the h calculus and sums of closed operators we develop a very general operator valued functional calcu lus for operators with an h 1 calculus we then apply this to the joint functional calculus of two commuting sectorial operators when one has an h 1 calculus using this we prove theorem of dore venni type on sums of commuting sectorial operators and apply our results to the problem of lp maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of ba nach spaces are essential here in the final section we exploit the special banach space structure of l1 spaces and c k spaces to obtain some more detailed results in this setting

quantization and dynamisation of trace poisson brackets the quantization problem for the trace bracket algebra derived from double poisson brackets is discussed we obtain a generalization of the boundary ybe or so called abcd algebra for the quantization of quadratic trace brackets a dynamical deformation is proposed on the lines of gervais neveu felder dynamical quantum algebras

the h infty calculus and sums of closed operators we develop a very general operator valued functional calculus for operators with an h infty calculus we then apply this to the joint functional calculus of two sectorial operators when one has an h infty calculus using this we prove theorem of dore venni type on sums of sectorial operators and apply our results to the problem of l p maximal regularity our main assumption is the r boundedness of certain sets of operators and therefore methods from the geometry of banach spaces are essential here in the final section we exploit the special banach space structure of l 1 spaces and c k spaces to obtain some more detailed results in this setting

poisson structures on moduli spaces of representations abstract we show that a poisson structure can be induced on the affine moduli space of semisimple representations of an associative algebra from a suitable lie algebra structure on the zeroth hochschild homology of the algebra in particular this applies to necklace lie algebra for path algebras of doubled quivers and preprojective algebras we call such structures h 0 poisson structures and show that they are well behaved for azumaya algebras and under morita equivalence

boundary layers on sobolev besov spaces and poisson s equation for the laplacian in lipschitz domains we study inhomogeneous boundary value problems for the laplacian in arbitrary lipschitz domains with data in sobolev besov spaces as such this is a natural continuation of work in jerison and kenig j funct anal 1995 16 219 where the inhomogeneous dirichlet problem is treated via harmonic measure techniques the novelty of our approach resides in the systematic use of boundary integral methods in this regard the key results are establishing the invertibility of the classical layer potential operators on scales of sobolev besov spaces on lipschitz boundaries for optimal ranges of indices applications tolp based helmholtz type decompositions of vector fields in lipschitz domains are also presented

double poisson algebras in this paper we develop poisson geometry for non commutative algebras this generalizes the bi symplectic geometry which was recently and independently introduced by crawley boevey etingof and ginzburg our quasi poisson brackets induce classical quasi poisson brackets on representation spaces as an application we show that the moduli spaces of representations associated to the deformed multiplicative preprojective algebras recently introduced by crawley boevey and shaw carry a natural poisson structure

resolvent estimates in l p for the stokes operator in lipschitz domains we establish the l p resolvent estimates for the stokes operator in lipschitz domains in mathbb r d d geqq 3 for frac 1 p frac 1 2 frac 1 2d varepsilon the result implies that the stokes operator in a three dimensional lipschitz domain generates a bounded analytic semigroup in l p for 3 2 e p 3 e this gives an affirmative answer to a conjecture of m taylor progr nonlinear differential equations appl vol 42 pp 320 334

r matrix quantization of the elliptic ruijsenaars schneider model it is shown that the classical l operator algebra of the elliptic ruijsenaars schneider model can be realized as a subalgebra of the algebra of functions on the cotangent bundle over the centrally extended current group in two dimensions it is governed by two dynamic and bar r matrices satisfying a closed system of equations the corresponding quantum r and bar r matrices are found as solutions to quantum analogues of these equations we present the quantum l operator algebra and show that the system of equations for r and bar r arises as the compatibility condition for this algebra it turns out that the r matrix is twist equivalent to the felder elliptic rf matrix with bar r playing the role of the twist the simplest representation of the quantum l operator algebra corresponding to the elliptic ruijsenaars schneider model is obtained the connection of the quantum l operator algebra to the fundamental relation rb ll llrb with the belavin elliptic r matrix is established as a by product of our construction we find a new n parameter elliptic solution to the classical yang baxter equation

solutions for semilinear parabolic equations in lp and regularity of weak solutions of the navier stokes system abstract we construct a unique local regular solution in l q 0 t l p for a class of semilinear parabolic equations which includes the semilinear heat equation u t u u u 0 and the navier stokes system here p and q are so chosen that the norm of l q 0 t l p is dimensionless or scaling invariant the main relation between p and q for the semilinear heat equation is 1 q 1 p 1 q n 2 provided that initial data are in l r with r n 2 1 where n is the space dimension applying our regular solutions to the navier stokes system we show that the k 2 dimensional hausdorff measure of possible time singularities of a turbulent solution is zero if the turbulent solution is in l q 0 t l p where k 2 q nq p p n 1 q we show moreover that a turbulent solution is regular if it is in c 0 t l n

