l p estimates for a singular integral operator motivated by calderxc3xb3n s second commutator abstract we prove a wide range of l p estimates for a trilinear singular integral operator motivated by dropping one average in calderon s second commutator for comparison by dropping two averages in calderon s second commutator one faces the trilinear hilbert transform the novelty in this paper is that in order to avoid difficulty of the level of the trilinear hilbert transform we choose to view the symbol of the operator as a non standard symbol the methods used come from time frequency analysis but must be adapted to the fact that our symbol is non standard

limited range multilinear extrapolation with applications to the bilinear hilbert transform we prove a limited range off diagonal extrapolation theorem that generalizes a number of results in the theory of rubio de francia extrapolation and use this to prove a limited range multilinear extrapolation theorem we give two applications of this result to the bilinear hilbert transform first we give sufficient conditions on a pair of weights w 1 w 2 for the bilinear hilbert transform to satisfy weighted norm inequalities of the form begin aligned bh l p 1 left w 1 p 1 right times l p 2 left w 2 p 2 right longrightarrow l p w p end aligned where w w 1w 2 and frac 1 p frac 1 p 1 frac 1 p 2 frac 3 2 this improves the recent results of culiuc et al by increasing the families of weights for which this inequality holds and by pushing the lower bound on p from 1 down to frac 2 3 the critical index from the unweighted theory of the bilinear hilbert transform second as an easy consequence of our method we obtain that the bilinear hilbert transform satisfies some vector valued inequalities with muckenhoupt weights this reproves and generalizes some of the vector valued estimates obtained by benea and muscalu in the unweighted case we also generalize recent results of carando et al on marcinkiewicz zygmund estimates for multilinear calderon zygmund operators

new maximal functions and multiple weights for the multilinear calderon zygmund theory a multi sub linear maximal operator that acts on the product of m lebesgue spaces and is smaller than the m fold product of the hardy littlewood maximal function is studied the operator is used to obtain a precise control on multilinear singular integral operators of calderon zygmund type and to build a theory of weights adapted to the multilinear setting a natural variant of the operator which is useful to control certain commutators of multilinear calderon zygmund operators with bmo functions is then considered the optimal range of strong type estimates a sharp end point estimate and weighted norm inequalities involving both the classical muckenhoupt weights and the new multilinear ones are also obtained for the commutators

multilinear calderon zygmund theory abstract a systematic treatment of multilinear calderon zygmund operators is presented the theory developed includes strong type and endpoint weak type estimates interpolation a multilinear t 1 theorem and a variety of results regarding multilinear multiplier operators

hardy spaces and a walsh model for bilinear cone operators the study of bilinear operators associated to a class of non smooth symbols can be reduced to ther study of certain special bilinear cone operators to which a time frequency analysis using smooth wave packets is performed in this paper we prove that when smooth wave packets are replaced by walsh wave packets the corresponding discrete walsh model for the cone operators is not only lp bounded as thiele has shown in his thesis for the walsh model corresponding to the bilinear hilbert transform but actually improves regularity as it maps into a hardy space the same result is expected to hold for the special bilinear cone operators

on a trilinear form related to the carleson theorem abstract demeter and thiele introduced the two dimensional bilinear hilbert transform in demeter and thiele 2010 3 and showed that the carleson operator can be identified in particular instances of the corresponding trilinear form demeter considered the walsh model of one such form in demeter 2012 2 in relation to the discussion of the walsh carleson theorem we prove boundedness of this trilinear form for a single triple of exponents at the boundary of the previously established range for this purpose we adapt the multilinear bellman function technique from kovac 2011 5 to the one and a half dimensional time frequency space

some remarks on bilinear littlewood paley theory abstract in this work some bilinear analogues of linear littlewood paley theory are explored paraproducts with functions decomposed at different scales are shown to be bounded on certain products of lebesgue spaces results concerning square functions associated with smooth and nonsmooth cutoffs are also obtained

reconstruction of singularities from full scattering data by new estimates of bilinear fourier multipliers we prove that the singularities of a complex valued potential q in the schrodinger hamiltonian q can be reconstructed from the linear born approximation for full scattering data by averaging in the extra variables we prove that with this procedure the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data in particular for q in w alpha 2 for 0 in 2d we recover the main singularity of q with an accuracy of one derivative in 3d the accuracy is epsilon 1 2 increasing with this gives a mathematical basis for diffraction tomography the proof is based on some new estimates for multidimensional bilinear fourier multipliers of independent interest

lp estimates for non smooth bilinear littlewood paley square functions on mathbb r in this work we study some non smooth bilinear analogues of linear littlewood paley square functions on the real line we prove boundedness properties in lebesgue spaces for them let us consider the functions phi n satisfying widehat phi n xi bf 1 n n 1 xi and define the bilinear operator s n f g x int f x y g x y phi n y dy these bilinear operators are closely related to the bilinear hilbert transforms then for exponents p q r in 2 infty satisfying frac 1 p frac 1 q frac 1 r we prove that left left sum n in mathbb z left s n f g right 2 right 1 2 right l r mathbb r lesssim f l p mathbb r g l q mathbb r

