vertical square functions and other operators associated with an elliptic operator abstract we study the vertical and conical square functions defined via elliptic operators in divergence form in general vertical and conical square functions are equivalent operators just in l 2 but when this square functions are defined through the heat or poisson semigroups that arise from an elliptic operator we are able to find open intervals containing 2 where the equivalence holds the intervals in question depend ultimately on the range where the semigroup is uniformly bounded or has off diagonal estimates as a consequence we obtain new boundedness results for some square functions besides we consider a non tangential maximal function associated with the poisson semigroup and extend the known range where that operator is bounded our methods are based on the use of extrapolation for muckenhoupt weights and change of angle estimates all our results are obtained in the general setting of a degenerate elliptic operator where the degeneracy is given by an a 2 weight in weighted lebesgue spaces of course they are valid in the unweighted and or non degenerate situations which can be seen as special cases and they provide new results even in those particular settings we also consider the square root of a degenerate elliptic operator in divergence form l w and improve the lower bound of the interval where this operator is known to be bounded on l p v d w finally we give unweighted boundedness results for the degenerate operators under consideration

uniform boundedness of conditional gauge and schrodinger equations we prove that for a bounded domaind rn withc2 boundary and q in k n loc n geqq 3 if e x exp int limits 0 tau d q x t dt mathop ddag limits infty ind then mathop sup limits mathop x in d limits z in partial d e z x exp int limits 0 tau d q x t dt infty xt brownian motion

conditional gauge and potential theory for the schrodinger operator on etend le theoreme de jauge conditionnelle a des operateurs plus generaux et a des domaines moins reguliers on obtient des resultats de theorie du potentiel pour l equation de schrodinger

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

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

the heat equation with rough boundary conditions and holomorphic functional calculus abstract in this paper we consider the laplace operator with dirichlet boundary conditions on a smooth domain we prove that it has a bounded h calculus on weighted l p spaces for power weights which fall outside the classical class of a p weights furthermore we characterize the domain of the operator and derive several consequences on elliptic and parabolic regularity in particular we obtain a new maximal regularity result for the heat equation with rough inhomogeneous boundary data

absolutely trianalytic tori in the generalized kummer variety we prove that a generic complex deformation of a generalized kummer variety contains no complex analytic tori

lp independence of spectral bounds of feynman kac semigroups by continuous additive functionals abstract we establish conditions for the l p independence of spectral bounds of feynman kac semigroup by continuous additive functionals whose revuz measures are smooth measures of kato class having non negative order green tightness our continuous additive functionals do not necessarily admit bounded variation in general examples of cauchy principal value and hilbert transform of brownian local time and for relativistic symmetric stable processes are presented

general gauge and conditional gauge theorems general gauge and conditional gauge theorems are established for a large class of not necessarily symmetric strong markov processes including brownian motions with singular drifts and symmetric stable processes furthermore new classes of functions are introduced under which the general gauge and conditional gauge theorems hold these classes are larger than the classical kato class when the process is brownian motion in a bounded c 1 1 domain

weighted norm inequalities gaussian bounds and sharp spectral multipliers abstract let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type assume that l generates a holomorphic semigroup e t l whose kernels p t x y have gaussian upper bounds but there is no assumption on the regularity in variables x and y in this article we study weighted l p norm inequalities for spectral multipliers of l we show that sharp weighted hormander type spectral multiplier theorems follow from gaussian heat kernel bounds and appropriate l 2 estimates of the kernels of the spectral multipliers these results are applicable to spectral multipliers for large classes of operators including laplace operators acting on lie groups of polynomial growth or irregular non doubling domains of euclidean spaces elliptic operators on compact manifolds and schrodinger operators with non negative potentials

