lp and bmo bounds of weighted hardy littlewood averages abstract this paper characterizes those nonnegative functions defined on 0 1 for which the weighted hardy littlewood average u f 0 1 f t t dt is bounded on either l p r n p 1 or bmo r n meanwhile the corresponding operator norms are worked out

irregular time dependent obstacles abstract we study the obstacle problem for the evolutionary p laplace equation when the obstacle is discontinuous and does not have regularity in the time variable two quite different procedures yield the same solution

lipschitz regularity of solutions for mixed integro differential equations we establish new holder and lipschitz estimates for viscosity solutions of a large class of elliptic and parabolic nonlinear integro differential equations by the classical ishii lions s method we thus extend the holder regularity results recently obtained by barles chasseigne and imbert 2011 in addition we deal with a new class of nonlocal equations that we term mixed integro differential equations these equations are particularly interesting as they are degenerate both in the local and nonlocal term but their overall behavior is driven by the local nonlocal interaction e g the fractional diffusion may give the ellipticity in one direction and the classical diffusion in the complementary one

the norm inequalities for the weighted cesaro mean operators this paper gives some necessary and sufficient conditions for the weighted cesaro mean operators to be bounded on herz spaces the corresponding new operator norm inequalities are obtained

parabolic control problems in measure spaces with sparse solutions optimal control problems in measure spaces lead to controls that have small support which is desirable e g in the context of optimal actuator placement for problems governed by parabolic partial differential equations well posedness is guaranteed in the space of square integrable measure valued functions which leads to controls with a spatial sparsity structure a conforming approximation framework allows one to derive numerically accessible optimality conditions as well as convergence rates in particular although the state is discretized the control problem can still be formulated and solved in the measure space numerical examples illustrate the structural features of the optimal controls

global weighted lorentz estimates to nonlinear parabolic equations over nonsmooth domains abstract an elaborate method is developed to establish a global calderon zygmund type theory in weighted lorentz spaces l q t for zero cauchy dirichlet problem of nonlinear parabolic equations of p laplacian type with p 2 d d 2 here we assume that the nonlinearity a a t x is merely measurable in the time variable t and has a mall bmo semi norm in the spatial variables x uniformly in variables while the boundary of bounded domain is flat in the sense of reifenberg our result not only develops nonlinear calderon zygmund theory under weak regularity assumptions on the datum but also extends the related framework from lebesgue spaces to refined weighted lorentz spaces

on lp estimates for a class of non local elliptic equations abstract we consider non local elliptic operators with kernel k y a y y d where 0 2 is a constant and a is a bounded measurable function by using a purely analytic method we prove the continuity of the non local operator l from the bessel potential space h p to l p and the unique strong solvability of the corresponding non local elliptic equations in l p spaces as a byproduct we also obtain interior l p estimates the novelty of our results is that the function a is not necessarily to be homogeneous regular or symmetric an application of our result is the uniqueness for the martingale problem associated to the operator l

about riesz transforms on the heisenberg groups see also m1 subsequently several other proofs have been given of probabilistic b ba see also g v or analytic d r p nature also p a meyer has given in m2 a similar result in the gaussian setting the aim of this work is to prove analogous inequalities for the family of heisenberg groups h n instead of the family of euclidean spaces r n the euclidean proofs do not adapt to the non commutative situation of h however our method is reminiscent of some ideas of the proofs by duoandikoetxearubio de francia d r and pisier p a new feature is however the central r6te played by the heat kernel and our proof heavily relies on its explicit expression in h n given by gavean g and hulanicki h let us recall finally that apart from any question of uniformity the boundedness of the riesz transforms on every nilpotent lie group is a well known issue see l v

global weighted estimates for quasilinear elliptic equations with non standard growth abstract in this paper we obtain a new global gradient estimates in weighted lorentz spaces for weak solutions of p x laplacian type equation with small bmo coefficients in a reifenberg flat domain the modified vitali covering lemma the maximal function technique and the appropriate localization method are the main analytical tools our results improve the known results for such equations

