bmo spaces associated with semigroups of operators we study bmo spaces associated with semigroup of operators on noncommutative function spaces i e von neumann algebras and apply the results to boundedness of fourier multipliers on non abelian discrete groups we prove an interpolation theorem for bmo spaces and prove the boundedness of a class of fourier multipliers on noncommutative lp spaces for all 1 p with optimal constants in p

sbv like regularity for hamilton jacobi equations with a convex hamiltonian in this paper we consider a viscosity solution u of the hamilton jacobi equation tu h dxu 0in 0 t rn where h is smooth and convex we prove that when d t hp dxu t hp h is bv for all t 0 t and suitable hypotheses on the lagrangian l hold the radon measure divd t can have cantor part only for a countable number of t s in 0 t this result extends a result of robyr for genuinely nonlinear scalar balance laws and a result of bianchini de lellis and robyr for uniformly convex hamiltonians

multiphase semiclassical approximation of an electron in a one dimensional crystalline lattice i homogeneous problems we present a computational approach for the wkb approximation of the wave function of an electron moving in a periodic one dimensional crystal lattice we derive a nonstrictly hyperbolic system for the phase and the intensity where the flux functions originate from the bloch spectrum of the schrodinger operator relying on the framework of the multibranch entropy solutions introduced by brenier and corrias we compute efficiently multiphase solutions using well adapted and simple numerical schemes in this first part we present computational results for vanishing exterior potentials which demonstrate the effectiveness of the proposed method

maximal theorems for the directional hilbert transform on the plane for a schwartz function f on the plane and a non zero v r 2 define the hilbert transform of f in the direction v to be h v f x p v r f x vy dy y let be a schwartz function with frequency support in the annulus 1 2 the l 2 estimate is sharp the method of proof is based upon techniques related to the pointwise convergence of fourier series indeed our main theorem implies this result on fourier series

pseudo localization of singular integrals and noncommutative calderon zygmund theory abstract the weak type 1 1 boundedness of singular integrals acting on matrix valued functions has remained open since the 1980s mainly because the methods provided by the vector valued theory are not strong enough in fact we can also consider the action of generalized calderon zygmund operators on functions taking values in any other von neumann algebra our main tools for its solution are two first the lack of some classical inequalities in the noncommutative setting forces to have a deeper knowledge of how fast a singular integral decreases l 2 sense outside of the support of the function on which it acts this gives rise to a pseudo localization principle which is of independent interest even in the classical theory second we construct a noncommutative form of calderon zygmund decomposition by means of the recent theory of noncommutative martingales this is a corner stone in the theory as application we obtain the sharp asymptotic behavior of the constants for the strong l p inequalities also unknown up to now our methods settle some basics for a systematic study of a noncommutative calderon zygmund theory

level set approach to mean curvature flow in arbitrary codimension we develop a level set theory for the mean curvature evolution of surfaces with arbitrary co dimension thus generalizing the previous work 8 15 on hypersurfaces the main idea is to surround the evolving surface of co dimension k in r by a family of hypersurfaces the level sets of a function evolving with normal velocity equal to the sum of the d k smallest principal curvatures the existence and the uniqueness of a weak level set solution is easily established using mainly the results of 8 and the theory of viscosity solutions for second order nonlinear parabolic equations the level set solutions coincide with the classical solutions whenever the latter exist the proof of this connection uses a careful analysis of the squared distance from the surfaces it is also shown that varifold solutions constructed by brakke 7 are included in the level set solutions the idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of de giorgi an introduction to the theory of barriers and his connection to the level set solutions is also provided

the sharp bound for the hilbert transform on weighted lebesgue spaces in terms of the classical ap characteristic we show that the norm of the hilbert transform as an operator in the weighted space l p r for 2 p oo is bounded by a constant multiple of the first power of the classical ap characteristic of this result is sharp we also prove a bilinear imbedding theorem with simple conditions

