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

an influence of unilateral sources and sinks in reaction diffusion systems exhibiting turing s instability on bifurcation and pattern formation abstract we consider a general reaction diffusion system exhibiting turing s diffusion driven instability in the first part of the paper we supplement the activator equation by unilateral integral sources and sinks of the type k u x k d k and k u x k d k these terms measure an average of the concentration over the set k and are active only when this average decreases below or increases above the value of the reference spatially homogeneous steady state which is shifted to the origin we show that the set of diffusion parameters in which spatially heterogeneous stationary solutions can bifurcate from the reference state is smaller than in the classical case without any unilateral integral terms this problem is studied for the case of mixed pure neumann and periodic boundary conditions in the second part of the paper we investigate the effect of both unilateral terms of the type u u and unilateral integral terms on the pattern formation using numerical experiments on the system with well known schnakenberg kinetics

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

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

unilateral sources and sinks of an activator in reaction diffusion systems exhibiting diffusion driven instability abstract a reaction diffusion system exhibiting turing s diffusion driven instability is considered the equation for an activator is supplemented by unilateral terms of the type s x u s x u describing sources and sinks active only if the concentration decreases below and increases above respectively the value of the basic spatially constant solution which is shifted to zero we show that the domain of diffusion parameters in which spatially non homogeneous stationary solutions can bifurcate from that constant solution is smaller than in the classical case without unilateral terms it is a dual information to previous results stating that analogous terms in the equation for an inhibitor imply the existence of bifurcation points even in diffusion parameters for which bifurcation is excluded without unilateral sources the case of mixed dirichlet neumann boundary conditions as well as that of pure neumann conditions is described

multiple weighted estimates for commutators of multilinear singular integrals with non smooth kernels abstract multilinear commutators and iterated commutators generated by the multilinear singular integrals with non smooth kernels and bmo functions are studied by the weighted estimates of a class of new variant maximal operators and the sharp maximal functions the multiple weighted norm inequalities for such operators are obtained in particular some previous results in anh and duong 2010 1 lerner et al 2009 4 perez et al 2011 5 are improved or extended significantly

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

bifurcation for a reaction diffusion system with unilateral and neumann boundary conditions abstract we consider a reaction diffusion system of activator inhibitor or substrate depletion type which is subject to diffusion driven instability if supplemented by pure neumann boundary conditions we show by a degree theoretic approach that an obstacle e g a unilateral membrane modeled in terms of inequalities introduces new bifurcation of spatial patterns in a parameter domain where the trivial solution of the problem without the obstacle is stable moreover this parameter domain is rather different from the known case when also dirichlet conditions are assumed in particular bifurcation arises for fast diffusion of activator and slow diffusion of inhibitor which is the difference from all situations which we know

on multilinear square function and its applications to multilinear littlewood paley operators with non convolution type kernels abstract let m 2 n 1 and x r n define the multilinear square function t by t f x 0 r n m k t x y 1 y m j 1 m f j y j d y 1 d y m 2 d t t 1 2 where the kernel k satisfies a class of integral smooth conditions which is much weaker than the standard calderon zygmund kernel conditions in this paper we first established the l p 1 w 1 l p m w m l p estimate of t when each p i 1 and weak type l p 1 w 1 l p m w m l p estimate of t when there is a p i 1 where i 1 m i p p i and each w i is a nonnegative function on r n as applications of the above results we obtained the boundedness of multilinear littlewood paley operators with non convolution type kernels including multilinear g function marcinkiewicz integral and g function

a solution of the parabolized navier stokes stability model in discrete space by two directional differential quadrature and application to swirl intense flows in this paper we propose a two directional differential quadrature discretization of the parabolized navier stokes differential system which models the hydrodynamic instability of non parallel swirling flows a specific operational differentiation matrix was derived based on the properties of the orthogonal base numerical experiments comparing the new approach with the standard collocation methods were also carried out for two models the trailing vortex model and a laboratory controlled swirl intense flow model numerical simulations are included that show good robustness properties of the proposed method the method proposed in this paper offers two major advantages affords an extended analysis of complex swirling flow that captures the streamwise development and the upstream extent of the instability and provides a reduced order model of the tollmien schlichting modes used for the global stability analysis the computational storage is significantly decreased compared with the related application of linear radial collocation approach

a two level method in time and space for solving the navier stokes equations based on newton iteration a two level method in time and space for solving the time dependent navier stokes equations based on newton iteration is considered in this paper considering that the small eddy components carry little part of the whole energy we use a larger time step size to obtain the small eddy component as compared with the large eddy components numerical analysis shows that it is possible to choose a proper time step size without compromising the overall accuracy moreover some numerical examples are provided to complement our theoretical analysis