smoothing methods and semismooth methods for nondifferentiable operator equations we consider superlinearly convergent analogues of newton methods for nondifferentiable operator equations in function spaces the superlinear convergence analysis of semismooth methods for nondifferentiable equations described by a locally lipschitzian operator in rn is based on rademacher s theorem which does not hold in function spaces we introduce a concept of slant differentiability and use it to study superlinear convergence of smoothing methods and semismooth methods in a unified framework we show that a function is slantly differentiable at a point if and only if it is lipschitz continuous at that point an application to the dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed

regularity results for nonlinear parabolic obstacle problems with subquadratic growth abstract in 26 it was shown that the spatial gradient of the solution u to the parabolic obstacle problem with superquadratic growth is local holder continuous provided the obstacle is regular enough in this paper we extend this regularity result to the subquadratic case this means we establish the local holder continuity of the spatial gradient of the solution u to the parabolic obstacle problem with subquadratic growth more precisely we prove that d u c loc 0 2 for some 0 1 provided the coefficients and the obstacle are regular enough moreover we use the local holder continuity to prove the local lipschitz continuity of the solution u i e u c loc 0 1 1 2

directional sparsity in optimal control of partial differential equations we study optimal control problems in which controls with certain sparsity patterns are preferred for time dependent problems the approach can be used to find locations for control devices that allow controlling the system in an optimal way over the entire time interval the approach uses a nondifferentiable cost functional to implement the sparsity requirements additionally bound constraints for the optimal controls can be included we study the resulting problem in appropriate function spaces and present two solution methods of newton type based on different formulations of the optimality system using elliptic and parabolic test problems we research the sparsity properties of the optimal controls and analyze the behavior of the proposed solution algorithms

weighted orlicz estimates for general nonlinear parabolic equations over nonsmooth domains we introduce a parabolic analogue of muckenhoupt weights to study optimal weighted regularity in orlicz spaces for a general nonlinear parabolic problem of p laplacian type in divergence form over a nonsmooth domain assuming that the nonlinearity is measurable with respect to the time variable and has a small bounded mean oscillation bmo with respect to the spatial variables that the lateral boundary of the parabolic cylinder is reifenberg flat and that the associated weight belongs to a suitable parabolic muckenhoupt class we obtain a global gradient estimate for such a nonlinear parabolic problem by essentially proving that the gradient of the weak solution is as globally integrable as the nonhomogeneous term in the weighted orlicz space our results extend the existing regularity estimates in lebesgue spaces to weighted orlicz spaces

hausdorff operators on the power weighted hardy spaces abstract in this paper we study the high dimensional hausdorff operators on the power weighted hardy spaces we establish their sharp boundedness on power weighted hardy spaces and local hardy spaces particularly our results solve an open question posted by liflyand in 16

regularity theory for fully nonlinear integro differential equations we consider nonlinear integro differential equations like the ones that arise from stochastic control problems with purely jump levy processes we obtain a nonlocal version of the abp estimate harnack inequality and interior c 1 regularity for general fully nonlinear integro differential equations our estimates remain uniform as the degree of the equation approaches two so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations

regularity results for stationary electro rheological fluids we prove regularity results for weak solutions to systems modelling electro rheological fluids in the stationary case as proposed in 27 31 a particular case of the system we consider is

boundedness of multidimensional hausdorff operators on h1 rn abstract in this paper we consider the h 1 boundedness of multidimensional hausdorff operators defined by h a f x r n u f a u x d u where l l o c 1 r n a u a i j u i j 1 n is an n n matrix and each a i j u is a measurable function of u let b i j 1 n b i j for the matrix b b i j u i j 1 n we prove that h a is bounded from the hardy space h 1 to itself if r n u det a 1 u ln 1 a 1 u n det a 1 u d u our result improves known results in addition we show that the above condition is optimal in the size condition

universal potential estimates abstract we prove a class of endpoint pointwise estimates for solutions to quasilinear possibly degenerate elliptic equations in terms of linear and nonlinear potentials of wolff type of the source term such estimates allow to bound size and oscillations of solutions and their gradients pointwise and entail in a unified approach virtually all kinds of regularity properties in terms of the given datum and regularity of coefficients in particular local estimates in holder lipschitz morrey and fractional spaces as well as calderon zygmund estimates follow as a corollary in a unified way moreover estimates for fractional derivatives of solutions by mean of suitable linear and nonlinear potentials are also implied the classical wolff potential estimate by kilpelainen mal and trudinger wang as well as recent wolff gradient bounds for solutions to quasilinear equations embed in such a class as endpoint cases

