new abstract hardy spaces the aim of this paper is to propose an abstract construction of spaces which keep the main properties of the already known hardy spaces h1 we construct spaces through an atomic or molecular decomposition we prove some results about continuity from these spaces into l1 and some results about interpolation between these spaces and the lebesgue spaces we also obtain some results on weighted norm inequalities finally we present partial results in order to understand a characterization of the duals of hardy spaces

new maximal functions and multiple weights for the multilinear calderon zygmund theory a multi sub linear maximal operator that acts on the product of m lebesgue spaces and is smaller than the m fold product of the hardy littlewood maximal function is studied the operator is used to obtain a precise control on multilinear singular integral operators of calderon zygmund type and to build a theory of weights adapted to the multilinear setting a natural variant of the operator which is useful to control certain commutators of multilinear calderon zygmund operators with bmo functions is then considered the optimal range of strong type estimates a sharp end point estimate and weighted norm inequalities involving both the classical muckenhoupt weights and the new multilinear ones are also obtained for the commutators

duality of hardy and bmo spaces associated with operators with heat kernel bounds the introduction and development of hardy and bmo spaces on euclidean spaces r in the 1960s and 1970s played an important role in modern harmonic analysis and applications in partial differential equations these spaces were studied extensively in 32 22 18 19 31 and many others an l function f on r is in the hardy spaceh r if the area integral function of the poisson integral e t f satisfies

a boundedness criterion for generalized calderon zygmund operators in 1965 a p calderon showed the l2 boundedness of the so called first calderon commutator this is one of the first examples of a non convolution operator associated to a kernel which satisfies certain size and smoothness properties comparable to those of the kernel of the hilbert transform these properties together with the l2 boundedness imply the lp boundedness for all p s in 1 oo many operators in analysis such as certain classes of pseudo differential operators and the cauchy integral operator on a curve are associated with kernels having these properties for these operators one of the major questions is if they are bounded on l2 we are going to give necessary and sufficient conditions for such an operator to be bounded on l2 they are essentially that the images of the function 1 under the actions of the operator and its adjoint both lie in bmo in the case of the aforementioned first commutator this can be checked by an integration by parts in the first section we present some basic notions and state the theorem which is proved in sections 2 and 3 in section 4 we show how to recover some classical results in sections 5 and 6 we construct a functional calculus for small perturbations of a and in section 7 we show a connection between the theory of calderon zygmund operators and kato s conjecture it is a pleasure to express our thanks to r r coifman and y meyer for suggesting many elegant simplifications in our proofs and most of the applications we also wish to thank stephen semmes for several pertinent remarks

abstract hardy sobolev spaces and interpolation the purpose of this work is to describe an abstract theory of hardy sobolev spaces on doubling riemannian manifolds via an atomic decomposition we study the real interpolation of these spaces with sobolev spaces and finally give applications to riesz inequalities

an operator valued tb theorem we establish conditions similar to the tb theorem of david journe and semmes which guarantee the boundedness of an integral transformation t with l x valued kernel on lxp rn where 1 p and x is a banach space with the unconditionality property of martingale differences umd

third order equation for harmonic generation complex canonical transformation and jwkb solution the quantum description of third harmonic generation can be formulated as an eigenvalue problem for a third order linear differential equation we perform a semiclassical study of this third order equation generalizing the familiar jwkb theory for the second order schrodinger equation and deriving explicit albeit approximate formulas for the eigenvalues within this semiclassical context a central role in this analysis is played by a nonlinear complex canonical transformation which permits a complete description of the classical motion generated by a complex polynomial hamiltonian function in the complexified position and momentum planes

a t 1 theorem for non integral operators let x be a space of homogeneous type and let l be a sectorial operator with bounded holomorphic functional calculus on l 2 x we assume that the semigroup e tl t 0 satisfies davies gaffney estimates associated with l are certain approximations of the identity we call an operator t a non integral operator if compositions involving t and these approximations satisfy certain weighted norm estimates the davies gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on t in calderon zygmund theory in this paper we show under the additional assumption that a vertical littlewood paley stein square function associated with l is bounded on l 2 x that a non integral operator t is bounded on l 2 x if and only if t 1 in bmo l x and t 1 in bmo l x here bmo l x and bmo l x denote the recently defined bmo x spaces associated with l that generalize the space bmo x of john and nirenberg generalizing a recent result due to f bernicot we show a second version of a t 1 theorem under weaker off diagonal estimates which gives a positive answer to a question raised by him as an application we prove l 2 x boundedness of a paraproduct operator associated with l we moreover study criterions for a t b theorem to be valid

