sample adaptive product quantization asymptotic analysis and examples vector quantization vq is an efficient data compression technique for low bit rate applications however the major disadvantage of vq is that its encoding complexity increases dramatically with bit rate and vector dimension even though one can use a modified vq such as the tree structured vq to reduce the encoding complexity it is practically infeasible to implement such a vq at a high bit rate or for large vector dimensions because of the huge memory requirement for its codebook and for the very large training sequence requirement to overcome this difficulty a structurally constrained vq called the sample adaptive product quantizer sapq has recently been proposed we extensively study the sapq that is based on scalar quantizers in order to exploit the simplicity of scalar quantization through an asymptotic distortion result we discuss the achievable performance and the relationship between distortion and encoding complexity we illustrate that even when sapq is based on scalar quantizers it can provide vq level performance we also provide numerical results that show a 2 3 db improvement over the lloyd max 1982 1960 quantizers for data rates above 4 b point

intrinsic ultracontractivity and eigenfunction estimates for schrodinger operators let d be a domain in rn n 2 and let h h0 v where h0 is a divergence form uniformly elliptic operator with dirichlet boundary conditions and v is in the kato class intrinsic ultracontractivity is proved for the semigroup of h when d is holder domain of order 0 or a uniformly holder domain of order for 0 2 for every 2 there exists a uniformly holder domain of order for which the dirichlet laplacian is not intrinsically ultracontractive for a large class of domains it is shown that the heat kernels for h0 and h decay at the same rate at the boundary applications are given to the lifetime of conditioned brownian motion some of our results seem to be new even for smooth domains

weighted regularity estimates for a class of higher order nonlinear parabolic and elliptic systems abstract in this paper we prove the weighted l p regularity estimates and the morrey regularity estimates for a class of higher order nonlinear parabolic systems which satisfy an asymptotically regular condition as a byproduct similar regularity estimates are also obtained for a corresponding class of higher order nonlinear elliptic systems

global calderon zygmund estimate for p laplacian parabolic system we establish a global calderon zygmund theory for the weak solution of the following p laplacian system begin aligned left begin array ll u t mathrm div left a x t nabla u p 2 nabla u right mathrm div f p 2 f f text in omega t u 0 text on partial omega times 0 t u u 0 text on omega times 0 end array right end aligned by proving that for every q ge p there holds begin aligned nabla u 0 q in l 1 omega text and f q f left p right frac q p in l 1 omega t longrightarrow nabla u q in l 1 omega t end aligned with the desired global calderon zygmund estimate where p frac p n 2 n is parabolic sobolev conjugate of p and left p right prime is holder conjugate of p

quantization of memoryless and gauss markov sources over binary markov channels joint source channel coding for stationary memoryless and gauss markov sources and binary markov channels is considered the channel is an additive noise channel where the noise process is an mth order markov chain two joint source channel coding schemes are considered the first is a channel optimized vector quantizer optimized for both source and channel the second scheme consists of a scalar quantizer and a maximum a posteriori detector in this scheme it is assumed that the scalar quantizer output has residual redundancy that can be exploited by the maximum a posteriori detector to combat the correlated channel noise these two schemes are then compared against two schemes which use channel interleaving numerical results show that the proposed schemes outperform the interleaving schemes for very noisy channels with high noise correlation gains of 4 5 db in signal to noise ratio are possible

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

solvability of the divergence equation implies john via poincare inequality abstract let r 2 be a bounded simply connected domain we show that for a fixed every p 1 the divergence equation div v f is solvable in w 0 1 p 2 for every f l 0 p if and only if is a john domain if and only if the weighted poincare inequality u x u q d x c u x q dist x q d x holds for some every q 1 this gives a positive answer to a question raised by russ 2013 in the case of bounded simply connected domains in higher dimensions similar results are proved under some additional assumptions on the domain in question

