on pro p analogues of limit groups via extensions of centralizers we begin a study of a pro p analogue of limit groups via extensions of centralizers and call mathcal l this new class of pro p groups we show that the pro p groups of mathcal l have finite cohomological dimension type fp and non positive euler characteristic among the group theoretic properties it is proved that they are free by torsion free nilpotent and if non abelian do not have a finitely generated non trivial normal subgroup of infinite index furthermore it is shown that every 2 generated pro p group in the class mathcal l is either free pro p or abelian

subgroup separability in residually free groups we prove that the finitely presentable subgroups of residually free groups are separable and that the subgroups of type mathrm fp infty are virtual retracts we describe a uniform solution to the membership problem for finitely presentable subgroups of residually free groups

the distribution function inequality and products of toeplitz operators and hankel operators abstract in this paper we study hankel operators and toeplitz operators through a distribution function inequality on the lusin area integral function and the littlewood paley theory a sufficient condition and a necessary condition are obtained for the boundedness of the product of two hankel operators they lead to a way to approach sarason s conjecture on products of toeplitz operators and shed light on the compactness of the product of hankel operators an elementary necessary and sufficient condition for the product of two toeplitz operators to be a compact perturbation of a toeplitz operator is obtained moreover a necessary condition is given for the product of hankel operators to be in the commutator ideal of the algebra generated by the toeplitz operators with symbols in a sarason algebra

the uncertainty principle on considere l existence et la regularite des solutions d equations aux derivees partielles la construction de solutions fondamentales explicites et les valeurs propres d operateurs de schrodinger

sharp two weight inequalities for singular integrals with applications to the hilbert transform and the sarason conjecture abstract we prove two weight norm inequalities for calderon zygmund singular integrals that are sharp for the hilbert transform and for the riesz transforms in addition we give results for the dyadic square function and for commutators of singular integrals as an application we give new results for the sarason conjecture on the product of unbounded toeplitz operators on hardy spaces

logarithmic bump conditions and the two weight boundedness of calderon zygmund operators abstract we prove that if a pair of weights u v satisfies a sharp a p bump condition in the scale of all log bumps or certain loglog bumps then haar shifts map l p v into l p u with a constant quadratic in the complexity of the shift this in turn implies the two weight boundedness for all calderon zygmund operators this gives a partial answer to a long standing conjecture we also give a partial result for a related conjecture for weak type inequalities to prove our main results we combine several different approaches to these problems in particular we use many of the ideas developed to prove the a 2 conjecture as a byproduct of our work we also disprove a conjecture by muckenhoupt and wheeden on weak type inequalities for the hilbert transform this is closely related to the recent counterexamples of reguera scurry and thiele

extrapolation with weights rearrangement invariant function spaces modular inequalities and applications to singular integrals abstract we present an extrapolation theory that allows us to obtain from weighted l p inequalities on pairs of functions for p fixed and all a weights estimates for the same pairs on very general rearrangement invariant quasi banach function spaces with a weights and also modular inequalities with a weights vector valued inequalities are obtained automatically without the need of a banach valued theory this provides a method to prove very fine estimates for a variety of operators which include singular and fractional integrals and their commutators in particular we obtain weighted and vector valued extensions of the classical theorems of boyd and lorentz shimogaki the key is to develop appropriate versions of rubio de francia s algorithm

weighted norm inequalities for the continuous square function we prove new weighted norm inequalities for real variable analogues of the lusin area function we apply our results to obtain new i weighted norm inequalities for singular integral operators ii weighted sobolev inequalities iii eigenvalue estimates for degenerate schrodinger operators

products of hankel and toeplitz operators on the bergman space we consider the question for which square integrable analytic functions f and g on the unit disk the densely defined products tftg are bounded on the bergman space we prove results analogous to those obtained by the second author 17 for such toeplitz products on the hardy space we furthermore obtain similar results for hankel products hfh g where f and g are square integrable on the unit disk and for the mixed haplitz products hftg and tgh f where f and g are square integrable on the unit disk and g is analytic

