vertical square functions and other operators associated with an elliptic operator abstract we study the vertical and conical square functions defined via elliptic operators in divergence form in general vertical and conical square functions are equivalent operators just in l 2 but when this square functions are defined through the heat or poisson semigroups that arise from an elliptic operator we are able to find open intervals containing 2 where the equivalence holds the intervals in question depend ultimately on the range where the semigroup is uniformly bounded or has off diagonal estimates as a consequence we obtain new boundedness results for some square functions besides we consider a non tangential maximal function associated with the poisson semigroup and extend the known range where that operator is bounded our methods are based on the use of extrapolation for muckenhoupt weights and change of angle estimates all our results are obtained in the general setting of a degenerate elliptic operator where the degeneracy is given by an a 2 weight in weighted lebesgue spaces of course they are valid in the unweighted and or non degenerate situations which can be seen as special cases and they provide new results even in those particular settings we also consider the square root of a degenerate elliptic operator in divergence form l w and improve the lower bound of the interval where this operator is known to be bounded on l p v d w finally we give unweighted boundedness results for the degenerate operators under consideration

the heat equation with rough boundary conditions and holomorphic functional calculus abstract in this paper we consider the laplace operator with dirichlet boundary conditions on a smooth domain we prove that it has a bounded h calculus on weighted l p spaces for power weights which fall outside the classical class of a p weights furthermore we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity in particular we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data

the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

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 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

regularity theory for solutions to second order elliptic operators with complex coefficients and the l p dirichlet problem abstract we establish a new theory of regularity for elliptic complex valued second order equations of the form l div a when the coefficients of the matrix a satisfy a natural algebraic condition a strengthened version of a condition known in the literature as l p dissipativity precisely the regularity result is a reverse holder condition for l p averages of solutions on interior balls and serves as a replacement for the de giorgi nash moser regularity of solutions to real valued divergence form elliptic operators in a series of papers cialdea and maz ya studied necessary and sufficient conditions for l p dissipativity of second order complex coefficient operators and systems recently carbonaro and dragicevic introduced a condition they termed p ellipticity and showed that it had implications for boundedness of certain bilinear operators that arise from complex valued second order differential operators their p ellipticity condition is exactly our strengthened version of l p dissipativity the regularity results of the present paper are applied to solve l p dirichlet problems for l div a b when a and b satisfy a carleson measure condition which previously was known only in the real valued case we show solvability of the l 2 dirichlet problem as well as solvability of the l p dirichlet boundary value problem for p in the range where a is p elliptic

the best circulant preconditioners for hermitian toeplitz systems ii the multiple zero case in 10 14 circulant type preconditioners have been proposed for ill conditioned hermitian toeplitz systems that are generated by nonnegative continuous functions with a zero of even order the proposed circulant preconditioners can be constructed without requiring explicit knowledge of the generating functions it was shown that the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers and that all eigenvalues are uniformly bounded away from zero therefore the conjugate gradient method converges linearly when applied to solving the circulant preconditioned systems in 10 14 it was claimed that this result can be the case where the generating functions have multiple zeros the main aim of this paper is to give a complete convergence proof of the method in 10 14 for this class of generating functions

on maximal regularity of type lp lq under minimal assumptions for elliptic non divergence operators we prove maximal regularity of type l p l q for operators in non divergence form with complex valued measurable coefficients on r n for a certain range of q which depends on dimension and the order of the operators this is done under the sole assumption that they generate an analytic semigroup in l q thus we give for this class of operators and this range of q a positive answer to brezis question whether generation of an analytic semigroup entails maximal l p regularity for other values of q we give several additional assumptions the proof relies on a result on maximal regularity under the assumption of suitable off diagonal bounds due to s blunck and the author which we improve here these off diagonal bounds are obtained by a modification of davies technique suited to cope with operators in non divergence form and they also imply new results on the scale of l q spaces in which a given operator generates an analytic semigroup we thus obtain a completely new approach to maximal l p regularity for operators of this kind in particular for those whose highest order coefficients belong to vmo moreover we obtain extensions of recent results due to kim and krylov for operators with measurable coefficients depending on one coordinate