the semigroup property of value functions in lagrange problems the lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form the binary operation of inf composition applied to the value functions of a lagrange problem equates the principle of optimality with a semigroup property this paper finds the infinitesimal generator of the semigroup by differentiating at t 0 the type of limit is epigraphical convergence in a uniform sense moreover the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed

repeated games with lack of information on one side we introduce the dual differential game of a repeated game with lack of information on one side as the natural continuous time version of the dual game introduced by de meyer 1996 a traditional way to study the value of differential games is through discrete time approximations here we follow the opposite approach we identify the limit value of a repeated game in discrete time as the value of a differential game namely we use the recursive structure for the finitely repeated version of the dual game to construct a differential game for which the upper values of the uniform discretization satisfy precisely the same property the value of the dual differential game exists and is the unique viscosity solution of a first order derivative equation which appears implicitely in de meyer and rosenberg 1999 with a limit condition we identify the solution by translating viscosity properties in the primal

central limit theorem for stochastic hamilton jacobi equations we study the asymptotic behavior of where u solves the hamilton jacobi equation u t h x u x 0 with h a stationary ergodic process in the x variable it was shown in rezakhanlou tarver rt that u converges to a deterministic function provided h x p is convex in p and the convex conjugate of h in the p variable satisfies certain growth conditions in this article we establish a central limit theorem for the convergence by showing that for a class of examples u x t can be stochastically represented as where z x t is a suitable random field in particular we establish a central limit theorem when the dimension is one and where is a random function that enjoys some mild regularity

solutions of vectorial hamilton jacobi equations and minimizers of nonquasiconvex functionals abstract we provide a unified approach to prove existence results for the dirichlet problem for hamilton jacobi equations and for the minimum problem for nonquasiconvex functionals of the calculus of variations with affine boundary data the idea relies on the definition of integro extremal solutions introduced in the study of nonconvex scalar variational problem

chaperoning state evolutions by variable durations mckendrick partial differential equations in demography provide the state of a population depending on chronological time t and age d and involve two time evolutions the chronological one t in 0 t mapsto t geq 0 and the calendar age t in t d t mapsto t t d geq 0 both with constant velocity equal to 1 the calendar age evolution chaperons so to speak the evolution t in t d t mapsto x t of the state of a system some physical biological and economic problems motivate the introduction of variable durations with variable velocities representing the fluidity of time offering mathematical metaphors of a subjective fleeting specious time passing more or less slowly variable durations are no longer prescribed but chosen among those available and regulated the joint evolution of the variable evolution and the state is assumed to be governed by a differential inclusion or control system and provides as a byproduct the unknown temporal windows on which they evolve toge

risk sensitive control with hara utility a control methodology based on the hyperbolic absolute risk averse hara utility function is presented as an alternative to the exponential of an integral approach to finding robust controllers the work is inspired by the intuition that hara controllers while being robust may give better performance than exponential controllers in normal situations the hara problem is shown to be equivalent to a certain differential game and the asymptotic properties of the hara problem and this differential game are studied as an example a linear quadratic hara problem is studied where the problem of finding a robust hara controller is proved to be equivalent to solving a standard linear quadratic problem for a system with a higher noise intensity this reveals an interesting relationship between robustness and uncertainty

spectral viscosity approximations to hamilton jacobi solutions the spectral viscosity approximate solution of convex hamilton jacobi equations with periodic boundary conditions is studied it is proved in this paper that the approximation and its gradient remain uniformly bounded formally spectral accurate and converge to the unique viscosity solution the l1 convergence rate of the order 1 varepsilon forall varepsilon 0 is obtained

the distance function from the boundary in a minkowski space let the space be endowed with a minkowski structure that is is the gauge function of a compact convex set having the origin as an interior point and with boundary of class and let be the asymmetric distance associated to given an open domain of class let be the minkowski distance of a point from the boundary of we prove that a suitable extension of to which plays the role of a signed minkowski distance to is of class in a tubular neighborhood of and that is of class outside the cut locus of that is the closure of the set of points of nondifferentiability of in in addition we prove that the cut locus of has lebesgue measure zero and that can be decomposed up to this set of vanishing measure into geodesics starting from and going into along the normal direction with respect to the minkowski distance we compute explicitly the jacobian determinant of the change of variables that associates to every point outside the cut locus the pair where denotes the unique projection of on and we apply these techniques to the analysis of pdes of monge kantorovich type arising from problems in optimal transportation theory and shape optimization