invertible toeplitz products we will discuss invertibility of toeplitz products tftg for analytic f and g on the bergman space and the hardy space we will furthermore describe when these toeplitz products are fredholm

two weight extrapolation via the maximal operator abstract we give several extrapolation theorems for pairs of weights of the form w m k w and w mw w r w where w is any non negative function r 1 and m k is the k th iterate of the hardy littlewood maximal operator as an application we show that our results can be used to extend and sharpen results for square functions and singular integral operators by chang et al 1985 comment math helv 60 217 246 chanillo and wheeden 1987 indiana univ math j 36 277 294 wilson 1987 duke math j 55 879 887 1989 trans amer math soc 314 661 692 1989 illinois j math 33 361 366 and uchiyama 1995 studia math 115 135 149 in the process we prove a conjecture due to wilson

on the finite presentation of subdirect products and the nature of residually free groups we establish it virtual surjection to pairs vsp as a general criterion for the finite presentability of subdirect products of groups if gamma 1 ldots gamma n are finitely presented and s we use the vsp criterion to characterize the finitely presented residually free groups we prove that the class of such groups is recursively enumerable we describe an algorithm that given a finite presentation of a residually free group constructs a canonical embedding into a direct product of finitely many limit groups we solve the multiple conjugacy problem and membership problem for finitely presentable subgroups of residually free groups we also prove that there is an algorithm that given a finite generating set for such a subgroup will construct a finite presentation new families of subdirect products of free groups are constructed including the first examples of finitely presented subgroups that are neither rm fp infty nor of stallings bieri type

weighted norm inequalities for the maximal operator on variable lebesgue spaces abstract we prove weighted strong and weak type norm inequalities for the hardy littlewood maximal operator on the variable lebesgue space l p our results generalize both the classical weighted norm inequalities on l p and the more recent results on the boundedness of the maximal operator on variable lebesgue spaces

extrapolation from a weights and applications abstract we generalize the a p extrapolation theorem of rubio de francia to a weights in the context of muckenhoupt bases our result has several important features first it can be used to prove weak endpoint inequalities starting from strong type inequalities something which is impossible using the classical result second it provides an alternative to the technique of good inequalities for proving l p norm inequalities relating operators third it yields vector valued inequalities without having to use the theory of banach space valued operators we give a number of applications to maximal functions singular integrals potential operators commutators multilinear calderon zygmund operators and multiparameter fractional integrals in particular we give new proofs which completely avoid the good inequalities of coifman s inequality relating singular integrals and the maximal operator of the fefferman stein inequality relating the maximal operator and the sharp maximal operator and the muckenhoupt wheeden inequality relating the fractional integral operator and the fractional maximal operator

irreducible affine varieties over a free group i irreducibility of quadratic equations and nullstellensatz the object of this paper which consists of two parts is to describe irreducible varieties over free groups and to characterize finitely generated fully residually free groups we prove that any variety over a free group f can be defined by a finite number of systems of equations s 1 in triangular form where quadratic words play the role of leading terms algebraically irreducible varieties are exactly the varieties whose coordinate groups are fully residually f the crucial point of the classification of fully residually free groups is to prove that the coordinate groups of irreducible varieties are embeddable into lyndon s free z x group f since every finitely generated fully residually free group is a free factor of the coordinate group of an irreducible variety and the group f is fully residually f we obtain a characterization of finitely generated fully residually free groups as subgroups of f the group f and its subgroups have been studied extensively during the last several years in particular every finitely generated subgroup of f hence every finitely generated fully residually free group can be obtained from free abelian groups of finite rank by finitely many free products with amalgamation and hnn extensions of the type where amalgamated and associated subgroups are free abelian of finite rank in particular this implies that every finitely generated fully residually free group is finitely presented there are three parts to this paper algebraic geometry over free groups the theory of free exponential groups and makanin razborov s machinery to deal with equations over free groups 11 14 13 the algebraic geometry approach has been shown to be very usefull in dealing with equations over groups it provides necessary topological means and a method to transcribe geometric notions into pure group theoretic language following baumslag myasnikov remeslennikov 1 we use the standard algebraic geometry notions such as variety zariski topology irreducibility of varieties radicals and coordinate groups some of the ideas of the algebraic geometry approach go back to r lyndon 9 e rips j stallings 16