higher integrability result for nonlinear elliptic systems with conormal boundary conditions abstract we prove a boundary higher integrability result for nonlinear elliptic systems with conormal boundary conditions in locally uniform domains to do it we derive a boundary version of gehring giaquinta modica lemma and sobolev poincare type inequality in locally uniform domains our result plays a key role for handling the coefficients when obtaining caldero n zygmund type estimates with conormal boundary conditions in nonsmooth domains

design of sample adaptive product quantizers for noisy channels channel optimized vector quantization covq has proven to be an effective joint source channel coding technique that makes the underlying quantizer robust to channel noise unfortunately covq retains the high encoding complexity of the standard vector quantizer vq for medium to high quantization dimensions and moderate to good channel conditions a technique called sample adaptive product quantization sapq was recently introduced by kim and shroff to reduce the complexity of the vq while achieving comparable distortions in this letter we generalize the design of sapq for the case of memoryless noisy channels by optimizing the quantizer with respect to both source and channel statistics numerical results demonstrate that the channel optimized sapq cosapq achieves comparable performance to the covq within 0 2 db while maintaining considerably lower encoding complexity up to half of that of covq and storage requirements robustness of the cosapq system against channel mismatch is also examined

gradient estimates for higher order elliptic equations on nonsmooth domains abstract we establish optimal gradient estimates in orlicz space for a nonhomogeneous elliptic equation of higher order with discontinuous coefficients on a nonsmooth domain our assumption is that for each point and for each sufficiently small scale the coefficients have small mean oscillation and the boundary of the domain is sufficiently close to a hyperplane as a consequence we prove the classical w m p m 1 2 1 p estimates for such a higher order equation our results easily extend to higher order elliptic and parabolic systems

generalized hardy inequality for the magnetic dirichlet forms we obtain lower bounds for the magnetic dirichlet form in dimensions d 2 for d 2 the results generalize a well known lower bound by the magnetic field strength we replace the actual magnetic field bby an non vanishing effective field which decays outside the support of bas dist x suppb 2 in the case d 3 we establish that the magnetic form is bounded from below by the magnetic field strength if one assumes that the field does not vanish and its direction is slowly varying

quantization based on a novel sample adaptive product quantizer sapq in this paper we propose a novel feedforward adaptive quantization scheme called the sample adaptive product quantizer sapq this is a structurally constrained vector quantizer that uses unions of product codebooks sapq is based on a concept of adaptive quantization to the varying samples of the source and is very different from traditional adaptation techniques for nonstationary sources sapq quantizes each source sample using a sequence of quantizers even when using scalar quantization in sapq we can achieve performance comparable to vector quantization with the complexity still close to that of scalar quantization we also show that important lattice based vector quantizers can be constructed using scalar quantization in sapq we mathematically analyze sapq and propose a algorithm to implement it we numerically study sapq for independent and identically distributed gaussian and laplacian sources through our numerical study we find that sapq using scalar quantizers achieves typical gains of 13 db in distortion over the lloyd max quantizer we also show that sapq can he used in conjunction with vector quantizers to further improve the gains

the design of joint source and channel trellis waveform coders the generalized lloyd algorithm is applied to the design of joint source and channel trellis waveform coders to encode discrete time continuous amplitude stationary and ergodic sources operating over discrete memoryless noisy channels experimental results are provided for independent and autoregressive gaussian sources binary symmetric channels and absolute error and squared error distortion measures performance of the joint codes is compared with the tandem combination of a trellis source code and a trellis channel code on the independent gaussian source using the squared error distortion measure operating over an additive white gaussian noise channel it is observed that the jointly optimized codes achieve performance close to or better than that of separately optimized tandem codes of the same constraint length performance improvement via a predictive joint source and channel trellis code is demonstrated for the autoregressive gaussian source using the squared error distortion measure

higher integrability in the obstacle problem for the fast diffusion equation abstract we prove local higher integrability of the spatial gradient for solutions to obstacle problems of porous medium type in the fast diffusion case m 1 the result holds for the natural range of exponents that is known from other regularity results for porous medium type equations we also cover the case of signed solutions