on maximal regularity of type lp lq under minimal assumptions for elliptic non divergence operators we prove maximal regularity of type l p l q for operators in non divergence form with complex valued measurable coefficients on r n for a certain range of q which depends on dimension and the order of the operators this is done under the sole assumption that they generate an analytic semigroup in l q thus we give for this class of operators and this range of q a positive answer to brezis question whether generation of an analytic semigroup entails maximal l p regularity for other values of q we give several additional assumptions the proof relies on a result on maximal regularity under the assumption of suitable off diagonal bounds due to s blunck and the author which we improve here these off diagonal bounds are obtained by a modification of davies technique suited to cope with operators in non divergence form and they also imply new results on the scale of l q spaces in which a given operator generates an analytic semigroup we thus obtain a completely new approach to maximal l p regularity for operators of this kind in particular for those whose highest order coefficients belong to vmo moreover we obtain extensions of recent results due to kim and krylov for operators with measurable coefficients depending on one coordinate

gaugeability and conditional gaugeability new kato classes are introduced for general transient borel right processes for which gauge and conditional gauge theorems hold these new classes are the genuine extensions of the green tight measures in the classical brownian motion case however the main focus of this paper is on establishing various equivalent conditions and consequences of gaugeability and conditional gaugeability we show that gaugeability conditional gaugeability and the subcriticality for the associated schrodinger operators are equivalent for transient borel right processes with strong duals analytic characterizations of gaugeability and conditional gaugeability are given for general symmetric markov processes these analytic characterizations are very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases connections with the positivity of the spectral radii of the associated schrodinger operators are also established

topological invariants of o grady s six dimensional irreducible symplectic variety we study o grady examples of irreducible symplectic varieties we establish that both of them can be deformed into lagrangian fibrations we analyze in detail the topology of the six dimensional example in particular we compute its euler characteristic and determine its beauville form

conditional gauge with unbounded potential let b be a ball in r d and q be a borel function on it we prove that if mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaacawg4bgaeyici4sabmoqayaaraaabeaa ieaakiaa bcadawdrbqaamaalaaabagaaiifaiaadghacaggoagaam yeaiaacmcacagg8baabagaaiifaiaadihacqghsislcawg5bgaaiif amaacaaaleqabagaamizaiabgkhitiaaikdaaaaaaogaa8hiagqaci aa rgacagf5baaleaacawgcbaabeqdcqghrii8aaaa 4fb6 mathop sup limits x in bar b int limits b frac q y x y d 2 dy is sma11 enough then mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0xe9gqpe0 rq0dxdd9vqaq jfrvkfhe9pgea0dxdar jb9hs0dxdbpyxe9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaamaaxababagaci 4caiaacwhacaggwbaaleaadawfqaqaaiaadihacqghiiizcawgcbaa meaacawg6bgaeyici4saeyoaiyraamoqaaqabaaaleqaagqaaogaa8 hiagqaciaa veadaqhaawcbagaa4neaaqaaiaa hhaaagcdawadaqa aigacwgacagg4bgaaiicaiaa bcadawdxbqaaiaa fhacawfoagaa4 heamaabaaaleaacagf0baabeaakiaacmcacawgkbgaamidaawcbaga a8hmaaqaaiaadshadawgaaadbagaamoqaaqabaaaniabguiiydaaki aawufacagldbaacawfgagaa8hpaiaa bcacawfragaa8hiaiabg6hi lcaa 5a6b mathop sup limits mathop x in b limits z in partial b e z x left exp int limits 0 t b q x t dt right infty this paper gives a new proof of one of the two main results by aizenman and simon in 1 by a simple and elementary method a basic theorem in chung and rao 2 is extended to the class of q treated in 1

weighted lp estimates for powers of selfadjoint operators abstract we prove l p and weighted l p estimates for bounded functions of a selfadjoint operator satisfying a pointwise gaussian estimate for its heat kernel as an application we obtain weighted estimates for fractional powers of an electromagnetic schrodinger operator with singular coefficients the proofs are based on a modification of techniques due to hebisch 1990 17 and auscher and martell 2006 4