sharp weighted estimates for classical operators abstract we give a general method based on dyadic calderon zygmund theory to prove sharp one and two weight norm inequalities for some of the classical operators of harmonic analysis the hilbert and riesz transforms the beurling ahlfors operator the maximal singular integrals associated to these operators the dyadic square function and the vector valued maximal operator in the one weight case we prove the sharp dependence on the a p constant by finding the best value for the exponent p such that t f l p w c n t w a p p f l p w for the hilbert transform the riesz transforms and the beurling ahlfors operator the sharp value of p was found by petermichl and volberg 2007 2008 2002 47 48 49 their proofs used approximations by the dyadic haar shift operators bellman function techniques and two weight norm inequalities our proofs again depend on dyadic approximation but avoid bellman functions and two weight norm inequalities we instead use a recent result due to a lerner 2010 34 to estimate the oscillation of dyadic operators by applying this we get a straightforward proof of the sharp dependence on the a p constant for any operator that can be approximated by haar shift operators in particular we provide a unified approach for the hilbert and riesz transforms the beurling ahlfors operator and their corresponding maximal singular integrals dyadic paraproducts and haar multipliers furthermore we completely solve the open problem of sharp dependence for the dyadic square functions and vector valued hardy littlewood maximal function in the two weight case we use the very same techniques to prove sharp results in the scale of a p bump conditions for the singular integrals considered above we show they map l p v into l p u 1 p if the pair u v satisfies sup q u 1 p a q v 1 p b q where a b p and b b p are orlicz functions this condition is sharp furthermore this condition characterizes in the scale of these a p bump conditions the corresponding two weight norm inequality for the hardy littlewood maximal operator m and its dual i e m l p v l p u and m l p u 1 p l p v 1 p muckenhoupt and wheeden conjectured that these two inequalities for m are sufficient for the hilbert transform to be bounded from l p v into l p u thus in the scale of a p bump conditions we prove their conjecture we prove similar sharp two weight results for the dyadic square function and the vector valued maximal operator

endpoint estimates for commutators of singular integral operators abstract we prove endpoint estimates for commutatotrs of singular integrals with bmo functions we first show that they satisfy l log l type inequalities and then that there is a hardy space type estimate where the usual atomic hardy space is replaced by an appropriate subspace

irreducible affine varieties over a free group ii systems in triangular quasi quadratic form and description of residually free groups abstract we shall prove the conjecture of myasnikov and remeslennikov 4 which states that a finitely generated group is fully residually free every finite set of nontrivial elements has nontrivial images under some homomorphism into a free group if and only if it is embeddable in the lyndon s exponential groupfz x which is thez x completion of the free group herez x is the ring of polynomials of one variable with integer coefficients historically lyndon s attempts to solve tarski s famous problem concerning the elementary equivalence of free groups of different ranks led him to introducefz x an free group is a groupgsuch that the class of formulas true ing is the same as the class of formulas true in a nonabelian free group a finitely generated group is free if and only if it is fully residually free 22 our result gives an algebraic description of free groups we shall give an algorithm to represent a solution set of an arbitrary system of equations overfas a union of finite number of irreducible components in the zariski topology onfn the solution set for every system is contained in the solution set of a finite number of systems in triangular form with quadratic words as leading terms the possibility of such a decomposition for a solution set was conjectured by razborov in 20 and also by rips we shall give a description of systems of equations determining irreducible components using methods developed in 13 19 it is possible to find some of these methods in 18 we are thankful to e rips for attracting our attention to these techniques