truncations of multilinear hankel operators we extend to multilinear hankel operators the fact that some truncations of bounded hankel operators are still bounded we prove and use a continuity property of bilinear hilbert transforms on products of lipschitz spaces and hardy spaces

a hormander type multiplier theorem for multilinear operators abstract in this paper we prove a hormander type multiplier theorem for multilinear operators as a corollary we can weaken the regularity assumption for multilinear fourier multipliers to assure the boundedness

on multilinear oscillatory integrals nonsingular and singular basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered l p norm inequalities are established for multilinear integral operators of calderon zygmund type which incorporate oscillatory factors e ip where p is a real valued polynomial

magnetoelectroelastic analysis of an arbitrary shape inclusion undergoing eigenfields and remote loadings abstract within the framework of the linear theory of magnetoelectroelasticity the problem of an arbitrary shape inclusion in an entire plane under remotely uniform in plane electromagnetic and anti plane mechanical loadings is investigated in this paper the inclusion is undergoing eigenfields and has different magnetoelectroelastic moduli as that of the matrix using complex variable and faber series method the complex potential for each component can be expressed in series form with unknown coefficients the continuity conditions of the interfaces are used to build up a set of linear equations to determine the unknown coefficients through solving these linear equations one can obtain the complex potentials both inside the inclusion and in the matrix numerical results are provided and shown that different combinations of piezoelectric piezomagnetic exhibit significant magnetoelectromechanical coupling effect that is not present in single phase piezoelectric or piezomagnetic material applying appropriate remote loadings and eigenfields can effectively reduce stress level and increase magnetoelectric response around the inclusion

multilinear singular integrals on non commutative l p spaces we prove l p bounds for the extensions of standard multilinear calderon zygmund operators to tuples of text umd spaces tied by a natural product structure the product can for instance mean the pointwise product in text umd function lattices or the composition of operators in the schatten von neumann subclass of the algebra of bounded operators on a hilbert space we do not require additional assumptions beyond text umd on each space in contrast to previous results we e g show that the rademacher maximal function property is not necessary the obtained generality allows for novel applications for instance we prove new versions of fractional leibniz rules via our results concerning the boundedness of multilinear singular integrals in non commutative l p spaces our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic probabilistic techniques in the text umd space setting

on the rate of decay of solutions of functional differential equations with infinite delay on etudie la vitesse de convergence vers zero des solutions d equations differentielles fonctionnelles a retard infini

estimates for trilinear flag paraproducts on l infty and hardy spaces the boundedness of trilinear flag paraproducts from products of hardy spaces h p times h q times h r to l s 1 p 1 q 1 r 1 s is discussed in the full range 0 p q r le infty the range of exponents to allow the boundedness is determined

asymptotic models of dilute composites with imperfectly bonded inclusions the asymptotic scheme for the analysis of dilute elastic composites which includes circular inclusions with imperfect bonding at the interface is presented interface is characterized by a discontinuous displacement field across it linearly related to the tractions the problem of a linear elastic circular inclusion with generic loading condition at infinity is solved and used to analyze effective elastic moduli of composite materials effects due to the interaction of a small circular defect and a crack are investigated it is shown that interfacial stiffness has a strong effect on the crack path and therefore may be an important design parameter for composites

wave packet frames on local fields of positive characteristic the objective of this paper is to construct wave packet frames on local fields of positive characteristic a necessary and sufficient condition for the wave packet system d p j t u n a e u m b j z m n n 0 to be a frame for l 2 k is given by means of the fourier transform

an embedding method for modeling micromechanical behavior and macroscopic properties of composite materials this paper presents a numerical method for modeling the micromechanical behavior and macroscopic properties of fiber reinforced composites and perforated materials the material is modeled by a finite rectangular domain containing multiple circular holes and elastic inclusions the rectangular domain is assumed to be embedded within a larger circular domain with fictitious boundary loading represented by truncated fourier series the analytical solution for the complementary problem of a circular domain containing holes and inclusions is obtained by using a combination of the series expansion technique with a direct boundary integral method the boundary conditions on the physical external boundary are satisfied by adopting an overspecification technique based on a least squares approximation all of the integrals arising in the method can be evaluated analytically as a result the elastic fields and effective properties can be expressed explicitly in terms of the coefficients in the series expansions several numerical experiments are conducted to verify the accuracy and efficiency of the numerical method and to demonstrate its application in determination of the macroscopic properties of composite materials

on calderon s conjecture with constants cfi p1 p2 depending only on fi p1 p2 and p p1p2 p1 p2 hold the flrst result of this type is proved in 4 and the purpose of the current paper is to extend the range of exponents p1 and p2 for which 2 is known in particular the case p1 2 p2 1 is solved to the a rmative this was

uniform estimates for bilinear hilbert transforms and bilinear maximal functions associated to polynomials we study the bilinear hilbert transforms and bilinear maximal functions associated to non flat polynomial curves and obtain uniform l r estimates for r d 1 over d where d is the degree of the polynomial in addition the lower bound of r is sharp up to the end point