gauss green theorem for weakly differentiable vector fields sets of finite perimeter and balance laws we analyze a class of weakly differentiable vector fields f w r n r n with the property that f 2 l 1 and div f is a signed radon measure these fields are called bounded divergence measure fields the primary focus of our inves tigation is to introduce a suitable notion of the normal trace of any divergence measure field f over the boundary of an arbitrary set of finite perimeter that en sures the validity of the gauss green theorem to achieve this we first establish a fundamental approximation theorem which states that given a signed radon measurethat is absolutely continuous with respect to h n 1 on r n a ny set of finite perimeter can be approximated by a family of sets with smooth boundary essentially from the measure theoretic interior of the set with respect to the mea sure kk the total variation measure we employ this approximation theorem to derive the normal trace of f on the boundary of any set of finite perimeter e as the limit of the normal traces of f on the boundaries of the approximate sets with smooth boundary so that the gauss green theorem for f holds one with these results we analyze the cauchy flux that is bounded by a nonnegative radon measure over any oriented surface i e an n 1 dimensional surface that is a part of the boundary of a set of finite perimeter and thereby develop a general mathematical formulation of the physical principle of the balance law through the cauchy flux finally we apply this framework to the derivation of systems of balance laws with measure valued source terms from the formulation of the balance law this framework also allows the recovery of cauchy entropy flux through the lax entropy inequality for entropy solutions of hyperbolic con servation laws c 2008 wiley periodicals inc

a theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity the energy functional of nonlinear plate theory is a curvature functional for surfaces rst proposed on physical grounds by g kirchhoff in 1850 we show that it arises as a 0 limit of three dimensional nonlinear elasticity theory as the thickness of a plate goes to zero a key ingredient in the proof is a sharp rigidity estimate for maps v v u r n u r n we show that the l 2 distance of rv from a single rotation matrix is bounded by a multiple of the l 2 distance from

uniformly fat sets in this paper we study closed sets e which are locally uniformly fat with respect to a certain nonlinear riesz capacity we show that e is actually locally uniformly fat with respect to a weaker riesz capacity two applications of this result are given the first application is concerned with proving sobolev type inequalities in domains whose complements are uniformly fat the second application is concerned with the fekete points of e introduction let x x x2 x be a point in euclidean n space rn with jxl the norm of x for a 0 let i x ixi a n denote the riesz kernel of order a and put i f x f ix y f n f y dy x e r rn ic u x fr yj c n d u y x e r rn when f is a lebesgue integrable function and u a borel measure on rn here dy denotes lebesgue measure on rn if e c rn 0 1 define the a p outer riesz capacity of e by ra p e inf jfllfp ia f 1 on el where if lip is the lebesgue p norm of f we note that a 1 p 2 is the classical newtonian capacity if ap n we let rc p e inf jfjjf j f 1 on el where jc is the truncated riesz kernel defined by je x ixi a n 100 a n x 100 0 lxl 100 we shall say that a property holds a p quasi everywhere abbreviated a p q e on a set e if it holds on e except perhaps for a set of rop capacity zero next we list some properties of riesz capacities to simplify the writing we state these properties only for 0 u l p l 1 a p q e on e and p 1 a p q e on supp u c e moreover e jjli pi ujj rop e received by the editors march 11 1987 1980 mathematic3 subject cla33ification 1985 revi3ion primary 31c15 31b15 1988 american mathematical society 0002 9947 88 1 00 25 per page

calderon zygmund theory for nonlinear elliptic problems with irregular obstacles abstract we consider a nonhomogeneous elliptic problem with an irregular obstacle involving a discontinuous nonlinearity over an irregular domain in divergence form of p laplacian type to establish the global calderon zygmund estimate by proving that the gradient of the weak solution is as integrable as both the gradient of the obstacle and the nonhomogeneous term under the bmo smallness of the nonlinearity and sufficient flatness of the boundary in the reifenberg sense