recursive based pcg methods for toeplitz systems with nonnegative generating functions in this paper we consider the solutions of symmetric positive definite but ill conditioned toeplitz systems an x b here we propose to solve the system by the recursive based preconditioned conjugate gradient method the idea is to use the inverse of am the principal submatrix of an with the gohberg semencul formula as a preconditioner for an the inverse of am can be generated recursively by using the formula until m is small enough the construction of the preconditioners requires only the entries of an and does not require the explicit knowledge of the generating function f of an we show that if f is a nonnegative bounded and piecewise continuous even function with a finite number of zeros of even order the spectra of the preconditioned matrices are uniformly bounded except for a fixed number of outliers hence the conjugate gradient method when applied to solving the preconditioned system converges very quickly numerical results are included to illustrate the effectiveness of our approach

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

vertical versus conical square functions we study the difference between vertical and conical square functions in the abstract and also in the specific case where the square functions come from an elliptic operator trans amer math soc 364 2012 no 10 5469 5489

extrapolation with weights rearrangement invariant function spaces modular inequalities and applications to singular integrals abstract we present an extrapolation theory that allows us to obtain from weighted l p inequalities on pairs of functions for p fixed and all a weights estimates for the same pairs on very general rearrangement invariant quasi banach function spaces with a weights and also modular inequalities with a weights vector valued inequalities are obtained automatically without the need of a banach valued theory this provides a method to prove very fine estimates for a variety of operators which include singular and fractional integrals and their commutators in particular we obtain weighted and vector valued extensions of the classical theorems of boyd and lorentz shimogaki the key is to develop appropriate versions of rubio de francia s algorithm

weighted lp estimates for powers of selfadjoint operators abstract we prove l p and weighted l p estimates for bounded functions of a selfadjoint operator satisfying a pointwise gaussian estimate for its heat kernel as an application we obtain weighted estimates for fractional powers of an electromagnetic schrodinger operator with singular coefficients the proofs are based on a modification of techniques due to hebisch 1990 17 and auscher and martell 2006 4

the kato problem for operators with weighted ellipticity we consider second order operators lw w 1 divawr with ellipticity controlled by a muckemphout a2 weight w we prove that the kato square root estimate l 1 2 w f l2 w kr fkl2 w holds in the weighted space l 2 w

on the extreme eigenvalues of hermitian block toeplitz matrices abstract we are concerned with the behavior of the minimum maximum eigenvalue 0 n n n of an n 1 n 1 hermitian toeplitz matrix tn where is an integrable real valued function kac murdoch and szego widom parter and r h chan obtained that 0 n min o 1 n2k in the case where c2k at least locally and inf has a zero of order 2k we obtain the same result under the second hypothesis alone moreover we develop a new tool in order to estimate the extreme eigenvalues of the mentioned matrices proving that the rate of convergence of 0 n to inf depends only on the order not necessarily even or integer or finite of the zero of inf with the help of this tool we derive an absolute lower bound for the minimal eigenvalues of toeplitz matrices generated by nonnegative l1 functions and also an upper bound for the associated euclidean condition numbers finally these results are extended to the case of hermitian block toeplitz matrices with toeplitz blocks generated by a bivariate integrable function

weighted norm inequalities off diagonal estimates and elliptic operators part i general operator theory and weights this is the first part of a series of four articles in this work we are interested in weighted norm estimates we put the emphasis on two results of different nature one is based on a good inequality with two parameters and the other uses calderon zygmund decomposition these results apply well to singular non integral operators and their commutators with bounded mean oscillation functions singular means that they are of order 0 non integral that they do not have an integral representation by a kernel with size estimates even rough so that they may not be bounded on all lp spaces for 1 p 8 pointwise estimates are then replaced by appropriate localized lp lq estimates we obtain weighted lp estimates for a range of p that is different from 1 8 and isolate the right class of weights in particular we prove an extrapolation theorem a la rubio de francia for such a class and thus vector valued estimates

the solution of the kato problem for degenerate elliptic operators with gaussian bounds we prove the kato conjecture for degenerate elliptic operators on r n more precisely we consider the divergence form operator lw w 1 diva where w is a muckenhoupt a2 weight and a is a complex valued n n matrix such that w 1 a is bounded and uniformly elliptic we show that if the heat kernel of the associated semigroup e tlw satisfies gaussian bounds then the weighted kato square root estimate kl 1 2 w f k l2 w k f k l2 w holds