subcriticality and gaugeability of the schrodinger operator we investigate a schrodinger operator 2 v in r d d 3 with a potential v in the class k d satisfying a similar kato condition at infinity and prove an equivalence theorem connecting various conditions on subcriticality strong positivity and gaugeability of the operator

lp norms of non critical schrodinger semigroups we consider schrodinger semigroups e th h v on rn with v c x 2 as x 0 c 12 n 2 2 with h 0 we determine the exact power law divergence of e th p p and of some e th q p as maps from lp to lq the results are expressed most naturally in terms of the power for which there exists a positive resonance such that h 0 x x

the structure of reverse holder classes on metric measure spaces abstract this work extends some euclidean results on the structure of the reverse holder classes to metric measure spaces with a doubling measure satisfying an annular decay condition the equivalence between the variants used to define the muckenhoupt class a and some standard correspondences between reverse holder classes and muckenhoupt classes are utilized it is established that the logarithm of the reverse holder classes is an open and convex subset of the space b m o and that the logarithm of the intersection of the reverse holder classes is a closed and convex subset of the space b m o as an intermediary result we also establish that the intersection of the reverse holder classes consists precisely of the pointwise multipliers of the muckenhoupt class a

weighted norm inequalities off diagonal estimates and elliptic operators part i general operator theory and weights this is the first part of a series of four articles in this work we are interested in weighted norm estimates we put the emphasis on two results of different nature one is based on a good inequality with two parameters and the other uses calderon zygmund decomposition these results apply well to singular non integral operators and their commutators with bounded mean oscillation functions singular means that they are of order 0 non integral that they do not have an integral representation by a kernel with size estimates even rough so that they may not be bounded on all lp spaces for 1 p 8 pointwise estimates are then replaced by appropriate localized lp lq estimates we obtain weighted lp estimates for a range of p that is different from 1 8 and isolate the right class of weights in particular we prove an extrapolation theorem a la rubio de francia for such a class and thus vector valued estimates

l p independence of spectral radius for generalized feynman kac semigroups under mild conditions on measures used in the perturbation we establish the l p independence of spectral radius for generalized feynman kac semigroups without assuming the irreducibility and the boundedness of the function appeared in the continuous additive functionals locally of zero energy in the framework of symmetric markov processes these results are obtained by using the gaugeability approach developed by the first named author as well as the recent progress on the irreducible decomposition for markov processes proved by the third author and on the analytic characterizations of gaugeability for generalized feynman kac functionals developed by the second and third authors

sharp decay estimates of lq norms for nonnegative schrodinger heat semigroups abstract let h v be a nonnegative schrodinger operator on l 2 r n where n 3 and v is a radially symmetric nonpositive function in r n decaying quadratically at the space infinity for any 1 p q we denote by e t h q p the operator norm of the schrodinger heat semigroup e t h from l p r n to l q r n in this paper under suitable conditions on v we give the exact and optimal decay rates of e t h q p as t for all 1 p q the decay rates of e t h q p depend on whether the operator h is subcritical or critical and on the behavior of the positive harmonic function for the operator h

lp norms of nonnegative schrodinger heat semigroup and the large time behavior of hot spots this paper is concerned with the cauchy problem for the heat equation with a potential p tu u v x uin rn 0 u x 0 x in rn where t t n 3 l2 rn and v v x is a smooth nonpositive and radially symmetric function having quadratic decay at the space infinity in this paper we assume that the schrodinger operator h v is nonnegative on l2 rn and give the exact power decay rates of lq norm q 2 of the solution e th of p as t furthermore we study the large time behavior of the solution of p and its hot spots

positive solutions and large time behaviors of schrodinger semigroups simon s problem let h v where v is a multiplication operator by a real valued function v x on rn which is uniformly holder continuous and 1 x 2 2 v x l rn for some 4 the relationship between existence of positive solutions with growth conditions of hg 0 and asymptotic behaviors as t of e th is established using it b simon s problem for h on r2 is solved