on the performance and complexity of channel optimized vector quantizers the performance and complexity of channel optimized vector quantizers are studied for the gauss markov source observations on the geometric structure of these quantizers are made which have an important implication on the encoding complexity for the squared error distortion measure it is shown that an operation equivalent to a euclidean distance measurement with respect to an appropriately defined set of points used to identify the encoding regions can be used to perform the encoding this implies that the encoding complexity is proportional to the number of encoding regions it is then demonstrated that for very noisy channels and a heavily correlated source when the codebook size is large the number of encoding regions is considerably smaller than the codebook size implying a reduction in encoding complexity

on the crossing rule abstract the paper gives conditions on a family of matrices which guarantee that some matrix in the family will have a multiple eigenvalue in particular the main theorem states exactly which dimensions admit k dimensional subspaces of matrices for which all nonzero elements have distinct eigenvalues this question arises naturally in the theory of first order hyperbolic systems of partial differential equations the main theorem in this context tells exactly for which integers n an n x n system in k space variables may be strictly hyperbolic author

soft decision demodulation design for covq over white colored and isi gaussian channels in this work the design of a q bit scalar and vector soft decision demodulator for gaussian channels with binary phase shift keying modulation is investigated the demodulator is used in conjunction with a soft decision channel optimized vector quantization covq system the covq is constructed for an expanded q 1 discrete channel consisting of the concatenation of the modulator the gaussian channel and the demodulator it is found that as the demodulator resolution q increases the capacity of the expanded channel increases resulting in an improvement of the covq performance consequently the soft decision demodulator is designed to maximize the capacity of the expanded channel three gaussian channel models are considered as follows 1 additive white gaussian noise channels 2 additive colored gaussian noise channels and 3 gaussian channels with intersymbol interference comparisons are made with a hard decision covq systems b covq systems which utilize interleaving and c an unquantized q spl infin soft decision decoder proposed by skoglund and hedelin 1999 it is shown that substantial improvements can be achieved over covq systems which utilize hard decision demodulation and or channel interleaving the performance of the proposed covq system is comparable with the system by skoglund and hedelin though its computational complexity is substantially less

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

joint source and noisy channel trellis encoding corresp in a trellis encoding communication system the decoder is a time invariant nonlinear filter with finite memory sliding block code and the encoder is a trellis search algorithm matched to the decoder a coding theorem is established for a trellis encoding of a stationary and ergodic source over a discrete memoryless noisy channel which shows that such communication systems can perform arbitrarily close to the source distortion rate function evaluated at the channel capacity

global gradient estimates for the p laplacian abstract we consider calderon zygmund type estimates for the non homogeneous p laplacian system div d u p 2 d u div g p 2 g where p is a variable exponent we show that g p l q r n l 1 r n implies d u p l q r n l 1 r n for any q 1 we also prove local estimates independent of the size of the domain and introduce new techniques to variable analysis the paper is an extension of the local estimates of acerbi mingione 2005

gradient continuity for p laplace systems abstract we prove the continuity of the gradient of weak solutions to nonhomogeneous p laplace systems with coefficients we assume that the nonhomogeneous term belongs to the lorentz space l n 1 the coefficient is dini continuous and the variable exponent p is logarithmic dini type continuous

global gradient estimates for the parabolic p x t laplacian equation abstract in this paper we consider the global gradient estimates for weak solutions of the parabolic p x t laplacian type equation in a reifenberg flat domain

holder estimates of p harmonic extension operators it is now a well known fact that for 1 p the p harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a 1 p poincare inequality are locally holder continuous in this note we provide a characterization of domains in such metric spaces for which p harmonic extensions of holder continuous boundary data are globally holder continuous we also provide a link between this regularity property of the domain and the uniform p fatness of the complement of the domain

self improvement of weighted pointwise inequalities on open sets abstract we prove a general self improvement property for a family of weighted pointwise inequalities on open sets including pointwise hardy inequalities with distance weights for this purpose we introduce and study the classes of p poincare and p hardy weights for an open set x where x is a metric measure space we also apply the self improvement of weighted pointwise hardy inequalities in connection with usual integral versions of hardy inequalities