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

pod deim nonlinear model order reduction of an adi implicit shallow water equations model in the present paper we consider a 2 d shallow water equations swe model on a b plane solved using an alternating direction fully implicit adi finite difference scheme on a rectangular domain the scheme was shown to be unconditionally stable for the linearized equations the discretization yields a number of nonlinear systems of algebraic equations we then use a proper orthogonal decomposition pod to reduce the dimension of the swe model due to the model nonlinearities the computational complexity of the reduced model still depends on the number of variables of the full shallow water equations model by employing the discrete empirical interpolation method deim we reduce the computational complexity of the reduced order model due to its depending on the nonlinear full dimension model and regain the full model reduction expected from the pod model to emphasize the cpu gain in performance due to use of pod deim we also propose testing an explicit euler finite difference scheme ee as an alternative to the adi implicit scheme for solving the swallow water equations model we then proceed to assess the efficiency of pod deim as a function of number of spatial discretization points time steps and pod basis functions as was expected our numerical experiments showed that the cpu time performances of pod deim schemes are proportional to the number of mesh points once the number of spatial discretization points exceeded 10000 and for 90 deim interpolation points the cpu time decreased by a factor of 10 in case of pod deim implicit swe scheme and by a factor of 15 for the pod deim explicit swe scheme in comparison with the corresponding pod swe schemes moreover our numerical tests revealed that if the number of points selected by deim algorithm reached 50 the approximation errors due to pod deim and pod reduced systems have the same orders of magnitude thus supporting the theoretical results existing in the literature

polynomial growth estimates for multilinear singular integral operators on demontre un theoreme pour des mesures de carleson et on l applique a l operateur de kato en dimension 1 on demontre un critere de limitation general on donne une application aux commutateurs de coolderon on etudie les d commutateurs

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

spectral collocation methods and polar coordinate singularities this paper considers the numerical solution of elliptic differential equations on the unit disk using polar coordinates the disk is mapped onto a rectangle the resulting transformed problem is solved by a method related to collocation since the origin is a coordinate singularity some natural trial functions are singular there and a special technique is applied to use zero as a collocation point for poisson and helmholtz equations a fast algorithm with an operation count of o n2 log n is presented numerical results show the different stability and convergence properties of the algorithms

compactness properties of commutators of bilinear fractional integrals commutators of a large class of bilinear operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be jointly compact under a similar commutation fractional integral versions of the bilinear hilbert transform yield separately compact operators

eigenvalues and bifurcation for problems with positively homogeneous operators and reaction diffusion systems with unilateral terms abstract reaction diffusion systems satisfying assumptions guaranteeing turing s instability and supplemented by unilateral terms of type v and v are studied existence of critical points and sometimes also bifurcation of stationary spatially non homogeneous solutions are proved for rates of diffusions for which it is excluded without any unilateral term the main tool is a general result giving a variational characterization of the largest eigenvalue for positively homogeneous operators in a hilbert space satisfying a condition related to potentiality and existence of bifurcation for equations with such operators the originally non variational non symmetric system is reduced to a single equation with a positively homogeneous potential operator and the abstract results mentioned are used

the structure of the reverse holder classes in this paper we study the structure of the class of functions rhs which satisfy the reverse holder inequality with exponent s i to do so we introduce a new operator the minimal operator which is analogous to the hardy littlewood maximal operator and a new class of functions rhcc which plays the same role for rhs that the class ax does for the ap classes

multilinear theory of strongly singular calderon zygmund operators and applications abstract the main purpose of this paper is to study the multilinear strongly singular calderon zygmund operator whose kernel is more singular near the diagonal than that of the standard multilinear calderon zygmund operator the sharp maximal estimate for this class of multilinear singular integrals is established and as applications its boundedness on product of weighted lebesgue spaces and product of variable exponent lebesgue spaces is obtained respectively moreover the endpoint estimates of the types l l b m o b m o b m o b m o and l m o l m o l m o are established for the multilinear strongly singular calderon zygmund operator these results improve the corresponding known ones for the standard multilinear calderon zygmund operator furthermore we remove the size condition assumption for the kernel of the multilinear strongly singular calderon zygmund operator which has been imposed in the literature for the case of the standard multilinear calderon zygmund operator to get the sharp maximal estimates see remarks 1 1 1 4 extra care is needed to deal with the mean oscillation over balls with small radius to overcome the stronger singularity in establishing such sharp maximal estimates and endpoint estimates