martin boundaries of elliptic skew products semismall perturbations and fundamental solutions of parabolic equations abstract we consider positive solutions of elliptic partial differential equations on non compact domains of riemannian manifolds we explicitly determine martin boundaries and martin kernels for a class of elliptic equations in skew product form by exploiting and developing perturbation theory for elliptic equations and short long time estimates for fundamental solutions of parabolic equations

riesz transforms fractional power and functional calculus of schrodinger operators on weighted lp spaces abstract we consider the schrodinger operator a v v on l p r n w d x where n 3 and w is a weight in some muckenhoupt class we study the boundedness of riesz transform type operators a 1 2 and v 1 2 a 1 2 on l p r n w d x our result extends the one of bui 2010 14 to signed potentials and treat the case where p 2 it also gives a weighted version of our earlier results assaad 2011 1 assaad and ouhabaz 2012 2 and of the result auscher and ben ali 2007 4 to weighted lebesgue spaces we study also the boundedness from l p r n w p d x to l q r n w q d x of the fractional power a 2 and the l p r n w d x boundedness of the h functional calculus of a

riesz transforms associated to schrodinger operators on weighted hardy spaces let w be some ap weight and enjoy reverse holder inequality and let l v be a schrodinger operator on rn where v lloc1 rn is a non negative function on rn in this article we introduce weighted hardy spaces hl w1 rn associated to l in terms of the area function characterization and prove their atomic characters we show that the riesz transform l 1 2 associated to l is bounded on lwp rn for 1 p 2 and bounded from hl w1 rn to the classical weighted hardy space hw1 rn

self improving properties of john nirenberg and poincare inequalities on spaces of homogeneous type abstract we give a condition which ensures that if one inequality of sobolev poincare type is valid then other stronger inequalities of a similar type also hold including weighted versions our main result includes many previously known results as special cases we carry out the analysis in the context of spaces of homogeneous type but the main result is new even in the usual euclidean setting

regularity of solutions to degenerate p laplacian equations abstract we prove regularity results for solutions of the equation div a x u x u p 2 2 a x u 0 1 p where x x 1 x m is a family of vector fields satisfying hormander s condition and a is an m m symmetric matrix that satisfies degenerate ellipticity conditions if the degeneracy is of the form w x 2 p 2 a x w x 2 p 2 w a p then we show that solutions are locally holder continuous if the degeneracy is of the form k x 2 p 2 a x k x 2 p 2 k a p r h where depends on the homogeneous dimension then the solutions are continuous almost everywhere and we give examples to show that this is the best result possible we give an application to maps of finite distortion

green function for schrodinger operator and conditioned feynman kac gauge consider the schrodinger operator h 2 q in a bounded c1 1 domain d with q klocd d q satisfies the condition sup spec 2 q d v x y g x y e y x exp 0 t q x t d t x y d where xt is the conditioned brownian motion starting at x and ending at y with the life time t and v g can be extended to a continuous and positive function on d d specifically we give the comparison theorem there exists a constant c c d q 0 such that 1 c g x y v x y c g x y x y d another result is that v nz x z x d z d exists and represents the poisson kernel for the dirichlet problem corresponding to schrodinger operator h

extrapolation of weights revisited new proofs and sharp bounds abstract we use an appropriate factorization of the a p weights to give another proof of the extrapolation theorem of rubio de francia it provides sharp bounds in terms of the a p constant of the weights then we extend the result to more general settings including off diagonal and partial range extrapolation among the applications we prove by iteration a multivariable extrapolation theorem and give a sharp bound for calderon zygmund operators on l p w for weights in a q q p