some characterizations of optimal trajectories in control theory several characterizations of optimal trajectories for the classical mayer problem in optimal control are provided for this purpose the regularity of directional derivatives of the value function is studied for instance it is shown that for smooth control systems the value function v is continuously differentiable along an optimal trajectory x t 0 1 to bf r n provided v is differentiable at the initial point t 0 x t 0 then the upper semicontinuity of the optimal feedback map is deduced the problem of optimal design is addressed obtaining sufficient conditions for optimality finally it is shown that the optimal control problem may be reduced to a viability one

an asymptotic analysis of hierarchical control of manufacturing systems under uncertainty this paper presents an asymptotic analysis of a hierarchical manufacturing system with machines subject to breakdown and repair the rate of change in machine states is much larger than the rate of fluctuation in demand and the rate of discounting of costs and this gives rise to a limiting problem in which the stochastic machine availability is replaced by the equilibrium mean availability the value function for the original problem converges to the value function of the limiting problem moreover the control for the original problem can be constructed from the optimal controls of the limiting problem in a way which guarantees asymptotic optimality of the value function the limiting problem is computationally more tractable and sometimes has a closed form solution

viscosity solutions of hamilton jacobi equations with unbounded nonlinear terms abstract the aim of this paper is to introduce a new definition of viscosity solutions for hamilton jacobi equations in infinite dimensions with unbounded nonlinear terms we prove existence and uniqueness results under usual assumptions regarding the bounded part of the hamiltonian

existence of solutions to the hamilton jacobi bellman equation under quadratic growth conditions abstract in this paper we show the existence of solutions with quadratic growth to hamilton jacobi bellman equations we assume that the hamiltonian has the quadratic growth both in x and p we obtain the explicit lipschitz bound of solutions in the weighted sup norm also we discuss the vanishing viscosity limit of solutions and show the existence of viscosity solutions to the resulting hamilton jacobi equation

rectifiability results for singular and conjugate points of optimal exit time problems we consider the value function v of optimal control problems with exit time under suitable assumptions through the study of the conjugate points we prove that the closure of the singular set of v is rectifiable moreover a sharper hausdorff estimate is given on the set of the conjugate nonsingular points

an explicit method for optimal control problems in this paper we have given the proof of an algorithm which transforms the hamilton jacobi bellman equation hjbe in its implicit form to a set of explicit ode s for the value function v once v is known the optimal control may be computed in feedback form from the hjbe

blow up time involved with perturbed hamilton jacobi equations in this paper we consider the evolution of positive bounded uniformly continuous data u 0 by perturbed equations like under general assumptions on u 0 existence uniqueness and regularity of the evolution u in the set are studied where the blow up function is given by the exact blow up rate of u is obtained uniqueness regularity decay at infinity of the function as well a s a representation formula for the case m 1 are also proved

twisted hilbert transforms vs kakeya sets of directions given a discrete group g and an orthogonal action g o n we study lp convergence of fourier integrals which are frequency supported on the semidirect product ro g given a unit u rn and 1 p 6 2 our main result shows that the twisted directional hilbert transform hu o idg is lp bounded iff the orbit o u is finite this is in sharp contrast with twisted riesz transforms ru o idg which are always bounded our result characterizes fourier summability in lp for this class of groups we also extend de leeuw s compactification theorem to this setting and obtain stronger estimates for functions with lacunary frequency support

groups with free regular length functions in ℤⁿ this is the first paper in a series of three where we take on the unified theory of non archimedean group actions length functions and infinite words our main goal is to show that group actions on zn trees give one a powerful tool to study groups all finitely generated groups acting freely on r trees also act freely on some zn trees but the latter ones form a much larger class the natural effectiveness of all constructions for zn actions which is not the case for r trees comes along with a robust algorithmic theory in this paper we describe the algebraic structure of finitely generated groups acting freely and regularly on zn trees and give necessary and sufficient conditions for such actions