quantitative estimates and extrapolation for multilinear weight classes in this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite this extends the recent extrapolation result of li martell and ombrosi we also obtain vector valued estimates including ell infty spaces and in particular we are able to reprove all the vector valued bounds for the bilinear hilbert transform obtained through the helicoidal method of benea and muscalu moreover our result is quantitative and in particular allows us to extend quantitative estimates obtained from sparse domination in the banach space setting to the quasi banach space setting our proof does not rely on any off diagonal extrapolation results and we develop a multilinear version of the rubio de francia algorithm adapted to the multisublinear hardy littlewood maximal operator as a corollary we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak type and mathrm bmo spaces

bifurcation points and eigenvalues of inequalities of reaction diffusion type on demontre certaines assertions concernant les points de bifurcation et les valeurs propres de systemes de reaction diffusion avec des conditions unilaterales

multiparameter riesz commutators it is shown that product bmo of s y a chang and r fefferman defined on the space bbb r d 1 otimes cdots otimes bbb r d t can be characterized by the multiparameter commutators of riesz transforms this extends a classical one parameter result of r coifman r rochberg and g weiss and at the same time extends the work of m lacey and s ferguson and m lacey and e terwilleger on multiparameter commutators with hilbert transforms

exponential decay estimates for singular integral operators the following subexponential estimate for commutators is proved begin aligned x in q b t f x tm 2f x le c e sqrt alpha t vert b vert bmo q qquad t 0 end aligned where c and alpha are absolute constants t is a calderon zygmund operator m is the hardy littlewood maximal function and f is any function supported on the cube q subset mathbb r n we also obtain that begin aligned x in q f x m f q tm lambda n q f x le c e alpha t q qquad t 0 end aligned where m f q is the median value of f on the cube q and m lambda n q is stromberg s local sharp maximal function with lambda n 2 n 2 as a consequence we derive karagulyan s estimate begin aligned x in q tf x tmf x le c e c t q qquad t 0 end aligned from 21 improving buckley s theorem 3 a completely different approach is used based on a combination of lerner s formula with some special weighted estimates of coifman fefferman type obtained via rubio de francia s algorithm the method is flexible enough to derive similar estimates for other operators such as multilinear calderon zygmund operators dyadic and continuous square functions and vector valued extensions of both maximal functions and calderon zygmund operators in each case m will be replaced by a suitable maximal operator

a dominated ergodic theorem for some bilinear averages abstract let t be a positive invertible linear operator with positive inverse on some l p 1 p where is a finite measure we study the convergence in the l p norm and the almost everywhere convergence of the bilinear operators a n f 1 f 2 1 2 n 1 i n n t i f 1 x 1 2 n 1 i n n t i f 2 x for functions f 1 l p 1 and f 2 l p 2 1 p 1 p 1 p 2 1 p 1 1 p 2 1 p it turns out to be that the convergence in l p is equivalent to the dominated estimate for the ergodic maximal operator associated to a n and to the uniform boundedness of the operators a n it is also shown that the convergence in the l p norm implies the almost everywhere convergence on the one hand the key facts to prove these results are transference arguments and the connection with a new class of weights recently introduced by lerner et al 2009 4 on the other hand our main result can be viewed as the ergodic counterpart of one of the main results in the above cited paper

on commutators of vector bmo functions and multilinear singular integrals with non smooth kernels abstract let t be a bounded multilinear operator on some product of l q r n spaces assume that t has a non smooth associated kernel which satisfies certain weak regularity conditions but not regular enough to fall under the scope of the standard multilinear calderon zygmund theory the main aim of this paper is to establish a sufficient condition on the kernel of t so that the commutator of a vector bmo function b and t is bounded on certain product l p r n spaces we obtain boundedness of the commutator of b and t by first proving certain pointwise estimates on the fefferman stein sharp maximal operator an important example of multilinear operators which satisfy our kernel conditions is the maximal m th order calderon commutator

weighted bilinear hardy inequalities abstract we characterize the weights w w 1 w 2 such that the weighted bilinear hardy inequality a b a x f q a x g q w x d x 1 q c a b f p 1 w 1 1 p 1 a b g p 2 w 2 1 p 2 holds for all nonnegative functions f and g with a positive constant c independent of f and g for all possible values of q p 1 and p 2 with 1 q p 1 p 2 we also characterize the good weights for the weighted bilinear n dimensional hardy inequality to hold

weak and strong type weighted estimates for multilinear calderon zygmund operators abstract in this paper we study the weighted estimates for multilinear calderon zygmund operators from l p 1 w 1 l p m w m to l p v w where 1 p p 1 p m with 1 p 1 1 p m 1 p and w w 1 w m is a multiple a p weight we give weak and strong type weighted estimates of mixed a p a type moreover the strong type weighted estimate is sharp whenever max i p i p m p 1