analyticity of layer potentials and l 2 solvability of boundary value problems for divergence form elliptic equations with complex l infty coefficients we consider divergence form elliptic operators of the form l dv a x nabla defined in r n 1 x t in r n times r n geq 2 where the l infty coefficient matrix a is n 1

extrapolation from a weights and applications abstract we generalize the a p extrapolation theorem of rubio de francia to a weights in the context of muckenhoupt bases our result has several important features first it can be used to prove weak endpoint inequalities starting from strong type inequalities something which is impossible using the classical result second it provides an alternative to the technique of good inequalities for proving l p norm inequalities relating operators third it yields vector valued inequalities without having to use the theory of banach space valued operators we give a number of applications to maximal functions singular integrals potential operators commutators multilinear calderon zygmund operators and multiparameter fractional integrals in particular we give new proofs which completely avoid the good inequalities of coifman s inequality relating singular integrals and the maximal operator of the fefferman stein inequality relating the maximal operator and the sharp maximal operator and the muckenhoupt wheeden inequality relating the fractional integral operator and the fractional maximal operator

the best circulant preconditioners for hermitian toeplitz systems in this paper we propose a new family of circulant preconditioners for ill conditioned hermitian toeplitz systems a x b the preconditioners are constructed by convolving the generating function f of a with the generalized jackson kernels for an n by n toeplitz matrix a the construction of the preconditioners requires only the entries of a and does not require the explicit knowledge of f when f is a nonnegative continuous function with a zero of order 2p the condition number of a is known to grow as o n2p we show however that our preconditioner is positive definite and the spectrum of the preconditioned matrix is uniformly bounded except for at most 2p 1 outliers moreover the smallest eigenvalue is uniformly bounded away from zero hence the conjugate gradient method when applied to solving the preconditioned system converges linearly the total complexity of solving the system is therefore of o n log n operations in the case when f is positive we show that the convergence is superlinear numerical results are included to illustrate the effectiveness of our new circulant preconditioners

riesz transforms fractional power and functional calculus of schrodinger operators on weighted lp spaces abstract we consider the schrodinger operator a v v on l p r n w d x where n 3 and w is a weight in some muckenhoupt class we study the boundedness of riesz transform type operators a 1 2 and v 1 2 a 1 2 on l p r n w d x our result extends the one of bui 2010 14 to signed potentials and treat the case where p 2 it also gives a weighted version of our earlier results assaad 2011 1 assaad and ouhabaz 2012 2 and of the result auscher and ben ali 2007 4 to weighted lebesgue spaces we study also the boundedness from l p r n w p d x to l q r n w q d x of the fractional power a 2 and the l p r n w d x boundedness of the h functional calculus of a

preconditioners for ill conditioned toeplitz systems constructed from positive kernels in this paper we are interested in the iterative solution of ill conditioned toeplitz systems generated by continuous nonnegative real valued functions f with a finite number of zeros we construct new w circulant preconditioners without explicit knowledge of the generating function f by approximating f by its convolution f kn with a suitable positive reproducing kernel kn by the restriction to positive kernels we obtain positive definite preconditioners moreover if f has only zeros of even order le 2s then we can prove that the property int pi pi t 2k k n t mbox d t le c n 2k k 0 hspace 1 5pt ldots s of the kernel is necessary and sufficient to ensure the convergence of the pcg method in a number of iteration steps independent of the dimension n of the system our theoretical results were confirmed by numerical tests

lack of natural weighted estimates for some singular integral operators we show that the classical hormander condition or analogously the l r hormander condition for singular integral operators t is not sufficient to derive coifman s inequality rn tf x p w x dx c rn mf x p w x dx where 0 p m is the hardy littlewood maximal operator w is any a weight and c is a constant depending upon p and the a constant of w this estimate is well known to hold when t is a calderon zygmlmd operator as a consequence we deduce that the following estimate does not hold rn t f x p w x dx c rn mf x p mw x dx where 0 p 1 and where w is an arbitrary weight however by a recent result due to a lerner this inequality is satisfied whenever t is a calderon zygmund operator one of the main ingredients of the proof is a very general extrapolation theorem for a weights