generalized viscosity solutions for hamilton jacobi equations with time measurable hamiltonians the objective of this paper is to extend the idea of viscosity solution for nonlinear first order hamilton jacobi equations u h t x d u 0 with time continuous h to time measurable hamiltonians we are primarily motivated by the fact that control problems for which the value function is the viscosity solution of a hamilton jacobi equation should not be restric ted to continuity in time in barron and jensen 1 we were confronted with a linear hamilton jacobi equation arising in the proof of the pon tryagin which had time measurable coeflicients the existing theory of viscosity solutions did not apply to even this case moreover important models of controlled first order hamiltonjacobi equations require time measurable controls for example models of controlled traffic flow or queueing processes might be of this type we are about then to extend the definition of viscosity solution to generalized viscosity solution applying to time measurable hamiltonians in this paper we give the definition and derive the corresponding uni queness result with an implicit domain of dependence consequent the fun damental technique used in the proof is the so called blow up method which has also been used elsewhere s although not in the generality discussed here this method effectively freezes the point to x0 under consideration and essentially removes it from the problem of course t is

dissipative control system for the stochastic nonlinear h problems abstract we characterize functions satisfying a dissipative inequality associated with a stochastic control problem such a characterization is provided in terms of an upper generalized gaussian solution or a viscosity supersolution to a partial differential equation called hamilton jacobi equation h j links between upper generalized gaussian solutions and viscosity supersolutions to hamilton jacobi equation are studied finally it shows that generalized gaussian solutions is identical to viscosity solutions to hamilton jacobi equation

isolated singularities for some types of curvature equations abstract we consider the removability of isolated singularities for the curvature equations of the form h k u 0 which is determined by the k th elementary symmetric function in an n dimensional domain we prove that for 1 k n 1 isolated singularities of any viscosity solutions to the curvature equations are always removable provided the solution can be extended continuously at the singularities we also consider the class of generalized solutions and prove the removability of isolated singularities

global structure and regularity of solutions to the eikonal equation the evolutionary eikonal equation is a hamilton jacobi equation with hamiltonian h p p which is not strictly convex nor smooth the regularizing effect of hamiltonian for the eikonal equation is much weaker than that of strictly convex hamiltonians therefore leading to new phenomena in this paper we study the set of singularity points of solutions in the upper half space for c1 or c2 initial data with emphasis on the countability of connected components of the set the regularity of solutions in the complement of the set of singularity points is also obtained

quasi variational inequalities and first order hamilton jacobi equations on considere des inegalites quasi variationnelles de hamilton jacobi dans r n on introduit pour ce probleme la notion de solution viscosite

second order numerical methods for first order hamilton jacobi equations we present practical numerical methods which produce provably second order approximations for a class of stationary first order hamilton jacobi partial differential equations using probabilistic methods we derive high order asymptotic expansions for a first order method and then use those results to design second order methods we prove second order convergence for the solution and for its gradient on a subset of the domain where the solution is smooth although we limit our attention to second order schemes in principle the techniques in this paper can be extended to arbitrarily high order methods examples illustrate the rate of convergence as well as global sharp resolution of discontinuities the hamilton jacobi equations we consider correspond to deterministic optimal control problems and our rate of convergence results are valid for the value functions and for the optimal feedback controls

persistence and stability of solutions of hamilton jacobi equations abstract given a convergent sequence of hamiltonians h n and a convergent sequence of initial data g n for the first order evolutionary hamilton jacobi equation we look for conditions ensuring that the sequences u n and v n of lax solutions and hopf solutions respectively converge the convergences we deal with are variational convergences we take advantage of several recent results giving criteria for the continuity of usual operations