fractional integrals on weighted and spaces we characterize the pairs of doubling weights u v on rn such that iiwiihj ch hj 0 0 is the fractional integral operator we also consider the behavior of an associated maximal function applications of the results to sobolev inequalities in weighted lp spaces are given 1 introduction and notation in this paper we will characterize the pairs of doubling weights it v on rn such that pafwm 0 is the fractional integral operator we shall also give applications of these results to sobolev inequalities in weighted lp spaces a weight function u is said to belong to ds p 1 if u tq 1 and every cube q c rn where u q denotes the u measure of q and tq is the cube with the same center as q but with t times the edgelength we write dx j idu analogously u g rdv v 0 if u tq ctnuu q for every t 1 and every cube q it is not hard to see that if u g d then u g rdv for some v 0 we write u rhr r 1 if 1 1 r 1 rprr u x t dx i x dx 0 and e s then we denote as usual f x t f 0t x where t x t n x t and n f x sup f y t vfithta x y t yern t 0 x y 0

feynman kac functionals and positive solutions of 1 2δu qu 0 let d be a bounded c 2 domain in d and let q be a bounded borel function in d for x d and z d suppose x t under the law p x z is brownian motion in d starting at x and conditioned to converge to z let be the lifetime of x t we show that if the quantity e x z left exp left int limits 0 t q x s ds right right is finite for one x d and one z d then this quantity remains bounded as x varies over d and z varies over d this may be considered one quantitative expression of the qualitative statement that no matter where brownian motion in d eventually hits d it goes all over d before it gets there we apply this result to show that if the equation 1 2 u qu 0 admits a non negative solution in d which is strictly positive on a subset of d of positive harmonic measure then for any non negative bounded borel function f on d it admits a unique bounded solution u satisfying u f on d and this solution u is non negative

the solution of the kato square root problem for second order elliptic operators on rn we prove the kato conjecture for elliptic operators on jfin more precisely we establish that the domain of the square root of a uniformly complex elliptic operator l div av with bounded measurable coefficients in ietn is the

lp independence of spectral bounds of schrodinger type semigroups let l be an m symmetric markov generator and a signed measure in the kato class we consider a schrodinger type operator h l on lp m we prove that under certain conditions for the markov semigroup generated by l and the potential the lp spectral bound of h is independent of p if and only if the l2 spectral bound is non positive

on positivity criticality and the spectral radius of the shuttle operator for elliptic operators in this paper we study the shuttle operator for a second order linear elliptic operator p on a noncompact manifold x zhao has introduced and studied the shuttle operator and its relation to the theory of positive solutions in the case of small perturbations of the laplacian in ir n 3 zhao was motivated by works of chung and varadhan which consider one dimensional schrodinger operators the main purpose of the paper is to extend the above studies we prove that in the general case the spectral radius of the shuttle operator is strictly less equal or strictly greater than 1 if and only if the operator p is respectively subcritical critical or supercritical in x we demonstrate the usefulness of the characterization for proving theorems our approach is purely analytic this research was partially supported by the israel science foundation administered by the israel academy of sciences and humanities and by the fund for the promotion of research at the technion

criticality and the λ property for elliptic equations abstract let l denote a nonnegative second order elliptic expression defined on c 0 and let l be an associated self adjoint operator l is said to have the property in iff l for some fixed positive function and any c 0 l is termed subcritical iff a generalized green s function may be defined for l in various applications of these and closely related concepts may be found in the literature it is the main purpose of this article to show that l is subcritical iff l has the property several corollaries and an example are given as consequences of this equivalence

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

quantitative estimates and extrapolation for multilinear weight classes in this paper we prove a quantitative multilinear limited range extrapolation theorem which allows us to extrapolate from weighted estimates that include the cases where some of the exponents are infinite this extends the recent extrapolation result of li martell and ombrosi we also obtain vector valued estimates including ell infty spaces and in particular we are able to reprove all the vector valued bounds for the bilinear hilbert transform obtained through the helicoidal method of benea and muscalu moreover our result is quantitative and in particular allows us to extend quantitative estimates obtained from sparse domination in the banach space setting to the quasi banach space setting our proof does not rely on any off diagonal extrapolation results and we develop a multilinear version of the rubio de francia algorithm adapted to the multisublinear hardy littlewood maximal operator as a corollary we obtain multilinear extrapolation results for some upper and lower endpoints estimates in weak type and mathrm bmo spaces