subgroups of direct products of limit groups if ti tn are limit groups and s c t x x tn is of type fp q then s contains a subgroup of finite index that is itself a direct product of at most n limit groups this answers a question of sela

cubulating malnormal amalgams let g be a group splitting as a graph of groups where each vertex group acts properly on a cat 0 cube complex when g is hyperbolic relative to virtually abelian subgroups we examine conditions producing a proper action of g on a cat 0 cube complex when g is hyperbolic this action is also cocompact

automorphisms of the category of free lie algebras we prove that every automorphism of the category of free lie algebras is a semi inner automorphism this solves problem 3 9 from g mashevitzky b plotkin e plotkin electron res announc amer math soc 8 2002 1 10 for lie algebras

finding all solutions of equations in free groups and monoids with involution this paper presents a pspace algorithm which yields a finite graph of exponential size that describes the set of all solutions of equations in free groups as well as the set of all solutions of equations with rational constraints in free monoids this became possible due to the recent recompression technique while this technique was successfully applied for pure word equations without involution or rational constraints it could not be used as a black box for free groups actually the presence of an involution and rational constraints complicates the situation and some additional analysis is necessary still the technique is general enough to accommodate both extensions in the end it simplifies proofs that satisfiability of word equations is in pspace and the corresponding result for equations in free groups with rational constraints as a byproduct we can decide in pspace whether the solution set is finite

complexity volumes of splittable groups abstract using graph of groups decompositions of finitely generated groups we define euler characteristic type invariants which are non zero in many interesting classes of hyperbolic limit and csa groups including elementarily free groups and one ended torsion free hyperbolic groups whose jsj decomposition contains a maximal hanging fuchsian vertex group

a two weight inequality for the hilbert transform assuming an energy hypothesis let and be locally finite positive borel measures on r subject to the pair of weights satisfying a side condition we characterize boundedness of the hilbert transform h from l2 to l2 in terms of the a2 condition i i i x xi 2d x i i i x xi 2d x 12 c i and the two testing conditions for all intervals i in r ih 1i x 2d x c id x ih 1i x 2d x c id x the proof uses the beautiful corona argument of nazarov treil and volberg there is a range of side conditions termed energy conditions at one endpoint the energy conditions are also a consequence of the testing conditions above and at the other endpoint they are the pivotal conditions of nazarov treil and volberg we detail an example which shows that the pivotal conditions are not necessary for boundedness of the hilbert transform

algebraic geometry over groups i algebraic sets and ideal theory abstract the object of this paper which is the first in a series of three is to lay the foundations of the theory of ideals and algebraic sets over groups

oe and w superrigidity results for actions by surface braid groups we show that several important normal subgroups of the mapping class group of a surface satisfy the following property any free ergodic probability measure preserving action y x is stablyoe superrigid these include the central quotients of most surface braid groups and most torelli groups and johnson kernels in addition we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice embeddings using these results in combination with previous results from cik13 we deduce that any free ergodic probabilitymeasure preserving action of almost any surface braid group is stablyw superrigid i e it can be completely reconstructed from its vonneumann algebra

on systems of equations over free products of groups using an analogue of the makanin razborov diagrams we give a description of the solution set of systems of equations over an equationally noetherian free product of groups g equivalently we give a parametrisation of the set hom h g of all homomorphisms from a finitely generated group h to g furthermore we show that every algebraic set over g can be decomposed as a union of finitely many images of algebraic sets of ntq systems if the universal horn theory of g the theory of quasi identities is decidable then our constructions are effective 2010 elsevier inc

the equation xp yq zr and groups that act freely on l trees let g be a finitely generated group that acts freely on a lambda tree where lambda is an ordered abelian group and let x y z be elements in g we show that if x p y q z r with integers p q r geq 4 then x y and z commute