to solve systems of linear integral equations in terms of chebyshev polynomials this method transforms the integral system into the matrix equation with the help of chebyshev collocation points and the unknown of this equation is a chebyshev coefficient matrix they are easily acquired by using the last matrix equation which corresponds to the system of linear algebraic equations as a result the finite chebyshev series approach is obtained

spectrally accurate nystrom solver error bounds for 1 d fredholm integral equations of the second kind abstract we present the theory underlying and computational implementation of analytical predictions of error bounds for the approximate solution of one dimensional fredholm integral equations of the second kind through asymptotic estimates of near supremal operator norms readily implementable formulae for the error bounds are computed explicitly using only the numerical solution of nystrom based methods on distributions of nodes at the roots or extrema of diverse orthogonal polynomials despite the predicted bounds demanding no a priori information about the exact solution they are validated to be spectrally accurate upon comparison with the explicit computational error accruing from the numerical solution of a variety of test problems some chosen to be challenging to approximation methods with known solutions potential limitations of the theory are discussed but these are shown not to arise in the numerical computations

triangular 3 subdivision schemes the regular case the paper deals with the investigation of triangular 3 subdivision schemes in the stationary shift invariant setting in section 2 we collect the available theory on refinable functions subdivision surfaces with emphasis on their sobolev and holder smoothness families of interpolatory and approximating 3 subdivision schemes are investigated in section 3 some dual 3 subdivision schemes which are related to vector valued refinable functions are also analyzed for this purpose we have developed matlab routines for numerically investigating properties of vector subdivision schemes

numerical solution of second kind fredholm integral equations system by using a taylor series expansion method in this paper we present a taylor series expansion method for a second kind fredholm integral equations system with smooth or weakly singular kernels this method reduce the system of integral equations to a linear system of ordinary differential equation after constructing boundary conditions this system reduce to a system of equations that can be solved easily with any of the usual methods finally we show the efficiency of the method using some numerical examples

projection methods and condition numbers in uniform norm for fredholm and cauchy singular integral equations in this paper the authors propose a numerical method for the approximate solution of some classes of fredholm and cauchy integral equations including the discrete collocation and collocation methods

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

a new cross subdivision scheme for surface design abstract in this paper a new primal approximation 2 subdivision scheme for surface design being named the cross scheme or c scheme for short is presented the c 2 continuity of the c scheme is mathematically analyzed the new scheme can be effectively applied to any 3d mesh of quadrilaterals extraordinary vertices and miscellaneous boundary scenarios are handled as well extensive implementation shows that it performs better than the classical catmull clark subdivision scheme both at nearby vertices and along the non vertical and non horizontal directions

on the lp theory of c0 semigroups associated with second order elliptic operators ii abstract we study positive c 0 semigroups on l p associated with second order uniformly elliptic divergence type operators with singular lower order terms subject to a wide class of boundary conditions we obtain an interval p min p max in the l p scale where these semigroups can be defined including the case 2 p min p max we present an example showing that the result is optimal we also show that the semigroups are analytic with angles of analyticity and spectra of the generators independent of p for the whole range of p where the semigroups are defined

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

triangular 7 and quadrilateral 5 subdivision schemes regular case abstract this paper is devoted to the study of triangular 7 and quadrilateral 5 surface subdivisions both approximation and interpolatory subdivision schemes are considered with illustrative examples of both scalar valued and matrix valued 7 and 5 subdivision masks that satisfy the sum rule of sufficiently high orders in particular optimal sobolev smoothness is determined and holder smoothness estimates are presented

numerical treatment of second kind fredholm integral equations systems on bounded intervals in this paper the authors propose numerical methods to approximate the solutions of systems of second kind fredholm integral equations they prove that such methods are stable and convergent error estimates in weighted l p norm 1

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

numerical solution of integral equations system of the second kind by block pulse functions abstract this paper endeavors to formulate block pulse functions to propose solutions for the fredholm integral equations system to begin with we describe the characteristic of block pulse functions and will go on to indicate that through this method a system of fredholm integral equations can be reduced to an algebraic equation numerical examples presented to illustrate the accuracy of the method

solution of second kind fredholm integral equations by means of gauss and anti gauss quadrature rules this paper is concerned with the numerical approximation of fredholm integral equations of the second kind a nystrom method based on the anti gauss quadrature formula is developed and investigated in terms of stability and convergence in appropriate weighted spaces the nystrom interpolants corresponding to the gauss and the anti gauss quadrature rules are proved to furnish upper and lower bounds for the solution of the equation under suitable assumptions which are easily verified for a particular weight function hence an error estimate is available and the accuracy of the solution can be improved by approximating it by an averaged nystrom interpolant the effectiveness of the proposed approach is illustrated through different numerical tests

kato s square root problem in banach spaces abstract let l be an elliptic differential operator with bounded measurable coefficients acting in bochner spaces l p r n x of x valued functions on r n we characterize kato s square root estimates l u p u p and the h functional calculus of l in terms of r boundedness properties of the resolvent of l when x is a banach function lattice with the umd property or a noncommutative l p space to do so we develop various vector valued analogues of classical objects in harmonic analysis including a maximal function for bochner spaces in the special case x c we get a new approach to the l p theory of square roots of elliptic operators as well as an l p version of carleson s inequality

some new function spaces and their applications to harmonic analysis abstract in this paper a family of spaces is introduced which seems well adapted for the study of a variety of questions related to harmonic analysis and its applications these spaces are the tent spaces they provide the natural setting for the study of such things as maximal functions the relevant space here is t p and also square functions where the space t2p is relevant as such these spaces lead to unifications and simplifications of some basic techniques in harmonic analysis thus they are closely related to lp and hardy spaces important parts of whose theory become corollaries of the description of tent spaces also as proc conf harmonic analysis cortona lect notes in math vol 992 springer verlag berlin new york 1983 already indicated where these spaces first appeared explicitly the tent spaces can be used to simplify some of the results related to the cauchy integral on lipschitz curves and multilinear analysis in retrospect one can recognize that various ideas important for tent spaces had been used if only implicitly for quite some time here one should mention carleson s inequality its simplifications and extensions the theory of hardy spaces and atomic decompositions

delta basis functions and their applications to systems of integral equations in the present paper delta functions dfs are proposed as a new set of basis functions their properties and relations to well known triangular functions tfs are described the simplicity and useful properties of newly proposed sets led us to use them with more accuracy and less computational burden furthermore dfs are applied to propose an efficient method for approximating the solution of integral equations systems convergence analysis and the rate of convergence have been considered as well some numerical examples are provided to illustrate the computational efficiency and accuracy of the method

new function spaces of bmo type the john nirenberg inequality interpolation and applications in this paper we introduce and develop some new function spaces of bmo bounded mean oscillation type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type the new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity and they generalize the classical bmo space we show that the john nirenberg inequality holds on these spaces and they interpolate with lp spaces by the complex interpolation method we then give applications on lp boundedness of singular integrals whose kernels do not satisfy the hormander condition 2005 wiley periodicals inc