on the associative analog of lie bialgebras abstract an infinitesimal bialgebra is at the same time an associative algebra and coalgebra in such a way that the comultiplication is a derivation this paper continues the basic study of these objects with emphasis on the connections with the theory of lie bialgebras it is shown that non degenerate antisymmetric solutions of the associative yang baxter equation are in one to one correspondence with non degenerate cyclic 2 cocycles the associative and classical yang baxter equations are compared it is studied when a solution to the first is also a solution to the second necessary and sufficient conditions for obtaining a lie bialgebra from an infinitesimal one are found in terms of a canonical map that behaves simultaneously as a commutator and a cocommutator the class of balanced infinitesimal bialgebras is introduced they have an associated lie bialgebra several well known lie bialgebras are shown to arise in this way the construction of drinfeld s double from earlier work by the author in press in contemp math amer math soc providence for arbitrary infinitesimal bialgebras is complemented with the construction of the balanced double for balanced ones this construction commutes with the passage from balanced infinitesimal bialgebras to lie bialgebras

double poisson cohomology of path algebras of quivers abstract in this note we give a description of the graded lie algebra of double derivations of a path algebra as a graded version of the necklace lie algebra equipped with the kontsevich bracket furthermore we formally introduce the notion of double poisson lichnerowicz cohomology for double poisson algebras and give some elementary properties we introduce the notion of a linear double poisson tensor on a quiver and show that it induces the structure of a finite dimensional algebra on the vector spaces v v generated by the loops in the vertex v we show that the hochschild cohomology of the associative algebra can be recovered from the double poisson cohomology then we use the description of the graded necklace lie algebra to determine the low dimensional double poisson lichnerowicz cohomology groups for three types of linear and nonlinear double poisson brackets on the free algebra c x y this allows us to develop some useful techniques for the computation of the double poisson lichnerowicz cohomology

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

the regularity of the stokes operator and the fujita kato approach to the navier stokes initial value problem in lipschitz domains abstract we study the regularity of the navier stokes equations in arbitrary lipschitz domains

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

l q l r estimates for solutions of the nonstationary stokes equations in an exterior domain and the navier stokes initial value problems inl q spaces

the interaction potential of the stationary measure of a high noise spinflip process a continuous time spinflip process started from an arbitrary state is proven to yield a gibbs state after any time t 0 if the noise is high enough this is done by approximating the process by discrete time processes the relative energies of those gibbs states are constructed explicitly via a high noise cluster expansion

on the mathrm l p theory of the navier stokes equations on three dimensional bounded lipschitz domains on a bounded lipschitz domain omega subset mathbb r d d ge 3 we continue the study of shen arch ration mech anal 205 2 395 424 2012 and of kunstmann and weis j evol equ 387 409 2016 of the stokes operator on mathrm l p sigma omega we employ their results in order to determine the domain of the square root of the stokes operator as the space mathrm w 1 p 0 sigma omega for frac 1 p frac 1 2 frac 1 2d varepsilon and some varepsilon 0 this characterization provides gradient estimates as well as mathrm l p mathrm l q mapping properties of the corresponding semigroup in the three dimensional case this provides a means to show the existence of solutions to the navier stokes equations in the critical space mathrm l infty 0 infty mathrm l 3 sigma omega whenever the initial velocity is small in the mathrm l 3 norm finally we present a different approach to the mathrm l p theory of the navier stokes equations by employing the maximal regularity proven by kunstmann and weis j evol equ 387 409 2016

commuting quantum traces for quadratic algebras consistent tensor products on auxiliary spaces hereafter denoted fusion procedures and commuting transfer matrices are defined for general quadratic algebras nondynamical and dynamical inspired by results on reflection algebras applications of these procedures then yield integer indexed families of commuting hamiltonians

the navier stokes equations in exterior lipschitz domains lp theory abstract we show that the stokes operator defined on l p for an exterior lipschitz domain r n n 3 admits maximal regularity provided that p satisfies 1 p 1 2 1 2 n e for some e 0 in particular we prove that the negative of the stokes operator generates a bounded analytic semigroup on l p for such p in addition l p l q mapping properties of the stokes semigroup and its gradient with optimal decay estimates are obtained this enables us to prove the existence of mild solutions to the navier stokes equations in the critical space l 0 t l 3 locally in time and globally in time for small initial data