verification theorems for hamilton jacobi bellman equations we study an optimal control problem in bolza form and we consider the value function associated to this problem we prove two verification theorems which ensure that if a function w satisfies some suitable weak continuity assumptions and a hamilton jacobi bellman inequality outside a countably h n rectifiable set then it is lower than or equal to the value function these results can be used for optimal synthesis approach

convergence of the optimal feedback policies in a numerical method for a class of deterministic optimal control problems we consider a markov chain based numerical approximation method for a class of deterministic nonlinear optimal control problems it is known that methods of this type yield convergent approximations to the value function on the entire domain these results do not easily extend to the optimal control which need not be uniquely defined on the entire domain there are however regions of strong regularity on which the optimal control is well defined and smooth typically the union of these regions is open and dense in the domain using probabilistic methods we prove that on the regions of strong regularity the markov chain method yields a convergent sequence of approximations to the optimal feedback control the result is illustrated with several examples

asymptotic behavior of the first order obstacle problem on considere l equation aux derivees partielles d ordre 1 non lineaire max u x x u x g x du x f x 0 x ou est un sous ensemble ouvert de r d est un parametre positif g g 1 g d est un champ vectoriel sur r d du est le gradient de la fonction inconnue u et f et sont des fonctions a valeurs reelles donnees on etudie le comportement de cette equation quand 0

value functions for differential games and control systems with discontinuous terminal cost this paper deals with mayer s problem for control systems and differential games with discontinuous terminal cost there are two main results in the paper the first one says that the value function for control systems can be characterized as the unique solution in suitable sense to the hamilton jacobi bellman equation without any regularity assumptions on the terminal cost for differential games satisfying isaacs s minmax condition the second main result says that the value function is the unique solution to the hamilton jacobi isaacs equation when the terminal cost is semicontinuous this allows to prove the existence of the value under isaacs s condition this paper extends some results already well known in the continuous case

non commutative martingale inequalities we prove the analogue of the classical burkholder gundy inequalites for non commutative martingales as applications we give a characterization for an ito clifford integral to be an lp martingale via its integrand and then extend the ito clifford integral theory in l2 developed by barnett streater and wilde to lp for all 1 p we include an appendix on the non commutative analogue of the classical fefferman duality between h1 and bmo

generalized one sided estimates for solutions of hamilton jacobi equations and applications on etudie les proprietes locales des solutions de l equation de hamilton jacobi u t h t x u 0 et plus generalement de l equation u t h t x u u 0 ou t o t et x r n

wiener s lemma for twisted convolution and gabor frames as a consequence ca is invertible and bounded on all p zd for 1 p oo simultaneously in this article we study several non commutative generalizations of wiener s lemma and their application to gabor theory the paper is divided into two parts the first part sections 2 and 3 is devoted to abstract harmonic analysis and extends wiener s lemma to twisted convolution the second part section 4 is devoted to the theory of gabor frames specifically to the design of dual windows with good time frequency localization in particular we solve a conjecture of janssen feichtinger and one of us 17 18 9 these two topics appear to be completely disjoint but they are not the solution of the conjectures about gabor frames is an unexpected application of methods from non commutative harmonic analysis to application oriented mathematics it turns out that the connection between twisted convolution and the heisenberg group and the theory of symmetric group algebras are precisely the tools needed to treat the problem motivated by signal analysis to be more concrete we formulate some of our main results first and will deal with the details and the technical background later

some properties of viscosity solutions of hamilton jacobi equations abstract recently m g crandall and p l lions introduced the notion of viscosity solutions of scalar nonlinear first order partial differential equations viscosity solutions need not be differentiable anywhere and thus are not sensitive to the classical problem of the crossing of characteristics the value of this concept is established by the fact that very general existence uniqueness and continuous dependence results hold for viscosity solutions of many problems arising in fields of application the notion of a viscosity solution admits several equivalent formulations here we look more closely at two of these equivalent criteria and exhibit their virtues by both proving several new facts and reproving various known results in a simpler manner moreover by forsaking technical generality we hereby provide a more congenial introduction to this subject than the original paper author