the dirichlet problem for higher order equations in composition form the present paper commences the study of higher order differential equations in composition form specifically we consider the equation lu divb adiva u 0 where a and b are elliptic matrices with complex valued bounded measurable coefficients and a is an accretive function elliptic operators of this type naturally arise for instance via a pull back of the bilaplacian 2 from a lipschitz domain to the upper half space more generally this form is preserved under a lipschitz change of variables contrary to the case of divergence form fourth order differential equations we establish well posedness of the dirichlet problem for the equation lu 0 with boundary data in l2 and with optimal estimates in terms of nontangential maximal functions and square functions

a survey and some extensions of t chan s preconditioner abstract for any given n by n matrix an t chan s circulant preconditioner cf an proposed by t chan t chan an optimal circulant preconditioner for toeplitz systems siam j sci statist comput 9 1988 766 771 is defined to be the solution of min c n a n c n f over all n by n circulant matrices cn the cf an called the optimal circulant preconditioner by t chan has been proved to be a good preconditioner for a large class of structured systems a generalization of t chan s circulant preconditioner was given by huckle t huckle circulant and skew circulant matrices for solving toeplitz matrix problems siam j matrix anal appl 13 1992 767 777 which can be used for solving some general linear systems in this paper we review some old and develop some new properties of t chan s circulant preconditioner

self improving properties of john nirenberg and poincare inequalities on spaces of homogeneous type abstract we give a condition which ensures that if one inequality of sobolev poincare type is valid then other stronger inequalities of a similar type also hold including weighted versions our main result includes many previously known results as special cases we carry out the analysis in the context of spaces of homogeneous type but the main result is new even in the usual euclidean setting

endpoint estimates for commutators of singular integral operators abstract we prove endpoint estimates for commutatotrs of singular integrals with bmo functions we first show that they satisfy l log l type inequalities and then that there is a hardy space type estimate where the usual atomic hardy space is replaced by an appropriate subspace

circulant preconditioners constructed from kernels circulant preconditioners for hermitian toeplitz systems are considered from the viewpoint of function theory it is shown that some well known circulant preconditioners can be derived from convoluting the generating function f of the toeplitz matrix with famous kernels like the dirichlet and the fejer kernels several circulant preconditioners are then constructed using this approach finally it is proven that if the convolution product converges to f uniformly then the circulant preconditioned toeplitz systems will have a clustered spectrum

a new approach to absolute continuity of elliptic measure with applications to non symmetric equations in the late 1950s and early 1960s the work of de giorgi degi and nash n and then moser mo initiated the study of regularity of solutions to divergence form elliptic equations with merely bounded measurable coefficients weak solutions in a domain 0 a priori only in a sobolev space w 1 loc 0 were shown to be ho lder continuous of some order depending just on ellipticity and maximum principles and harnack inequalities were established the dirichlet problem for such operators with continuous data on the boundary was established in lsw this in turn paved the way for a more systematic and detailed study of the properties of the elliptic measures d l associated to l div a on a domain 0 the classical properties of existence of non tangential limits of solutions fatou type theorems and comparison principles appeared in cfms but owed a great deal to the earlier work of carleson ca and hunt and wheeden h w on harmonic functions in lipschitz domains all the results mentioned above were carried out for elliptic operators l div a where the matrix a aij has bounded measurable coefficients and is symmetric however it turns out that the symmetry of the matrix is not needed to get these results morrey mor first observed this in connection with the de giorgi nash moser theory for the results in cfms this fact has not been formally observed until now with appropriate reformulation in terms of adjoint operators and adjoint green s functions the results of cfms are valid without the symmetry assumption see section 1 doi 10 1006 aima 1999 1899 available online at http www idealibrary com on

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

circulant preconditioners with unbounded inverses abstract the eigenvalue and singular value distributions for matrices s 1 n a n and c 1 n a n are examined where a n s n and c n are toeplitz matrices simple circulants and optimal circulants generated by the fourier expansion of some function f recently it has been proven that a cluster at 1 exists whenever f is from the wiener class and strictly positive both restrictions are now weakened a proof is given for the case when f may take the zero value and hence the circulants are to have unbounded inverses the main requirements on f are that it belong to l 2 and be in some sense sparsely vanishing specifically if f is nonnegative and circulants s n or c n are positive definite then the eigenvalues of s 1 n a n or c 1 n a n are clustered at 1 if f is complex valued and s n or c n are nonsingular then the singular values of s 1 n a n or c 1 n a n are clustered at 1 as well also proposed and studied are the improved circulants it is shown that improved simple circulants can be much more advantageous than optimal circulants this depends crucially on the smoothness properties of f further clustering on theorems are given that pertain to multilevel toeplitz matrices preconditioned by multilevel simple and optimal circulants