periodic homotopy groups of exceptional lie groups torsion free cases the v 1 periodic homotopy groups v 1 751 x p are computed explicitly for all pairs x p where x is an exceptional lie group whose integral homology has no p torsion this yields new lower bounds for p exponents of actual homotopy groups of these spaces delicate calculations with the unstable novikov spectral sequence are required in the proof

large time behavior of the lp norm of schrodinger semigroups abstract letv 0 v c0 rv with v 3 be such that h 1 2 v 0 but for any e 0 1 2 1 e v is not positive we determine the exact rate of divergence of the norm of e th as a map from l to l a number of related problems are discussed

classes of weights related to schrodinger operators abstract in this work we obtain boundedness on weighted lebesgue spaces on r d of the semi group maximal function riesz transforms fractional integrals and g function associated to the schrodinger operator v where v satisfies a reverse holder inequality with exponent greater than d 2 we consider new classes of weights that locally behave as muckenhoupt s weights and actually include them the notion of locality is defined by means of the critical radius function of the potential v given in shen 1995 8

weighted estimates for powers and smoothing estimates of schrodinger operators with inverse square potentials abstract let l a be a schrodinger operator with inverse square potential a x 2 on r d d 3 the main aim of this paper is to prove weighted estimates for fractional powers of l a the proof is based on weighted hardy inequalities and weighted inequalities for square functions associated to l a as an application we obtain smoothing estimates regarding the propagator e i t l a

criticality and ground states for second order elliptic equations abstract let p be a second order elliptic operator in a domain r n suppose that p admits a positive minimal green function g p in it is the main purpose of this paper to prove that if w c is a small perturbation of p in then max w x 0 min w x 0 t u u g p outside a compact set in

weighted norm inequalities off diagonal estimates and elliptic operators part iii harmonic analysis of elliptic operators this is the third part of a series of four articles on weighted norm in equalities o diagonal estimates and elliptic operators for l in some class of elliptic operators we study weighted norm l p inequalities for singular non integral opera tors arising from l those are the operators l for bounded holomorphic functions the riesz transforms rl 1 2 or 1 2 l 1 2 and its inverse l 1 2 1 2 some quadratic functionals gl and gl of littlewood paley stein type and also some vector valued inequalities such as the ones involved for maximal l p regularity for each we obtain sharp or nearly sharp ranges of p using the general theory for bound edness of part i and the o diagonal estimates of part ii we also obtain commutator results with bmo functions

lp self improvement of generalized poincare inequalities in spaces of homogeneous type abstract in this paper we study self improving properties in the scale of lebesgue spaces of generalized poincare inequalities in spaces of homogeneous type in contrast with the classical situation the oscillations involve approximation of the identities or semigroups whose kernels decay fast enough and the resulting estimates take into account their lack of localization the techniques used do not involve any classical poincare or sobolev poincare inequalities and therefore they can be used in general settings where these estimates do not hold or are unknown we apply our results to the case of riemannian manifolds with doubling volume form and assuming gaussian upper bounds for the heat kernel of the semigroup e t with being the laplace beltrami operator we obtain generalized poincare inequalities with oscillations that involve the semigroup e t and with right hand sides containing either or 1 2

differentiability of spectral functions for symmetric alpha stable processes let be a signed radon measure in the kato class and define a schrodinger type operator h 1 2 2 on r d we show that its spectral bound c inf h is differentiable if d 2 and is green tight

structure of positive solutions to δ v u 0 in rn on etudie la structure de toutes les solutions positives d une equation de schrodinger stationnaire ou le potentiel v est suppose etre une fonction a valeur reelle appartenant a l p loc r n avec p n 2 pour n 2 et p 1 pour n 1 on considere une formule de representation explicite pour des solutions positives leurs asymptotes a l infini et le probleme de dirichlet avec une condition aux limites a l infini