extension of multilinear fractional integral operators to linear operators on mixed norm lebesgue spaces in kenig and stein math res lett 6 1 1 15 1999 https doi org 10 4310 mrl 1999 v6 n1 a1 the following type of multilinear fractional integral begin aligned int mathbb r mn frac f 1 l 1 x 1 ldots x m x cdots f m 1 l m 1 x 1 ldots x m x x 1 cdots x m lambda dx 1 ldots dx m end aligned was studied where l i are linear maps from mathbb r m 1 n to mathbb r n satisfying certain conditions they proved the boundedness of such multilinear fractional integral from l p 1 times cdots times l p m 1 to l q when the indices satisfy the homogeneity condition in this paper we show that the above multilinear fractional integral extends to a linear operator for functions in the mixed norm lebesgue space l mathbf p which contains l p 1 times cdots times l p m 1 as a subset under less restrictions on the linear maps l i we give a complete characterization of the indices mathbf p q and lambda for which such an operator is bounded from l mathbf p to l q and for m 1 or n 1 we give necessary and sufficient conditions on l 1 ldots l m 1 mathbf p p 1 ldots p m 1 q and lambda such that the operator is bounded

thermal conduction of a circular inclusion with variable interface parameter abstract an imperfect bonding problem associated with a solitary circular inclusion embedded in an infinite matrix under a remotely applied uniform intensity is considered specifically we study the effect of imperfect interfaces which are either of weakly or of highly conducting type and that the interface parameter could vary arbitrarily along the interface by using the orthogonality properties of the trigonometric series we show that the solution field is governed by a linear set of algebraic equations with an infinite number of unknowns the governing matrix for the unknowns is primarily composed of elements which are simple combinations of the fourier coefficients of the interface parameter solutions of the boundary value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of circular inclusions with equal size the effective properties solely depend on two particular constants among an infinite number of unknowns it is demonstrated that even for a composite with isotropic dispersions of inclusions the composite may become effectively anisotropic due to the presence of a variable interface parameter further we present two microstructure independent properties regarding the effective conductivity of the considered system we first show that the effective conductivity tensor for a composite with variably imperfect interfaces is always diagonally symmetric this is accomplished by means of a reciprocal relation that is established in such systems next we present dual relations for the effective conductivities of two dimensional composites with variably imperfect interfaces the latter result is a direct consequence of the existence of a dual relation for the local fields in such composites as pointed out by benveniste and miloh benveniste y miloh t 1999 j mech phys solids 47 1873 1892

transverse conductivity and longitudinal shear of elliptic fiber composite with imperfect interface abstract the paper addresses the problem of calculating the local fields and effective transport properties and longitudinal shear stiffness of elliptic fiber composite with imperfect interface the rayleigh type representative unit cell approach has been used the micro geometry of composite is modeled by a periodic structure with a unit cell containing multiple elliptic inclusions the developed method combines the superposition principle the technique of complex potentials and certain new results in the theory of special functions an appropriate choice of the potentials provides reducing the boundary value problem to an ordinary well posed set of linear algebraic equations the exact finite form expression of the effective stiffness tensor has been obtained by analytical averaging the local gradient and flux fields the convergence of solution has been verified and the parametric study of the model has been performed the obtained accurate statistically meaningful results illustrate a substantial effect of imperfect interface on the effective behavior of composite

bilinear integral operators with certain hypersingularities abstract in this paper we establish the sharp l p boundedness for the bilinear integral operators with certain hypersingularities that generalize the bilinear hilbert transform

a uniform estimate in this article we study bilinear operators whose input consists of two functions on the real line and whose output is again a function on the real line we shall be concerned with a priori estimates for these operators for simplicity we initially require the two input functions to be schwartz functions and allow the output function to be a tempered distribution by abstract generalities such a bilinear operation b can be formally written as

a circular inclusion with circumferentially inhomogeneous interface in antiplane shear a general method is developed for the rigorous solution of a problem associated with a circular inclusion embedded within an infinite matrix in antiplane shear the bonding at the inclusion matrix interface is assumed to be imperfect most significant is the fact that the imperfection in the interface is assumed to be circumferentially inhomogeneous using analytic continuation the basic boundary value problem for two analytic functions is reduced to a first order differential equation for a single analytic function and the closed form solution is obtained the method is illustrated using several specific examples the results from these examples are compared to the corresponding results when the imperfection in the interface is homogeneous these comparisons illustrate how the circumferential variation of the parameter describing the imperfection has a pronounced effect on the stresses induced within the inclusion