kakeya sets and directional maximal operators in the plane we completely characterize the boundedness of planar directional maximal operators on lp more precisely if is a set of directions we show that m the maximal operator associated to line segments in the directions is unbounded on lp for all p 1

noncommutative riesz transforms a probabilistic approach for 2 p 0 in the classical setting this is certainly satisfied for the maps tt f f 1 t where t is a semigroup of measure preserving maps on a probability space in that case n l then the maps tt act on all noncommutative lp spaces lp n and in particular on the hilbert space l2 l2 n let a be the negative generator i e tt e ta for technical reasons we have to assume that there is a weakly dense algebra a n which lies in the domain of the unbounded operator a i e a a a the algebra a plays the role of smooth functions in the classical setting this assumption might be too restrictive for some classical applications but it is easily verified in the new noncommutative examples for elements x y a we may define the gradient form 2 x y a x y x a y a x y the correct definition of is the one given by sauvageot as a bilinear form with values in l1 again our arguments require stronger assumptions in fact in all our examples we will assume that the tt s are selfadjoint i e ttxy xtty and unital i e tt 1 1 then a is indeed a positive unbounded operator under these circumstances we can formulate p a meyer s problem is it true that x x 2 p c p a 1 2 x p 0 1 holds for all selfadjoint elements x a let us illustrate this question by considering the laplace operator a f f where i 2 x2 i then it is easily verified that

completely bounded multipliers of the fourier algebra of a simple lie group of real rank one the fourier algebra a g of a locally compact group g is the space of matrix coefficients of the regular representation and is the predual of the yon neumann algebra vn g generated by the regular representation of g on l 2 g a multiplier m of a g is a bounded operator on a g given by pointwise multiplication by a function on g also denoted m we say m is a completely bounded multiplier ofa g if the transposed operator on vn g is completely bounded definition below it may be possible to find a net ofa g functions m i ie i say such that mi tends to

viability property for a backward stochastic differential equation and applications to partial differential equations in the present paper we study conditions under which the solutions of a backward stochastic differential equation remains in a given set of constraints this property is the so called viability property in a separate section this condition is translated to a class of partial differential equations

cauchy dirichlet problem for first order nonlinear systems abstract let r nbe open u r mand thus the gradient matrixdu r m n we lete r m nbe compact and denote byrcoeandpcoethe rank one convex and polyconvex hull ofe respectively we show that ifrcoe pcoe and two other hypotheses named the segment property and the extreme points property and if c1 r m is such that d x e int pcoe x then there exists a dense set of u w1 r m such that formula we apply this existence theorem to some relevant examples studied in the literature as well as to problems with x u dependence

regularity results for solutions of a class of hamilton jacobi equations the regularity of the gradient of viscosity solutions of first order hamilton jacobi equations is studied under a strict convexity assumption on h t x estimates on the discontinuity set of du are derived such estimates imply that solutions of the above problem are smooth in the complement of a closed n rectifiable set in particular it follows that du belongs to the classsbv i e d2u is a measure with no cantor part

an adaptive method with rigorous error control for the hamilton jacobi equations part ii the two dimensional steady state case in this paper we devise and study an adaptive method for finding approximations to the viscosity solution of hamilton jacobi equations the method which is an extension to two space dimensions of a similar method previously proposed for one space dimension is studied in the framework of steady state hamilton jacobi equations with periodic boundary conditions it seeks numerical approximations whose l distance to the viscosity solution is no bigger than a prescribed tolerance a thorough numerical study is carried out which shows that a strict error control is achieved and that the method exhibits an optimal computational complexity which does not depend on the value of the tolerance or on the type of hamiltonian

remarks regarding the gap between continuous lipschitz and differentiable storage functions for dissipation inequalities appearing in h control abstract this paper deals with the regularity of solutions of the hamilton jacobi inequality which arises in h control it shows by explicit counterexamples that there are gaps between existence of continuous and locally lipschitz positive definite and proper solutions and between lipschitz and continuously differentiable ones on the other hand it is shown that it is always possible to smooth out solutions provided that an infinitesimal increase in gain is allowed