conditional gaugeability and subcriticality of generalized schrodinger operators abstract we obtain a necessary and sufficient condition for conditional gaugeability and show the equivalence between conditional gaugeability and subcriticality of generalized schrodinger type operators we apply the condition to concrete examples

analytic characterization of conditional gaugeability for non local feynman kac transforms abstract an analytic characterization of gaugeability and conditional gaugeability is given for non local or discontinuous feynman kac transforms of general symmetric markov processes this analytic characterization is very useful in determining whether a process perturbed by a potential is gaugeable or conditionally gaugeable in concrete cases

drift transforms and green function estimates for discontinuous processes in this paper we consider girsanov transforms of pure jump type for discontinuous markov processes we show that under some quite natural conditions the green functions of the girsanov transformed process are comparable to those of the original process as an application of the general results the drift transform of symmetric stable processes is studied in detail in particular we show that the relativistic a stable process in a bounded c 1 1 smooth open set d can be obtained from symmetric a stable process in d through a combination of a pure jump girsanov transform and a feynman kac transform from this we deduce that the green functions for these two processes in d are comparable

tent space boundedness via extrapolation we study the action of operators on tent spaces such as maximal operators calderon zygmund operators riesz potentials we also consider singular non integral operators we obtain boundedness as an application of extrapolation methods in the banach range in the non banach range boundedness results for calderon zygmund operators follow by using an appropriate atomic theory we end with some consequences on amalgam spaces

the atomic decomposition of weighted hardy spaces associated to self adjoint operators on product spaces abstract let l be a non negative self adjoint operator acting on l 2 r n satisfying a pointwise gaussian estimate for its heat kernel let w be an a p weight on r n r n 1 p in this article we obtain a weighted q atomic decomposition with q p for the weighted hardy space h l w 1 r n r n associated to l

singular symplectic moduli spaces moduli spaces of semistable sheaves on a k3 or abelian surface with respect to a general ample divisor are shown to be locally factorial with the exception of symmetric products of a k3 or abelian surface and the class of moduli spaces found by o grady consequently since singular moduli space that do not belong to these exceptional cases have singularities in codimension 4 they do no admit projective symplectic resolutions

precise asymptotics of small eigenvalues of reversible diffusions in the metastable regime we investigate the close connection between metastability of the reversible diffusion process x defined by the stochastic differential equation d xt f xt d t 2e d wt e 0 and the spectrum near zero of its generator l f where f d and w denotes brownian motion on d for generic f to each local minimum of f there corresponds a metastable state we prove that the distribution of its rescaled relaxation time converges to the exponential distribution as 0 with optimal and uniform error estimates each metastable state can be viewed as an eigenstate of l with eigenvalue which converges to zero exponentially fast in 1 modulo errors of exponentially small order in 1 this eigenvalue is given as the inverse of the expected metastable relaxation time the eigenstate is highly concentrated in the basin of attraction of the corresponding trap

boundedness results for commutators with bmo functions via weighted estimates a comprehensive approach we present a unified method to obtain unweighted and weighted estimates of linear and multilinear commutators with bmo functions that is amenable to a plethora of operators and functional settings our approach elaborates on a commonly used cauchy integral trick recovering many known results but yielding also numerous new ones in particular we solve a problem about the boundedness of the commutators of the bilinear hilbert transform with functions in bmo

on critical and subcritical sturm liouville operators we study critical and subcritical sturm liouville operators k 1 ddx p ddx v in l2 a b k dx a b r defined as friedrichs extensions under most general conditions on k p and v in particular we prove the equivalence of the notions of criticality as introduced by simon and murata for an important class of potentials v

weighted norm inequalities off diagonal estimates and elliptic operators part iv riesz transforms on manifolds and weights this is the fourth article of our series here we study weighted norm inequalities for the riesz transform of the laplace beltrami operator on riemannian manifolds and of subelliptic sum of squares on lie groups under the doubling volume property and gaussian upper bounds