a sensitive interval of imperfect interface parameters based on the analysis of general solution for anisotropic matrix containing an elliptic inhomogeneity abstract a general solution is proposed to solve the plane problem for infinite anisotropic medium containing an elliptic inhomogeneity with imperfect interface the imperfect interface is usually described as a spring model with vanishing thickness based on the assumption that the tractions are continuous but the normal or tangential displacements are discontinuous due to a jump at the interface by means of the series expansion of the complex stress functions and the factor functions for the imperfect elliptic interface a general procedure to determine the coefficients in the series is illustrated and the convergent solutions are obtained by truncating finite number of terms in the series the present solutions are verified with available analytical results for the cases of perfect interface and debonded interface or hole the patterns of the stresses in the anisotropic medium or matrix and inhomogeneity due to the eigenstrains and far filed stresses are presented respectively a sensitive interval of interface parameters is suggested in which the influence of the change of interface parameters on the stress field is very obvious monotonic and non monotonic change of peak stresses on a subset of interface parameters is also discussed the method and the procedure proposed in this work can be used in the analysis of strength and failure of anisotropic materials containing an elliptic inhomogeneity with imperfect interface

two circular inclusions with inhomogeneously imperfect interfaces in plane elasticity abstract this paper is concerned with the problem of two circular inclusions with circumferentially inhomogeneously imperfect interfaces embedded in an infinite matrix in plane elastostatics infinite series form solutions to this problem are derived by applying complex variable techniques the numerical results demonstrate that the interface imperfection interface inhomogeneity and interaction among neighboring inclusions fibers will exert a significant influence on the stresses along the interfaces and average stresses within the inclusions

loosening of elastic inclusions abstract a numerical method is presented for simulating the occurrence of localized slip and separation along the interfaces of multiple randomly distributed circular elastic inclusions in an infinite elastic plane the method is an extension of a direct boundary integral approach previously described elsewhere for solving problems involving perfectly bonded circular inclusions here we allow displacement discontinuities to develop along the inclusion matrix interfaces in accordance with a linear mohr coulomb yield condition combined with a tensile strength cut off the displacements tractions and displacement discontinuities on the inclusion boundaries are all represented by truncated fourier series and an explicit iterative algorithm is adopted to determine zones of slip and separation under the prevailing loading conditions several examples are given to demonstrate the accuracy and generality of the approach

on the boundedness of the bilinear hilbert transform along non flat smooth curves we are proving l 2 bbb r times l 2 bbb r rightarrow l 1 bbb r bounds for the bilinear hilbert transform h gamma along curves gamma t gamma t with gamma being a smooth non flat curve near zero and infinity

a galerkin boundary integral method for multiple circular elastic inclusions with homogeneously imperfect interfaces a galerkin boundary integral method is presented to solve the problem of an infinite isotropic elastic plane containing a large number of randomly distributed circular elastic inclusions with homogeneously imperfect interfaces problems of interest might involve thousands of inclusions with no restrictions on their locations except that the inclusions may not overlap sizes and elastic properties the tractions are assumed to be continuous across the interfaces and proportional to the corresponding displacement discontinuities the analysis is based on a numerical solution of a complex hypersingular integral equation with the unknown tractions and displacement discontinuities at each circular boundary approximated by truncated complex fourier series the method allows one to calculate the stress and displacement fields everywhere in the matrix and inside the inclusions numerical examples are included to demonstrate the effectiveness of the approach

a class of bilinear multipliers given by littlewood paley decomposition abstract in this paper we study a particular class of bilinear multipliers which are given by littlewood paley decompositions in the first part of the paper we show that if is a bilinear multiplier for p q r 1 p q satisfying the holder s condition 1 p 1 q 1 r and have support inside 0 1 then its periodization j z j is also a bilinear multiplier for the same triplet p q r further we show that for a given triplet p q r of exponents outside the local l 2 range there exists sequence j j z of uniformly bounded bilinear multipliers so that the function j z j is not a bilinear multiplier for the triplet p q r in the second part we describe several results for bilinear multipliers of the type m which are similar to the first part in nature in particular we point out that the results described by p honzik 2014 13 can be generalized to a more general setting

multilinear interpolation between adjoint operators abstract multilinear interpolation is a powerful tool used in obtaining strong type boundedness for a variety of operators assuming only a finite set of restricted weak type estimates a typical situation occurs when one knows that a multilinear operator satisfies a weak l q estimate for a single index q which may be less than one and that all the adjoints of the multilinear operator are of similar nature and thus they also satisfy the same weak l q estimate under this assumption in this note we give a general multilinear interpolation theorem which allows one to obtain strong type boundedness for the operator and all of its adjoints for a large set of exponents the key point in the applications we discuss is that the interpolation theorem can handle the case q 1 when q 1 weak l q has a predual and such strong type boundedness can be easily obtained by duality and multilinear interpolation cf interpolation spaces an introduction springer new york 1976 math ann 319 2001 151 in function spaces and applications lund 1986 lecture notes in mathematics vol 1302 springer berlin new york 1988 j amer math soc 15 2002 469 proc amer math soc 21 1969 441

the bilinear hilbert transform in umd spaces we prove l p bounds for the bilinear hilbert transform acting on functions valued in intermediate umd spaces such bounds were previously unknown for umd spaces that are not banach lattices our proof relies on bounds on embeddings from bochner spaces l p mathbb r x into outer lebesgue spaces on the time frequency scale space mathbb r 3