a counterexample to weighted estimates for multilinear fourier multipliers with sobolev regularity abstract the problem whether the weighted estimates for multilinear fourier multipliers with sobolev regularity hold under weak condition on weights is considered

reaction diffusion systems destabilizing effect of unilateral conditions on considere un systeme de reaction diffusion dans un domaine r n soit v une solution stationnaire et spatialement homogene avec les conditions aux limites de neumann on etudie la stabilite linearisee de v

interior estimates in morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients abstract let us consider the nondivergence form elliptic equation a ij u x i x j in this paper we show that if the known term belongs to the morrey space l p then the second derivatives of the w 2 p solution u belong to the same space next we derive a c 1 loc regularity result that is related to a recent work by caffarelli

on some pointwise inequalities in a recent paper studia math 138 2000 285 291 we proved pointwise estimates relating some classical maximal and singular integral operators here we show that in a sense there are more flexible inequalities which not only imply the previously known results but also give something new in particular they hold for the multilinear calderon zygmund operators this result gives a new approach to a recent work by grafakos and torres and unifies some classical inequalities by cotlar and coifman and fefferman

multilinear calderon zygmund operators with kernels of dini s type and applications abstract the main purpose of this paper is to establish a number of results concerning boundedness of multi linear calderon zygmund operators with kernels of mild regularity let t be a multilinear calderon zygmund operator of type t with being nondecreasing and dini 1 but without assuming to be concave we obtain the end point weak type estimates for multilinear operator t the multiple weighted norm inequalities for multilinear operator t and multilinear commutators of t with b m o functions are also established as applications multiple weighted norm estimates for para products and bilinear pseudo differential operators with mild regularity and their commutators are obtained moreover some boundedness properties of the multilinear operators are also established on variable exponent lebesgue spaces our results improve most of the earlier ones in the literature by removing the assumption of concavity of t and weakening the assumption of dini 1 2 to dini 1

a novel two grid method for semilinear elliptic equations a new finite element discretization technique based on two coarse and fine subspaces is presented for a semilinear elliptic boundary value problem the solution of a nonlinear system on the fine space is reduced to the solution of two small one linear and one nonlinear systems on the coarse space and a linear system on the fine space it is shown both theoretically and numerically that the coarse space can be extremely coarse and still achieve asymptotically optimal approximation as a result the numerical solution of such a nonlinear equation is not significantly more expensive than the solution of one single linearized equation

unilateral regulation breaks regularity of turing patterns we consider a reaction diffusion system undergoing turing instability and augment it by an additional unilateral source term we investigate its influence on the turing instability and on the character of resulting patterns the nonsmooth positively homogeneous unilateral term v has favorable properties but the standard linear stability analysis cannot be performed we illustrate the importance of the nonsmoothness by a numerical case study which shows that the turing instability can considerably change if we replace this term by its arbitrarily precise smooth approximation however the nonsmooth unilateral term and all its approximations yield qualitatively similar patterns although not necessarily developing from small disturbances of the spatially homogeneous steady state further we show that the unilateral source breaks the approximate symmetry and regularity of the classical patterns and yields asymmetric and irregular patterns moreover a given system with a unilateral source produces spatial patterns even for diffusion parameters with ratios closer to 1 than the same system without any unilateral term

2 solvability of the dirichlet problem for nondivergence elliptic equations with vmo coefficients we prove a well posedness result in the class w 2 p w 0 1 p for the dirichlet problem lu f a e in u 0 on we assume the coefficients of the elliptic nondivergence form equation that we study are in vmo l

multiple weighted norm inequalities for maximal multi linear singular integrals with non smooth kernels we obtain weighted norm inequalities for maximal truncated operators of multi linear singular integrals with non smooth kernels in the sense of duong et al this class of operators extends the class of multi linear calderon zygmund operators introduced by coifman and meyer and includes the higher order commutators of calderon the weighted norm inequalities obtained in this work are with respect to the new class of multiple weights of lerner et al the key ingredient in the proof is the introduction of a new multi sublinear maximal operator that plays the role of the hardy littlewood maximal function in a version of cotlar s inequality as applications of these results new weighted estimates for the m th order calderon commutators and their maximal counterparts are deduced

existence and boundedness of multilinear littlewood paley operators on campanato spaces abstract in this paper we established the boundedness of multilinear littlewood paley operators on campanato spaces including the multilinear g function multilinear lusin s area integral and multilinear littlewood paley g function we proved that if the above operators are finite for one point then they are finite almost everywhere moreover norm inequalities do hold for the above operators on the corresponding campanato spaces