reverse holder inequalities for singular parabolic equations near the boundary abstract we show that weak solutions to a singular parabolic partial differential equation globally belong to a higher sobolev space than assumed a priori to this end we prove that the gradients satisfy a reverse holder inequality near the boundary the results extend to singular parabolic systems as well motivation for studying reverse holder inequalities comes partly from applications to regularity theory

global gradient estimates for nonlinear equations of p laplace type from composite materials abstract we study global gradient estimates for the weak solution to divergence form nonlinear elliptic equations with p growth nonlinearities from composite materials we assume that the domain is bounded and is composed of disjoint reifenberg flat domains and the nonlinearities have small bmo seminorms in each subdomain under these assumptions based on our new geometric observation for disjoint reifenberg domains and gradient estimates for nonlinear equations with measurable p growth nonlinearities we establish global w 1 q estimates for p q

elliptic equations with measurable nonlinearities in nonsmooth domains abstract we study elliptic equations with measurable nonlinearities in nonsmooth domains we establish an optimal global w 1 p estimate under the condition that the associated nonlinearity is allowed to be merely measurable in one variable but has a sufficiently small bmo semi norm in the other variables while the underlying domain is sufficiently flat in the reifenberg sense that the boundary of the domain is locally trapped between two narrow strips

sharp hardy inequalities on the solid torus abstract in this paper we establish the classical hardy inequality in the solid torus and some variants of it the general idea is to use the fact that sobolev embeddings can be improved in the presence of symmetries in all cases using techniques that exploit the symmetry presented by the solid torus we calculate the displayed best constants and we prove that they are the same as the standard hardy best constants which appear in convex domains although the solid torus is not convex

ap weight and integrability of solutions in obstacle problems integrability of solutions to obstacle problems is obtained under assumption that the obstacles belong to the a p weight class this result is used to prove an existence result for a p weight problems c 2011 elsevier ltd all rights reserved

hyperbolic conservation laws with umbilic degeneracy i in this paper a compactness framework for approximate solutions to nonlinear hyperbolic systems with umbilic degeneracy is established by combining techniques of compensated compactness with some classical methods and by a detailed analysis of a highly singular equation of euler poisson darboux type then this framework is successfully applied to prove the convergence of the viscosity method and to prove the existence of global entropy solutions for the cauchy problem with large initial data for a canonical class of the systems with quadratic flux form

nonstrictly hyperbolic nonlinear systems on etudie le caractere bien pose du probleme aux valeurs initiales localement en temps pour une classe de systemes hyperboliques non lineaires

regularity and free boundary regularity for the p laplace operator in reifenberg flat and ahlfors regular domains regularity and free boundary regularity for the p laplace operator in reifenberg flat and ahlfors regular domains

global integrability theorems for a harmonic tensors abstract we extend global integrability theorems for the gradients of a harmonic functions to the exterior derivative of differential forms satisfying rather general nonhomogeneous elliptic equations these include the usual a harmonic equations geometric conditions on the boundary of the domains of integration imply a corresponding exponent of integrability in the process we generalize the weak reverse holder inequality to such differential forms

large time existence for 3d water waves and asymptotics we rigorously justify in 3d the main asymptotic models used in coastal oceanography including shallow water equations boussinesq systems kadomtsev petviashvili kp approximation green naghdi equations serre approximation and full dispersion model we first introduce a variable nondimensionalized version of the water waves equations which vary from shallow to deep water and which involves four dimensionless parameters using a nonlocal energy adapted to the equations we can prove a well posedness theorem uniformly with respect to all the parameters its validity ranges therefore from shallow to deep water from small to large surface and bottom variations and from fully to weakly transverse waves