discrete decompositions for bilinear operators and almost diagonal conditions this is the published version also available here http dx doi org 10 1090 s0002 9947 01 02912 9 first published in transaction of american mathematical society in 2002 published by the american mathematical society

a galerkin boundary integral method for multiple circular elastic inclusions the problem of an infinite isotropic elastic plane containing an arbitrary number of circular elastic inclusions is considered the analysis procedure is based on the use of a complex singular integral equation the unknown tractions at each circular boundary are approximated by a truncated complex fourier series a system of linear algebraic equations is obtained by using the classical galerkin method and the gauss seidel algorithm is used to solve the system several numerical examples are considered to demonstrate the effectiveness of the approach copyright 2001 john wiley sons ltd

calderon zygmund operators with non doubling measures in this survey the authors review some basic results on calderon zygmund theory on r d with the non doubling radon measure instead of the usual doubling condition the measure is only assumed to satisfy some growth condition that is there exist positive con stants n 0 d a ndc such that b x r cr n for all x r d and r 0 in this setting the results that we review are the following the hardy space h 1 and the regularized bmo space rbmo some interpolation results related to h 1 a nd rbmo the calderon zygmund decomposition the t 1 theorem and the properties of the calderon zygmund op erator on lebesgue spaces and hardy spaces cotlar s inequality and the boundedness of the maximal calderon zygmund operator the behaviors of the multilinear calderon zygmund op erator on lebesgue spaces weighted norm inequalities for the calderon zygmund operator the boundedness of the commutator generated by the calderon zygmund operator and rbmo function moreover some new progress obtained recently is also included

the quartile operator and pointwise convergence of walsh series the bilinear hilbert transform is given by h f g x p v f x t g x t t it satisfies estimates of the type h f g r c s t f s g t in this paper we prove such estimates for a discrete model of the bilinear hilbert transform involving the walsh fourier transform we also reprove the well known fact that the walsh fourier series of a function in lp 0 1 with 1 p converges pointwise almost everywhere the purpose of this exposition is to clarify the connection between these two results and to present an easy approach to recent methods of time frequency analysis

a galerkin boundary integral method for multiple circular elastic inclusions with uniform interphase layers abstract the paper presents a numerical method for solving the problem of an infinite isotropic elastic plane containing a large number of randomly distributed circular elastic inclusions with uniform interphase layers the bonds between the inclusions and the interphases as well as between the interphases and the matrix are assumed to be perfect in general the inclusions may have different elastic properties and sizes the thicknesses of the interphases and their elastic properties are arbitrary the analysis is based on a numerical solution of a complex singular integral equation with the unknown tractions at each circular boundary approximated by a truncated complex fourier series the galerkin technique is used to obtain a system of linear algebraic equations the resulting numerical method allows one to calculate the elastic fields everywhere in the matrix and inside the inclusions and the interphases using the assumption of macro isotropic behavior in a plane section one can find the effective elastic moduli for an equivalent homogeneous material the method can be viewed as an extension of our previous work int j solids struct 39 2002 4723 where simpler spring like interface conditions were modeled the problem of overlapping of the fibers and matrix inherent to spring type interface is discussed in the context of the present model numerical examples are included to demonstrate the effectiveness of the new approach

multilinear weighted convolution of l 2 functions and applications to nonlinear dispersive equations the x s b spaces as used by beals bourgain kenig ponce vega klainerman machedon and others are fundamental tools to study the low regularity behavior of nonlinear dispersive equations it is of particular interest to obtain bilinear or multilinear estimates involving these spaces by plancherel s theorem and duality these estimates reduce to estimating a weighted convolution integral in terms of the l 2 norms of the component functions in this paper we systematically study weighted convolution estimates on l 2 as a consequence we obtain sharp bilinear estimates for the kdv wave and schrodinger x s b spaces

iterated trilinear fourier integrals with arbitrary symbols abstract we prove l p estimates for trilinear multiplier operators with singular symbols these operators arise in the study of iterated trilinear fourier integrals which are trilinear variants of the bilinear hilbert transform specifically we consider trilinear operators determined by multipliers that are products of two functions m 1 1 2 and m 2 2 3 such that the singular set of m 1 lies in the hyperplane 1 2 and that of m 2 lies in the hyperplane 2 3 while previous work 8 requires that the multipliers satisfy 1 2 2 3 our results allow for the case of the arbitrary multipliers which have common singularities

improved bound for the bilinear bochner riesz operator we study l p times l q rightarrow l r bounds for the bilinear bochner riesz operator mathcal b alpha alpha 0 in mathbb r d d ge 2 which is defined by open image in new window we make use of a decomposition which relates the estimates for mathcal b alpha to the square function estimates for the classical bochner riesz operators in consequence we significantly improve the previously known bounds

on calderon s conjecture for the bilinear hilbert transform abstract we show that the bilinear hilbert transform defined by maps lp lq into lr for 1 p q 1 p 1 q 1 r and 2 3 r