use of abstract hardy spaces real interpolation and applications to bilinear operators this paper can be considered as the sequel of bernicot and zhao j func anal 255 1761 1796 2008 where the authors have proposed an abstract construction of hardy spaces h1 they shew an interpolation result for these hardy spaces with the lebesgue spaces here we describe a more precise result using the real interpolation theory and we clarify the use of hardy spaces then with the help of the bilinear interpolation theory we then give applications to study bilinear operators on lebesgue spaces these ideas permit us to study singular operators with singularities similar to those of bilinear calderon zygmund operators in a far more abstract framework as in the euclidean case

counterintuitive transitions in the multistate landau zener problem with linear level crossings we generalize the brundobler elser hypothesis in the multistate landau zener problem to the case when instead of a state with the highest slope of the diabatic energy level there is a band of states with an arbitrary number of parallel levels having the same slope we argue that the probabilities of counterintuitive transitions among such states are exactly zero

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

the poincare inequality is an open ended condition let p 1 and let x d be a complete metric measure space with borel and doubling that admits a 1 p poincare inequality then there exists e 0 such that x d admits a 1 q poincare inequality for every q p e quantitatively

a bifurcation phenomenon of stokes curves around a double turning point and influence of virtual turning points upon the transition probabilities for three level systems abstract we study the non adiabatic transition problem of three level systems from the viewpoint of the exact wkb analysis we show the importance of virtual turning points and new stokes curves in this problem by presenting some concrete examples where the contribution of virtual turning points to transition probabilities becomes greater than that of ordinary turning points in the course of our discussion we encounter a bifurcation phenomenon of stokes curves around a double turning point and we investigate this issue as well

proof of the landau zener formula in an adiabatic limit with small eigenvalue gaps we consider a smooth operator valued functionh t that has two isolated non degenerate eigenvaluese a t ande b t for 0 we assume these eigenvalues are bounded away from the rest of the spectrum ofh t but have an avoided crossing with one another with a closest approach that iso as tends to zero under these circumstances we study the small e limit for the adiabatic schrodinger equation i varepsilon frac partial psi partial t h t varepsilon 1 2 psi we prove that the landau zener formula correctly describes the coupling between the adiabatic states associated with the eigenvaluese a t ande b t as the system propagates through the avoided crossing

the tb theorem on non homogeneous spaces 0 introduction main objects and results 3 0 1 main results 4 0 2 an application of t1 heorem electric intensity capacity 7 0 3 how to interpret calderon zygmund operator t 9 0 3 1 bilinear form is defined on lipschitz functions 10 0 3 2 bilinear form is defined for smooth functions 11 0 3 3 apriori boundedness 12 0 4 plan of the paper 15

exact wkb analysis of non adiabatic transition probabilities for three levels the schrodinger equation id dt h with a large parameter and an appropriate 3 3 matrix h is studied by the exact wkb method i e wkb analysis based on the borel resummation transition probabilities of solutions of the equation are explicitly calculated with the help of the connection formula for wkb solutions since a 3 3 system is considered addition of new stokes curves is necessary to obtain the correct connection formula berk nevins roberts observation and we present a practically useful procedure for finding required new stokes curves

heat semigroup and singular pdes we provide in this work a semigroup approach to the study of singular pdes in the line of the paracontrolled approach developed recently by gubinelli imkeller and perkowski starting from a heat semigroup we develop a functional calculus and introduce a paraproduct based on the semigroup for which commutator estimates and schauder estimates are proved together with their paracontrolled extensions this machinery allows us to investigate singular pdes in potentially unbounded riemannian manifolds under mild geometric conditions as an illustration we study the generalized parabolic anderson model equation and prove under mild geometric conditions its well posed character in small time on a potentially unbounded 2 dimensional riemannian manifold for an equation driven by a coloured noise and for all times for the linear parabolic anderson model equation in 2 dimensional unbounded manifolds this machinery can be extended to an even more singular setting and deal with sobolev spaces rather than holder spaces

virtual turning points and bifurcation of stokes curves for higher order ordinary differential equations for a higher order linear ordinary differential operator p its stokes curve bifurcates in general when it hits another turning point of p this phenomenon is most neatly understandable by taking into account stokes curves emanating from virtual turning points together with those from ordinary turning points this understanding of the bifurcation of a stokes curve plays an important role in resolving a paradox recently found in a system of linear differential equations associated with the fourth painleve equation

hardy and bmo spaces associated to divergence form elliptic operators consider a second order divergence form elliptic operator l with complex bounded measurable coefficients in general operators based on l such as the riesz transform or square function may lie beyond the scope of the calderon zygmund theory they need not be bounded in the classical hardy bmo and even some l p spaces in this work we develop a theory of hardy and bmo spaces associated to l which includes in particular a molecular decomposition maximal and square function characterizations duality of hardy and bmo spaces and a john nirenberg inequality