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

the neumann problem for elliptic equations with non smooth coefficients in this paper we study the neumann and regularity problems for divergence form symmetric elliptic operators l div av where a aifix is a matrix of bounded measurable functions and ellipticity is the existence of a 0 such that 2 114 2 ais ir 21412 for all i r we assume that l is defined in the unit ball b but the methods of proof are sufficiently general to allow b to be a bounded starlike lipschitz domain the neumann problem lu 0 in b avun q f

analysis of preconditioning techniques for ill conditioned toeplitz matrices several preconditioning techniques for solving toeplitz systems are known in literature but their convergence features are completely understood only in the well conditioned case we study the application of tau circulant and hartley preconditioners to ill conditioned toeplitz matrices by proving that only the first class realizes a rate of convergence not depending on the dimension of the system

new results on the exponent set of primitive nearly reducible matrices abstract for a primitive nearly reducible matrix j a ross defined e n to be the least integer greater than 5 that no n n matrix has as its exponent and he proved n 2 e n n 2 4n 6 recently j y shao proved e n n 2 2n 10 9 1 and yang shangjun and j p barker proved e n n 2 4 n 2 3 2 if a certain conjecture in number theory is true in this paper we obtain the following new results e n n 2 n 2 1 1 for n 5 and e n m 2 n m 1 2 where m is any given positive integer and n is large enough in fact we have determined e n asymptotically lim n e n n 2 1

weighted sobolev spaces and pseudodifferential operators with smooth symbols absmacr let u i be the fefferman stein sharp function of u and for 1 0 such that the pointwise estimate au x cm u x holds for all x e ri and all schwartz functions u this estimate implies the boundedness of 0 order pseudodifferential operators on weighted lp spaces whenever the weight function belongs to muckenhoupt s class ap having established this we construct weighted sobolev spaces of fractional order in r and on a compact manifold prove a version of sobolev s theorem and exhibit coercive weighted estimates for elliptic pseudodifferential operators

weighted norm inequalities off diagonal estimates and elliptic operators part iii harmonic analysis of elliptic operators this is the third part of a series of four articles on weighted norm in equalities o diagonal estimates and elliptic operators for l in some class of elliptic operators we study weighted norm l p inequalities for singular non integral opera tors arising from l those are the operators l for bounded holomorphic functions the riesz transforms rl 1 2 or 1 2 l 1 2 and its inverse l 1 2 1 2 some quadratic functionals gl and gl of littlewood paley stein type and also some vector valued inequalities such as the ones involved for maximal l p regularity for each we obtain sharp or nearly sharp ranges of p using the general theory for bound edness of part i and the o diagonal estimates of part ii we also obtain commutator results with bmo functions

necessary and sufficient conditions for absolute continuity of elliptic harmonic measure our purpose in this paper is to give a necessary and sufficient condition on the modulus of continuity of the coefficients of an elliptic operator in divergence form in order that the corresponding poisson kernel for the dirichlet problem exist this condition on the coefficients is global continuity together with the additional property that the modulus of continuity along some nontangential direction at each boundary point be bounded uniformly in these points and directions by a function 7 t satisfying the dini type condition

an explicit formula for the inverse of band triangular toeplitz matrix abstract in order to estimate the condition number of the preconditioned matrix proposed in f r lin w k ching inverse toeplitz preconditioners for hermitian toeplitz systems numer linear algebra appl 12 2005 221 229 we study the inverse of band triangular toeplitz matrix we derive an explicit formula for the entries of the inverse of band lower triangular toeplitz matrix by means of divided difference and use the formula to estimate the condition number of the preconditioned matrices in particular we prove that the minimal eigenvalue of preconditioned matrix is well separated from the origin

square roots of elliptic second order divergence operators on strongly lipschitz domains l p theory we study l p estimates for square roots of second order elliptic non necessarily selfadjoint operators in divergence form l mathrm div a nabla on lipschitz domains subject to dirichlet or to neumann boundary conditions pursuing our work 4 where we considered operators on mathbb r n we obtain among other things l 1 2 f p le c nabla f p for all 1 p infty if l is real symmetric and the domain bounded which is new for 1 p 2 we also obtain similar results for perturbations of constant coefficients operators our methods rely on a singular integral representation calderon zygmund theory and quadratic estimates a feature of this study is the use of a commutator between the resolvent of the laplacian dirichlet and neumann and partial derivatives which carries the geometry of the boundary