lorentz estimates for degenerate and singular evolutionary systems we prove estimates of calderon zygmund type for evolutionary p laplacian systems in the setting of lorentz spaces we suppose the coefficients of the system to satisfy only a vmo condition with respect to the space variable our results hold true mutatis mutandis also for stationary p laplacian systems

gradient estimates for a class of parabolic systems we establish local calderon zygmund type estimates for a class of parabolic problems whose model is the nonhomogeneous degenerate singular parabolic p laplacian system ut div du p 2du div f p 2f proving that f lqloc du lqloc q p we also treat systems with discontinuous coefficients of vanishing mean oscillation vmo type

global lorentz estimates with variable power to linear parabolic integro differential equations abstract we shall establish a global estimate in the framework of the lorentz spaces for variable power of the fractional derivative of solutions to the cauchy dirichlet problems of linear parabolic integro differential equations with smooth coefficients m t x y 1 2 in a bounded domain here we assume that some higher order derivatives of m t x y in y are merely measurable in the time variable t and satisfy uniformly continuous in the spatial variables x the boundaries of domains are c 1 1 and the variable exponents satisfy log holder continuity

the p adic weighted hardy cesaro operator and an application to discrete hardy inequalities abstract this paper aims to investigate the boundedness of the p adic analog of the weighted hardy cesaro operator u s f z p f s t t d t on weighted lebesgue spaces and weighted bmo spaces in each case we obtain the corresponding operator norms u s in particular these results have a surprising relevance to discrete hardy inequalities on the real field we prove a reverse bmo hardy inequality and give a necessary condition on so that the commutator of u s is bounded on l r q p n with symbols in bmo q p n

global gradient estimates for general nonlinear parabolic equations in nonsmooth domains we establish the natural calderon zygmund theory for a nonlinear parabolic equation of p laplacian type in divergence form 0 1 ut diva du x t div f p 2f in t by essentially proving that 0 2 f p lq t du p lq t for every q 1 the equation under consideration is of general type and not necessarily of variation form the involved nonlinearity a a x t is assumed to have a small bmo semi norm with respect to x t variables and the lateral boundary of the domain is assumed to be reifenberg flat as a consequence we are able to not only relax the known regularity requirements on the nonlinearity for such a regularity theory but also extend local results to a global one in a nonsmooth domain whose boundary has a fractal property we also find an optimal regularity estimate in orlicz sobolev spaces for such nonlinear parabolic problems

gradient estimates below the duality exponent we show sharp local a priori estimates and regularity results for possibly degenerate non linear elliptic problems with data not lying in the natural dual space we provide a precise non linear potential theoretic analog of classical potential theory results due to adams duke math j 42 765 778 1975 and adams and lewis studia math 74 169 182 1982 concerning morrey spaces imbedding regularity properties for this we introduce a technique allowing for a non local representation of solutions via riesz potentials in turn yielding optimal local estimates simultaneously in both rearrangement and non rearrangement invariant function spaces in fact we also derive sharp estimates in lorentz spaces covering borderline cases which remained open for some while

multivariate hausdorff operators on the spaces lp rn abstract the operators h indicated in the title are characterized by a finite borel measure on r n a borel measurable function c on r n and an n n a e nonsingular matrix a whose entries are also borel measurable functions on r n and h f is defined by means of a lebesgue stieltjes integral with respect to we give simple sufficient conditions in order that these operators be bounded on the lebesgue spaces l p r n for some 1 p these sufficient conditions are exact even in the well known special cases of the cesaro and copson operators we also determine the hausdorff operator h which is adjoint to h in a certain sense we reveal interrelations among these operators and the fourier transform of a function f in l 1 r n on closing we present further special hausdorff operators