littlewood paley theory and the t 1 theorem with non doubling measures let be a radon measure on rd which may be non doubling the only condition that must satisfy is b x r crn for all x rd r 0 and for some fixed 0 n d in this paper littlewood paley theory for functions in lp is developed one of the main difficulties to be solved is the construction of reasonable approximations of the identity in order to obtain a calderon type reproducing formula moreover it is shown that the t 1 theorem for n dimensional calderon zygmund operators without doubling assumptions can be proved using the littlewood paley type decomposition that is obtained for functions in l2 as in the classical case of homogeneous spaces

a t1 theorem for integral transformations with operator valued kernel we consider generalized calderon zygmund operators whose kernel k x y takes values in l x continuous linear operators on the banach space x and satis es variants of the classical standard estimates involving rboundedness which has recently become a crucial notion in connection with operator valued singular integrals boundedness criteria in the spirit of the t1 theorem of david and journe are proved for such operators on the bochner spaces lp rn x where 1 p and x is a umd space for some results x is also required to have pisier s property in the special case t1 t 1 0 we obtain an essentially complete analogue of the scalar valued theorem we also provide su cient conditions for the general case but they are stronger in general than the necessary t1 t 1 bmo type conditions although they reduce to them in the classical situation a counterexample is given to show that the natural necessary conditions are not su cient in general

new stokes line in wkb theory the wkb theory for differential equations of arbitrary order or integral equations in one dimension is investigated the rules previously stated for the construction of stokes lines for nth order differential equations n 3 or integral equations are found to be incomplete because these rules lead to asymptotic forms of the solutions that depend on path this paradox is resolved by the demonstration that new stokes lines can arise when previously defined stokes lines cross a new formulation of the wkb problem is given to justify the new stokes lines with the new stokes lines the asymptotic forms can be shown to be independent of path in addition the wkb eigenvalue problem is formulated and the global dispersion relation is shown to be a functional of loop integrals of the action

hardy spaces and divergence operators on strongly lipschitz domains of rn abstract let be a strongly lipschitz domain of r n consider an elliptic second order divergence operator l including a boundary condition on and define a hardy space by imposing the non tangential maximal function of the extension of a function f via the poisson semigroup for l to be in l1 under suitable assumptions on l we identify this maximal hardy space with h 1 r n if r n with h r 1 under the dirichlet boundary condition and with h z 1 under the neumann boundary condition

new function spaces of bmo type the john nirenberg inequality interpolation and applications in this paper we introduce and develop some new function spaces of bmo bounded mean oscillation type on spaces of homogeneous type or measurable subsets of spaces of homogeneous type the new function spaces are defined by variants of maximal functions associated with generalized approximations to the identity and they generalize the classical bmo space we show that the john nirenberg inequality holds on these spaces and they interpolate with lp spaces by the complex interpolation method we then give applications on lp boundedness of singular integrals whose kernels do not satisfy the hormander condition 2005 wiley periodicals inc

the quest for solvable multistate landau zener models recently integrability conditions ics in mutistate landau zener mlz theory were proposed 1 they describe common properties of all known solved systems with linearly time dependent hamiltonians here we show that ics enable efficient computer assisted search for new solvable mlz models that span complexity range from several interacting states to mesoscopic systems with many body dynamics and combinatorially large phase space this diversity suggests that nontrivial solvable mlz models are numerous in addition we refine the formulation of ics and extend the class of solvable systems to models with points of multiple diabatic level crossing

proportional hazards model with random effects we propose a general proportional hazards model with random effects for handling clustered survival data this generalizes the usual frailty model by allowing a multivariate random effect with arbitrary design matrix in the log relative risk in a way similar to the modelling of random effects in linear generalized linear and non linear mixed models the distribution of the random effects is generally assumed to be multivariate normal but other preferably symmetrical distributions are also possible maximum likelihood estimates of the regression parameters the variance components and the baseline hazard function are obtained via the em algorithm the e step of the algorithm involves computation of the conditional expectations of functions of the random effects for which we use markov chain monte carlo mcmc methods approximate variances of the estimates are computed by louis formula and posterior expectations and variances of the individual random effects can be obtained as a by product of the estimation the inference procedure is exemplified on two data sets

s matrix for generalized landau zener problem the schrodinger equation i psi a bt psi with constant hermitian matrices a and b is studied in the form where b is diagonalized this equation is a generalization of the landau zener problem to an arbitrary number of crossing energy levels an approach the independent crossing approximation leading to a partial understanding of the general case in terms of the two level problem is introduced it is found that certain s matrix elements and thus the corresponding transition probabilities for the general problem are exactly given by formulae of unexpected simplicity suggesting that some kind of general analytic solution might exist the full s matrix is calculated for a solvable special case of the problem the solution of another special case previously discovered for systems with three states is generalized to any number of states the asymptotic behaviour of the system is discussed in general and given explicitly to lowest order in 1 t