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

a note on the exponent set of primitive minimally strong digraphs abstract we obtain a lower bound for e n the least integer greater than or equal to 5 that is not the exponent of any nxn primitive nearly reducible matrix the main result is that under certain hypotheses about the distance between n and the nearest prime number we have that e n n2 3

the connections between dirichlet regularity and neumann problems for second order elliptic operators with complex bounded measurable coefficients abstract the present paper discusses relations between regularity dirichlet and neumann problems we investigate the boundary problems for block operators and prove in particular that the solvability of the regularity problem does not imply the solvability of the dual dirichlet problem for general elliptic operators with complex bounded measurable coefficients this is strikingly different from the case of real operators for which such an implication was established in 1993 by c kenig j pipher invent math 113 3 1993 447 509 and since then has served as an integral part of many results

lp self improvement of generalized poincare inequalities in spaces of homogeneous type abstract in this paper we study self improving properties in the scale of lebesgue spaces of generalized poincare inequalities in spaces of homogeneous type in contrast with the classical situation the oscillations involve approximation of the identities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization the techniques used do not involve any classical poincare or sobolev poincare inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown we apply our results to the case of riemannian manifolds with doubling volume form and assuming gaussian upper bounds for the heat kernel of the semigroup e t with being the laplace beltrami operator we obtain generalized poincare inequalities with oscillations that involve the semigroup e t and with right hand sides containing either or 1 2

on the exponent of a primitive minimally strong digraph abstract we consider n n primitive nearly reducible matrices for n 5 as defined by ross let e n be the least integer such that no such n n matrix has this integer as its exponent the investigation of e n is the first open problem in ross s paper here we offer a method to compute e n for small n then we generalize ross s estimate which is e n n 1 to e n p 1 n p for n 2p 1 and p 11 a prime less than 100 000 there are extant various estimates and conjectures concerning the difference of successive primes if one of the most hopeful of these conjectures be true then our lower bound for e n holds for all p 11 and in fact we obtain e n n 2 4 n 2 3 2

tent space boundedness via extrapolation we study the action of operators on tent spaces such as maximal operators calderon zygmund operators riesz potentials we also consider singular non integral operators we obtain boundedness as an application of extrapolation methods in the banach range in the non banach range boundedness results for calderon zygmund operators follow by using an appropriate atomic theory we end with some consequences on amalgam spaces

the atomic decomposition of weighted hardy spaces associated to self adjoint operators on product spaces abstract let l be a non negative self adjoint operator acting on l 2 r n satisfying a pointwise gaussian estimate for its heat kernel let w be an a p weight on r n r n 1 p in this article we obtain a weighted q atomic decomposition with q p for the weighted hardy space h l w 1 r n r n associated to l

the regularity problem for second order elliptic operators with complex valued bounded measurable coefficients the present paper establishes a certain duality between the dirichlet and regularity problems for elliptic operators with t independent complex bounded measurable coefficients t being the transversal direction to the boundary to be precise we show that the dirichlet boundary value problem is solvable in l p subject to the square function and non tangential maximal function estimates if and only if the corresponding regularity problem is solvable in l p moreover the solutions admit layer potential representations in particular we prove that for any elliptic operator with t independent real possibly non symmetric coefficients there exists a p 1 such that the regularity problem is well posed in l p

boundedness results for commutators with bmo functions via weighted estimates a comprehensive approach we present a unified method to obtain unweighted and weighted estimates of linear and multilinear commutators with bmo functions that is amenable to a plethora of operators and functional settings our approach elaborates on a commonly used cauchy integral trick recovering many known results but yielding also numerous new ones in particular we solve a problem about the boundedness of the commutators of the bilinear hilbert transform with functions in bmo