a study of vector quantization for noisy channels several issues related to vector quantization for noisy channels are discussed an algorithm based on simulated annealing is developed for assigning binary codewords to the vector quantizer code vectors it is shown that this algorithm could result in dramatic performance improvements as compared to randomly selected codewords a modification of the simulated annealing algorithm for binary codeword assignment is developed for the case where the bits in the codeword are subjected to unequal error probabilities resulting from unequal levels of error protection an algorithm for the design of an optimal vector quantizer for a noisy channel is briefly discussed and its robustness under channel mismatch conditions is studied numerical results for a stationary first order gauss markov source and a binary symmetric channel are provided it is concluded that the channel optimized vector quantizer design algorithm if used carefully can result in a fairly robust system with no additional delay the case in which the communication channel is nonstationary as in mobile radio channels is studied and some preliminary ideas for quantizer design are presented

hadamard based soft decoding for vector quantization over noisy channels we present an estimator based or soft vector quantizer decoder for communication over a noisy channel the decoder is optimal according to the mean square error criterion and hadamard based in the sense that a hadamard transform representation of the vector quantizer is utilized in the implementation of the decoder an efficient algorithm for optimal decoding is derived we furthermore investigate suboptimal versions of the decoder providing good performance at lower complexity the issue of joint encoder decoder design is considered both for optimal and suboptimal decoding results regarding the channel distortion and the structure of a channel robust code are also provided through numerical simulations soft decoding is demonstrated to outperform hard decoding in several aspects

self improvement of uniform fatness revisited we give a new proof for the self improvement of uniform p fatness in the setting of general metric spaces our proof is based on rather standard methods of geometric analysis and in particular the proof avoids the use of deep results from potential theory and analysis on metric spaces that have been indispensable in the previous proofs of the self improvement a key ingredient in the proof is a self improvement property for local hardy inequalities

well posedness in sobolev spaces of the full water wave problem in 2 d we consider the motion of the interface of 2 d irrotational incompressible inviscid water wave with air above water and surface tension zero we show that the interface is always not accelerating into the water region normal to itself as rapidly as the normal acceleration of gravity as long as the interface is nonself intersect we therefore obtain the existence and uniqueness of solutions of the full water wave problem locally in time for any initial interface which is nonself intersect including the case that the interface is of multiple heights

in between the inequalities of sobolev and hardy abstract we establish both sufficient and necessary conditions for the validity of the so called hardy sobolev inequalities on open sets of the euclidean space these inequalities form a natural interpolating scale between the weighted sobolev inequalities and the weighted hardy inequalities the assouad dimension of the complement of the open set turns out to play an important role in both sufficient and necessary conditions

well posedness of the water waves equations 1 1 presentation of the problem the water waves problem for an ideal liquid consists of describing the motion of the free surface and the evolution of the velocity field of a layer of perfect incompressible irrotational fluid under the influence of gravity in this paper we restrict our attention to the case when the surface is a graph parameterized by a function x where t denotes the time variable and x xi xd rd the horizontal spatial variables the method developed here works equally well for any integer d 1 but the only physically relevant cases are of course d 1 and d 2 the layer of fluid is also delimited from below by a not necessarily flat bottom parameterized by a time independent function b x we denote by q t the fluid domain at time t the incompressibility of the fluids is expressed by 1 1 div v 0 in qu t 0

gradient estimates for nonlinear elliptic double obstacle problems abstract we study a nonlinear elliptic double obstacle problem with irregular data and establish an optimal calderon zygmund theory the partial differential operator is of the p laplacian type and includes merely measurable coefficients in one variable we prove that the gradient of a weak solution is as integrable as both the gradient of assigned two obstacles and the nonhomogeneous divergence term under a small bmo semi norm assumption on the coefficients in the other variables

weighted universal image compression we describe a general coding strategy leading to a family of universal image compression systems designed to give good performance in applications where the statistics of the source to be compressed are not available at design time or vary over time or space the basic approach considered uses a two stage structure in which the single source code of traditional image compression systems is replaced with a family of codes designed to cover a large class of possible sources to illustrate this approach we consider the optimal design and use of two stage codes containing collections of vector quantizers weighted universal vector quantization bit allocations for jpeg style coding weighted universal bit allocation and transform codes weighted universal transform coding further we demonstrate the benefits to be gained from the inclusion of perceptual distortion measures and optimal parsing the strategy yields two stage codes that significantly outperform their single stage predecessors on a sequence of medical images weighted universal vector quantization outperforms entropy coded vector quantization by over 9 db on the same data sequence weighted universal bit allocation outperforms a jpeg style code by over 2 5 db on a collection of mixed test and image data weighted universal transform coding outperforms a single data optimized transform code which gives performance almost identical to that of jpeg by over 6 db