holder estimates for nonlinear degenerate parabolic sytems on etablit des estimations de holder pour le gradient d espace u des solutions locales faibles des systemes paraboliques degeneres on etablit la continuite de holder des solutions non negatives d une equation parabolique degeneree

an end point orlicz type estimate for nonlinear elliptic equations abstract we study nonlinear elliptic equations of p laplacian type in divergence form to establish a natural calderon zygmund type theory of an orlicz space type where the lebesgue space is the special case with t t q p f p l 1 q 2 r d u p l 1 q r in the previous results the estimates obtained were strictly above the natural exponents and the function such as t t log 1 t was ruled out for the candidate of but with our approach can be selected up to the end point case of the estimates and the functions t t and t t log 1 t can be additionally selected for

higher integrability for solutions to parabolic problems with irregular obstacles and nonstandard growth abstract the aim of this paper is to derive the self improving property of integrability for the spatial gradient of solutions to degenerate parabolic obstacle problem with irregular obstacles and p x t nonstandard growth more precisely we prove that the spatial gradient of the solution is integrable to a larger power than the natural one determined by the structural assumptions on the involved differential operator

quasiconformal mappings on the heisenberg group on etudie les applications conformes et quasi conformes sur le groupe de heisenberg la metrique invariante a gauche naturelle sur cet espace est non isotrope sous des dilatations et ne peut se decrire par une metrique de riemann ceci engendre de nouveaux phenomenes

on w1 p estimates for elliptic equations in divergence form first we will apply the method to study w 1 p regularity for a nonlinear elliptic operator in divergence form we would like to point out that in the particular case of a linear elliptic equation this method gives an alternative proof to the classical one which uses the general theory of singular integrals the utility of the method that we describe below is that hypotheses a and b are obtained directly by studying the deviation of the coefficients of a from the coefficients of a0 and this is usually not a difficult task a more interesting application of this kind of approximation method is to elliptic homogenization problems for which we obtain results that give w 1 p estimates with a weak hypothesis of regularity in the coefficients for instance we are able to study the following cases

lorentz estimates for obstacle parabolic problems we prove that the spatial gradient of variational solutions to parabolic obstacle problems of p laplacian type enjoys the same regularity of the data and of the derivatives of the obstacle in the scale of lorentz spaces

optimality conditions and error analysis of semilinear elliptic control problems with l 1 cost functional semilinear elliptic optimal control problems involving the l 1 norm of the control in the objective are considered necessary and sufficient second order optimality conditions are derived a priori finite element error estimates for piecewise constant discretizations for the control and piecewise linear discretizations of the state are shown error estimates for the variational discretization of the problem in the sense of m hinze comput optim appl 30 2005 pp 45 61 are also obtained numerical experiments confirm the convergence rates

regularity results for fully nonlinear parabolic integro differential operators in this paper we consider the regularity theory for fully nonlinear parabolic integro differential equations with symmetric kernels we are able to find parabolic versions of alexandrov backelman pucci estimate with 0 sigma 2 and we show a harnack inequality holder regularity and c 1 alpha regularity of the solutions by obtaining decay estimates of their level sets

proximal point method and elliptic regularization abstract a generalized proximal point method for solving variational inequalities with maximal monotone operators is developed it admits a successive approximation of the feasible set and of a symmetric component of the operator as well as an inexact solving of the regularized problems the conditions on the approximation are coordinated with the properties of finite element methods for solving problems in mathematical physics the choice of the regularizing functional exploits a possible reserve of monotonicity of the operator in the variational inequality for the minimal surface problem and related variational inequalities as well as for the convection diffusion problem the studied method extends the principle of elliptic regularization a special convergence analysis shows a more qualitative convergence of the method applied to these problems than it follows from the general theory of proximal point methods also applications to some variational inequalities from the elasticity theory are investigated

hardy space estimates of hausdorff operators on the heisenberg group abstract this is a continuation of a series of works about hausdorff operator on hardy spaces in the setting of the heisenberg group in this paper we obtain the boundedness from lebesgue spaces to hardy spaces for fractional hausdorff operators and their compositions with riesz transforms we also establish the boundedness for two kinds of special hausdorff operators the hausdorff poisson operator and the hausdorff gauss operator on hardy spaces