generalizations of the carleson hunt theorem i the classical singularity case in this article we prove l p estimates for a general maximal operator which extend both the classical coifman meyer and carleson hunt theorems in harmonic analysis

on the use of somigliana s formula and fourier series for elasticity problems with circular boundaries this paper considers the problem of an infinite isotropic elastic plane containing an arbitrary number of non overlapping circular holes and isotropic elastic inclusions the holes and inclusions are of arbitrary size and the elastic properties of all of the inclusions can if desired be different the analysis is based on the two dimensional version of somigliana s formula which gives the displacements at a point inside a region v in terms of integrals of the tractions and displacements over the boundary s of this region we take v to be the infinite plane and s to be an arbitrary number of circular holes within this plane any or all of the holes can contain an elastic inclusion and we assume for simplicity that all inclusions are perfectly bonded to the material matrix the displacements and tractions on each circular boundary are represented as truncated fourier series and all of the integrals involved in somigliana s formula are evaluated analytically an iterative solution algorithm is used to solve the resulting system of linear algebraic equations several examples are given to demonstrate the accuracy and efficiency of the numerical method copyright 2003 john wiley sons ltd

the bilinear maximal functions map into l p for 2 3 p 1 the bilinear maximal operator defined below maps lp x lq into lr provided 1 0 2t t in particular mfg is integrableif f and g are square integrable answering a conjecture posed by alberto calder6n

one modification of the martingale transform and its applications to paraproducts and stochastic integrals abstract in this paper we introduce a variant of burkholder s martingale transform associated with two martingales with respect to different filtrations even though the classical martingale techniques cannot be applied we show that the discussed transformation still satisfies some expected l p estimates then we apply the obtained inequalities to general dilation twisted paraproducts particular instances of which have already appeared in the literature as another application we construct stochastic integrals 0 t h s d x s y s associated with certain continuous time martingales x t t 0 and y t t 0 the process x t y t t 0 is shown to be a good integrator although it is not necessarily a semimartingale or even adapted to any convenient filtration

on boundedness of discrete multilinear singular integral operators let m be a measurable locally bounded function defined in r2 let 1 p1 q1 p2 q2 such that pi 1 implies qi let also 0 p3 q3 and 1 p 1 p1 1 p2 1 p3 we prove the following transference result the operator cm f g x r rf g m e2 ix d d initially defined for integrable functions with compact fourier support extends to a bounded bilinear operator from lp1 q1 r lp2 q2 r into lp3 q3 r if and only if the family of operators dm t p a b n t1p 1212 1212p q m t t e2 in d d initially defined for finite sequences a ak1 k1 z b bk2 k2 z where p k1 zak1e 2 ik1 and q k2 zbk2e 2 ik2 extend to bounded bilinear operators from lp1 q1 z lp2 q2 z into lp3 q3 z with norm bounded by uniform constant for all t 0 we apply this result to prove boundedness of the discrete bilinear hilbert transforms and other related discrete multilinear singular integrals including the endpoints

variational estimates for the bilinear iterated fourier integral abstract we prove pointwise variational l p bounds for a bilinear fourier integral operator in a large but not necessarily sharp range of exponents this result is a joint strengthening of the corresponding bounds for the classical carleson operator the bilinear hilbert transform the variation norm carleson operator and the bi carleson operator terry lyon s rough path theory allows for extension of our result to multilinear estimates we consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations

an arbitrary piezoelectric inclusion with weakly and highly conducting imperfect interface abstract in the current study we rigorously analyze an arbitrarily shaped piezoelectric inclusion surrounded by an infinite isotropic piezoelectric matrix subject to antiplane shear and in plane electric field loadings the inclusion and matrix are separated by a homogeneously imperfect interface that characterizes a spring type interaction between the elastic and electric interfacial boundary conditions furthermore the boundary conditions for a mechanically compliant weakly conducting and mechanically compliant highly conducting interface are incorporated into the analysis using complex variable techniques the potential function inside the inclusion is formulated as a faber series expansion and a system of linear algebraic equations for a closed form solution is developed for the corresponding faber coefficients under a finite number of terms under this approach expressions for both the elastic and electric fields are developed for the inclusion and matrix the results are presented in exact form for an elliptic inclusion and numerically simulated for a finite number of terms for purposes of verification additionally the cases of a square and star inclusion geometry are analyzed and results are presented numerically the results clearly demonstrate that not only is the stress distribution inside the inclusion interface non uniform but that the magnitude of the peak stresses are highly dependent on the inclusion shape and imperfect interface condition

rough bilinear singular integrals abstract we study the rough bilinear singular integral introduced by coifman and meyer 8 t f g x p v r n r n y z 2 n y z y z f x y g x z d y d z when is a function in l q s 2 n 1 with vanishing integral and 2 q when q we obtain boundedness for t from l p 1 r n l p 2 r n to l p r n when 1 p 1 p 2 and 1 p 1 p 1 1 p 2 for q 2 we obtain that t is bounded from l 2 r n l 2 r n to l 1 r n for q between 2 and infinity we obtain the analogous boundedness on a set of indices around the point 1 2 1 2 1 to obtain our results we introduce a new bilinear technique based on tensor type wavelet decompositions