weighted hardy inequalities abstract for bounded lipschitz domains d in r n it is known that if 1 p then for all 0 0 where 0 p 1 0 there is a constant c with d u x p dist x d p d x c d u x p dist x d d x for all u c 0 d we show that if d is merely assumed to be a bounded domain in r n that satisfies a whitney cube counting condition with exponent and has plump complement then the same inequality holds with 0 now taken to be p n n p n p 2 n further we extend the known results see h brezis m marcus hardy s inequalities revisited dedicated to ennio de giorgi ann scuola norm sup pisa cl sci 4 25 1997 1998 217 237 m hoffmann ostenhof t hoffmann ostenhof a laptev a geometrical version of hardy s inequality j funct anal 189 2002 537 548 j tidblom a geometrical version of hardy s inequality for w 1 p proc amer math soc 132 2004 2265 2271 concerning the improved hardy inequality d u x p dist x d p d x d p n d u x p d x c d u x p d x c c n p by showing that the class of domains for which the inequality holds is larger than that of all bounded convex domains

the gehring lemma dimension free estimates abstract we present a dimension free estimate for the exponent of higher integrability in gehring s lemma for an inhomogeneous weak reverse holder inequality

global gradient estimates for parabolic equations with measurable nonlinearities abstract we establish a global calderon zygmund theory of nonlinear parabolic equations with measurable nonlinearities in divergence form by proving that the spatial gradient of a weak solution is as integrable as the inhomogeneous term nonlinearity a x t is assumed to be only measurable in one spatial variable and has locally small bmo semi norm in the other spatial and time variables uniformly with respect to variable the boundary of the bounded domain can be beyond the lipschitz category but it is well trapped in two narrow strips at each point and at each scale

global gradient estimates for non quadratic vector valued parabolic quasi minimizers abstract we establish local and global higher integrability estimates for the gradient of vector valued parabolic quasi minimizers of integral functionals with polynomial p growth p 2 n n 2 on a parabolic domain t in r n 1 n 2 and under a weak regularity assumption on the parabolic boundary of t

the equivalence between pointwise hardy inequalities and uniform fatness we prove an equivalence result between the validity of a pointwise hardy inequality in a domain and uniform capacity density of the complement this result is new even in euclidean spaces but our methods apply in general metric spaces as well we also present a new transparent proof for the fact that uniform capacity density implies the classical integral version of the hardy inequality in the setting of metric spaces in addition we consider the relations between the above concepts and certain hausdorff content conditions

adaptive scalar quantization without side information in this paper we introduce a novel technique for adaptive scalar quantization adaptivity is useful in applications including image compression where the statistics of the source are either not known a priori or will change over time our algorithm uses previously quantized samples to estimate the distribution of the source and does not require that side information be sent in order to adapt to changing source statistics our quantization scheme is thus backward adaptive we propose that an adaptive quantizer can be separated into two building blocks namely model estimation and quantizer design the model estimation produces an estimate of the changing source probability density function which is then used to redesign the quantizer using standard techniques we introduce nonparametric estimation techniques that only assume smoothness of the input distribution we discuss the various sources of error in our estimation and argue that for a wide class of sources with a smooth probability density function pdf we provide a good approximation to a universal quantizer with the approximation becoming better as the rate increases we study the performance of our scheme and show how the loss due to adaptivity is minimal in typical scenarios in particular we provide examples and show how our technique can achieve signal to noise ratios within 0 05 db of the optimal lloyd max quantizer for a memoryless source while achieving over 1 5 db gain over a fixed quantizer for a bimodal source