strongly singular convolution operators on the heisenberg group we consider the l2 mapping properties of a model class of strongly singular integral operators on the heisenberg group hn these are convolution operators on hn whose kernels are too singular at the origin to be of calderon zygmund type this strong singularity is compensated for by introducing a suitably large oscillation our results are obtained by utilizing the group fourier transform and uniform asymptotic forms for laguerre functions due to erdelyi

generalized poincare embeddings and weighted hardy operator on qpα q spaces abstract the poincare embeddings w 1 n r n bmo r n and the john nirenberg inequality in bmo r n are classical results in modern analysis and partial differential equations this paper studies the generalized poincare embeddings and the john nirenberg inequality in q p q r n spaces furthermore for the weighted hardy operator u and its adjoint v this paper gives sufficient and necessary conditions on weight function to ensure the boundedness of them on q p q r n spaces

boundedness of hausdorff operators on real hardy spaces h1 over locally compact groups abstract results of liflyand and collaborators on the boundedness of hausdorff operators on the hardy space h 1 over finite dimensional real space are generalized to the case of locally compact groups that are spaces of homogeneous type special cases and examples of compact lie groups homogeneous groups in particular the heisenberg group and finite dimensional spaces over division rings are considered in conclusion we solve for the space l 2 an open question on compactness of a hausdorff operator posed by liflyand

linear potentials in nonlinear potential theory pointwise gradient bounds via riesz potentials such as those available for the linear poisson equation actually hold for general quasilinear degenerate equations of p laplacean type the regularity theory of such equations completely reduces to that of the classical poisson equation up to the c1 level

w 2 p cdot regularity for elliptic equations in nondivergence form with bmo coefficients we establish an optimal w 2 p cdot estimate to the dirichlet problem for an elliptic equation in nondivergence form with discontinuous coefficients on a c 1 1 bounded domain for every variable exponent p cdot with log holder continuity the matrix of the coefficients is assumed to have a small bmo semi norm depending on the exponent the boundary of the domain and the matrix itself

w 1 p regularity for elliptic equations with measurable coefficients in nonsmooth domains we establish global w 1 p estimates for second order elliptic equations in divergence form under the natural assumption that p is log holder continuous to this end we assume that the coefficients are measurable in one variable and have small bmo semi norms in the other variables and the boundary of the domain is reifenberg flat our work is an optimal and natural extension of w 1 p regularity for such equations with merely measurable coefficients beyond lipschitz domains

global estimates for quasilinear elliptic equations on reifenberg flat domains new global regularity estimates are obtained for solutions to a class of quasilinear elliptic boundary value problems the coefficients are assumed to have small bmo seminorms and the boundary of the domain is sufficiently flat in the sense of reifenberg the main regularity estimates obtained are in weighted lorentz spaces other regularity results in lorentz morrey morrey and holder spaces are shown to follow from the main estimates

lorentz estimate with a variable power for parabolic obstacle problems with non standard growths abstract we prove a global lorentz estimate for the variable power of the gradients of weak solution to parabolic obstacle problems with p t x growth over a bounded nonsmooth domain it is mainly assumed that the variable exponents p t x satisfy a strong type log holder continuity the associated nonlinearities are merely measurable in the time variable and have small bmo semi norms in the spatial variables while the underlying domain is quasiconvex

proximal schemes for parabolic optimal control problems with sparsity promoting cost functionals abstractfast first order proximal methods that solve linear and bilinear parabolic optimal control problems with a sparsity cost functional are discussed weak convergence of these methods is proved and for benchmarking purposes the proposed inexact proximal schemes are compared to an inexact semi smooth newton method results of numerical experiments are presented to demonstrate the computational effectiveness of proximal schemes and to validate the theoretical estimates

estimates of weighted hardy littlewood averages on the p adic vector space abstract in the p adic vector space q p n we characterize those non negative functions defined on z p w q p 0 w p 1 for which the weighted hardy littlewood average u f z p f t t d t is bounded on l r q p n 1 r and on bmo q p n also in each case we find the corresponding operator norm u