new estimates of the maximal bochner riesz operator in the plane we prove new l p estimates with 1 p 2 for the maximal bochner riesz operator in the plane

bilinear interpolation theorems and applications abstract interpolation of bilinear operators is studied the celebrated littlewood mixed norm inequality is used to prove interpolation theorems for bilinear operators defined on couples of c 0 weighted sequence spaces generated by parameters of quasi concave functions it is shown that these results can be lifted to a wider class of abstract methods of interpolation the abstract results are used to prove bilinear interpolation theorems to the setting of calderon lozanovskii spaces as an application the boundedness of the bilinear hilbert transform between orlicz spaces including zygmund classes is proved

on the asymptotic stability of solutions of volterra integro differential equations nonlinear differential delay equations are investigated by means of their associated semigroups conditions are found for which solutions have nonexponential decay but nevertheless behave asymptotically as an inverse power of t

bilinear oscillatory integrals and boundedness for new bilinear multipliers abstract we consider bilinear oscillatory integrals i e pseudo product operators whose symbol involves an oscillating factor lebesgue space inequalities are established which give decay as the oscillation becomes stronger this extends the well known linear theory of oscillatory integral in some directions the proof relies on a combination of time frequency analysis of coifman meyer type with stationary and non stationary phase estimates as a consequence of this analysis we obtain lebesgue estimates for new bilinear multipliers defined by non smooth symbols

the disc as a bilinear multiplier a classical theorem of c fefferman says that the characteristic function of the unit disc is not a fourier multiplier on l p r 2 unless p 2 in this article we obtain a result that brings a contrast with the previous theorem we show that the characteristic function of the unit disc in r 2 is the fourier multiplier of a bounded bilinear operator from l p 1 r l p 2 r into l p r when 2 p 1 p 2 and inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i the proof of this result is based on a new decomposition of the unit disc and delicate orthogonality and combinatorial arguments this result implies norm convergence of bilinear fourier series and strengthens the uniform boundedness of the bilinear hilbert transforms as it yields uniform vector valued bounds for families of bilinear hilbert transforms

l p estimates for the biest ii the fourier case we prove l p estimates theorem 1 2 for the biest a trilinear multiplier operator with singular symbol the methods used are based on the treatment of the walsh analogue of the biest in the prequel 13 of this paper but with additional technicalities due to the fact that in the fourier model one cannot obtain perfect localization in both space and frequency

on the boundedness of bilinear operators on products of besov and lebesgue spaces abstract we prove mapping properties of the form t b p 1 1 q 1 l p 2 b p 3 2 q 2 and t b p 1 1 q 1 b p 2 2 q 2 l p 3 for certain related indices p 1 p 2 p 3 q 1 q 2 1 2 r where t is a bilinear hormander mihlin multiplier or a molecular paraproduct applications to bilinear littlewood paley theory are discussed

on a biparameter maximal multilinear operator abstract it is well known that estimates for maximal operators and questions of pointwise convergence are strongly connected in recent years convergence properties of so called non conventional ergodic averages have been studied by a number of authors including assani austin host kra tao etc in particular much is known regarding convergence in l 2 of these averages but little is known about pointwise convergence in this spirit we consider the pointwise convergence of a particular ergodic average and study the corresponding maximal trilinear operator over r thanks to a transference principle lacey in 15 and demeter tao and thiele in 6 have studied maximal multilinear operators previously however the maximal operator we develop has a novel biparameter structure which has not been previously encountered and cannot be estimated using their techniques we will carve this biparameter maximal multilinear operator using a certain taylor series and produce non trivial holder type estimates for one of the two main terms by treating it as a singular integral the symbol of which has singular set given by two intersecting planes similarly to that of the biest operator studied by muscalu tao and thiele in 24 and 25

the hormander multiplier theorem for multilinear operators in this paper we provide a version of the mikhlin hormander multiplier theorem for multilinear operators in the case where the target space is l p for pe 1 this extends a recent result of tomita 15 who proved an analogous result for p 1

an elliptic inclusion with imperfect interface in anti plane shear abstract a semi analytic solution is developed for the problem associated with an elliptic inclusion embedded within an infinite matrix in anti plane shear the bonding at the inclusion matrix interface is assumed to be homogeneously imperfect the interface is modeled as a spring interphase layer with vanishing thickness the behaviour of this interphase layer is based on the assumption that tractions are continuous but displacements are discontinuous across the interface complex variable techniques are used to obtain infinite series representations of the stresses induced within the inclusion the results obtained demonstrate how the non uniform stress field and the average stresses inside the inclusion vary with the aspect ratio of the inclusion and the parameter describing the imperfect interface in addition it is shown that in some cases depending on the aspect ratio of the ellipse it is possible to identify specific values of the interface parameter which correspond to maximum peak stress along the interface

endpoint bounds for the bilinear hilbert transform we study the behavior of the bilinear hilbert transform bht at the boundary of the known boundedness regionh a sample of our results is the estimate bht f1 f2 f3 c f1 3 4 f2 3 4 f3 1 2 log log ee f3 min f1 f2 valid for all tuples of sets f j r of finite measure and functions f j such that f j 1f j j 1 2 3 with the additional restriction that f3 be supported on a major subset f 3 of f3 that depends on f j j 1 2 3 the use of subindicator functions in this fashion is standard in the given context see 25 the double logarithmic term improves over the single logarithmic term obtained in 1 whether the double logarithmic term can be removed entirely as is the case for the quartile operator discussed in 7 remains open we employ our endpoint results to describe the blow up rate of weak type and strong type estimates for bht as the tuple approaches the boundary ofh we also discuss bounds on lorentz orlicz spaces near l 2 3 improving on results of 5 the main technical novelty in our article is an enhanced version of the multi frequency calderon zygmund decomposition of 26

multilinear operators with non smooth kernels and commutators of singular integrals we obtain endpoint estimates for multilinear singular integral operators whose kernels satisfy regularity conditions significantly weaker than those of the standard calder6n zygmund kernels as a consequence we deduce endpoint l 1 x l 1 to weak l 1 m estimates for the mth order commutator of calderon our results reproduce known estimates for m 1 2 but are new for m 3 we also explore connections between the 2nd order higher dimensional commutator and the bilinear hilbert transform and deduce some new off diagonal estimates for the former

finite plane deformations of a three phase circular inhomogeneity matrix system abstract we consider finite plane deformations of a three phase circular inhomogeneity matrix system in which the inhomogeneity the interphase layer and the matrix belong to the same class of compressible hyperelastic materials of harmonic type but with each phase possessing its own distinct material properties we obtain the complete solution when the system is subjected to general classes of remote piola stress specifically remote stress distributions characterized by stress functions described by general polynomials of order n 1 in the corresponding complex variable z used to describe the matrix as a particular case of the aforementioned analysis we establish an eshelby type result namely that for this class of harmonic materials a three phase circular inhomogeneity under uniform remote stress and eigenstrain admits an internal uniform stress field when subjected to plane deformations

extrapolation for multilinear muckenhoupt classes and applications abstract in this paper we solve a long standing problem about the multivariable rubio de francia extrapolation theorem for the multilinear muckenhoupt classes a p which were extensively studied by lerner et al and which are the natural ones for the class of multilinear calderon zygmund operators as well as for some bilinear rough singular integral operators furthermore we go beyond the classes a p and extrapolate within the classes a p r which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear hilbert transform we give several applications which can be easily obtained using extrapolation first weighted norm inequalities scalar and vector valued for the bilinear rough singular integral operators are established second for the bilinear hilbert transform one can extrapolate from the recent result of culiuc et al who considered the banach range extend the estimates to the quasi banach range and furthermore prove trivially vector valued inequalities we also extend recent results of carando et al on marcinkiewicz zygmund estimates for multilinear calderon zygmund operators finally our last application gives new weighted estimates scalar and vector valued for the commutators of multilinear calderon zygmund operators bilinear rough singular integral operators and for the bilinear hilbert transform with bmo functions using ideas from benyi et al

the multilinear marcinkiewicz interpolation theorem revisited the behavior of the constant abstract we provide a self contained proof of the multilinear extension of the marcinkiewicz real method interpolation theorem with initial assumptions a set of restricted weak type estimates considering possible degenerate situations that may arise the advantage of this proof is that it yields a logarithmically convex bound for the norm of the operator on the intermediate spaces in terms of the initial restricted weak type bounds it also provides an explicit estimate in terms of the exponents of the initial estimates the constant blows up like a negative power of the distance from the intermediate point to the boundary of the convex hull of the initial points

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

maximal estimates for the bilinear spherical averages and the bilinear bochner riesz operators abstract we study the maximal estimates for the bilinear spherical average and the bilinear bochner riesz operator first we obtain l p l q l r estimates for the bilinear spherical maximal function on the optimal range thus we settle the problem which was previously considered by geba greenleaf iosevich palsson and sawyer later barrionevo grafakos d he honzik and oliveira and recently heo hong and yang secondly we consider l p l q l r estimates for the maximal bilinear bochner riesz operators and improve the previous known results for the purpose we draw a connection between the maximal estimates and the square function estimates for the classical bochner riesz operators

lp estimates for the bilinear hilbert transform for the bilinear hilbert transform given by we announce the inequality h fg p3 kp1 p2 f p1 g p2 provided 2 p1 p2 1 p3 1 p1 1 p2 and 1 p3 2

on the two dimensional bilinear hilbert transform we investigate the bilinear hilbert transform in the plane and the pointwise convergence of bilinear averages in ergodic theory arising from bf z 2 actions our techniques combine novel one and a half dimensional phase space analysis with more standard one dimensional theory