applications of mixed martingale hardy spaces in fourier analysis abstract in this paper we consider the martingale hardy spaces defined with the help of the mixed l p norm using atomic decomposition and doob s inequality some applications will be shown in fourier analysis such as the uniform boundedness of the partial sums of the walsh fourier series on l p 1 p and the boundedness of the fejer maximal operator from h p to l p 1 2 p as a consequence of the boundedness we get some almost everywhere and norm convergence results

the linear algebra of the generalized pascal matrix abstract this paper discusses three kinds of generalized pascal matrix and generalizes the results of r brawer and m pirovino

orlicz lorentz hardy martingale spaces abstract martingale hardy spaces are widely studied in the field of mathematical physics and probability in this paper we develop the theory of orlicz lorentz hardy martingale spaces which are much more wider than the classical lorentz hardy martingale spaces more precisely we first investigate several basic properties of orlicz lorentz spaces and then construct the atomic decomposition theorems of these martingale function spaces also we establish the dual theorem of orlicz lorentz hardy spaces for martingales furthermore we study the boundedness of generalized fractional integral operators i in this new framework where is a non negative concave function the results partially extend the very recent results jiao et al 2017 21

martingale hardy spaces for 0 p 1 the relation between the usually considered martingale hardy spaces is examined it will be shown that if the stochastic basis is regular then all these hardy spaces are equivalent an atomic description of the spacesh andp p will be given for 0 p 1 and with the help of this we shall consider the duals of these spaces moreover the dual space ofvmo p generated by the dual space ofh p will be determined in the case when all algebras are generated by countably many atoms furthermore we shall show in this case that the dual ofvmo 1 ish 1

global surfaces of section in non regular convex energy levels of hamiltonian systems in this paper we prove the existence of global sections of disk type in non regular and strictly convex energy levels of integrable and near integrable hamiltonian systems with two degrees of freedom this extends a result of hofer et al in ann math 2 148 1 197 289 1998 where the same statement is true provided the energy level is regular

loop products and closed geodesics the critical points of the length function on the free loop space m of a compact riemannian manifold m are the closed geodesics on m the length function gives a filtration of the homology of m and we show that the chas sullivan product hi hj hi j n is compatible with this filtration we obtain a very simple expression for the associated graded homology ring grh m when all geodesics are closed or when all geodesics are nondegenerate we also interpret sullivan s coproduct see su1 su2 on c as a product in cohomology h 0 h 0 hi j n 1 0 where 0 m is the constant loop we show that is also compatible with the length filtration and we obtain a similar expression for the ring grh 0 the nonvanishing of products n and n is shown to be determined by the rate at which the morse index grows when a geodesic is iterated we determine the full ring structure h 0 for spheres m s n 3

hamiltonian pseudo rotations of projective spaces the main theme of the paper is the dynamics of hamiltonian diffeomorphisms of mathbb c mathbb p n with the minimal possible number of periodic points equal to n 1 by arnold s conjecture called here hamiltonian pseudo rotations we prove several results on the dynamics of pseudo rotations going beyond periodic orbits using floer theoretical methods one of these results is the existence of invariant sets in arbitrarily small punctured neighborhoods of the fixed points partially extending a theorem of le calvez and yoccoz and of franks and misiurewicz to higher dimensions the other is a strong variant of the lagrangian poincare recurrence conjecture for pseudo rotations we also prove the c 0 rigidity of pseudo rotations with exponentially liouville mean index vector this is a higher dimensional counterpart of a theorem of bramham establishing such rigidity for pseudo rotations of the disk

multiple closed geodesics on 3 spheres abstract this paper is devoted to a study on closed geodesics on finsler and riemannian spheres we call a prime closed geodesic on a finsler manifold rational if the basic normal form decomposition cf y long bott formula of the maslov type index theory pacific j math 187 1999 113 149 of its linearized poincare map contains no 2 2 rotation matrix with rotation angle which is an irrational multiple of or irrational otherwise we prove that if there exists only one prime closed geodesic on a d dimensional irreversible finsler sphere with d 2 it cannot be rational then we further prove that there exist always at least two distinct prime closed geodesics on every irreversible finsler 3 dimensional sphere our method yields also at least two geometrically distinct closed geodesics on every reversible finsler as well as riemannian 3 dimensional sphere we prove also such results hold for all compact simply connected 3 dimensional manifolds with irreversible or reversible finsler as well as riemannian metrics

two elliptic closed geodesics on positively curved finsler spheres abstract in this paper we prove that for every finsler n dimensional sphere s n f with reversibility and flag curvature k satisfying 1 2 k 1 either there exist infinitely many closed geodesics or there exist at least two elliptic closed geodesics and each linearized poincare map has at least one eigenvalue of the form e 1 with being an irrational multiple of

braid floer homology abstract area preserving diffeomorphisms of a 2 disc can be regarded as time 1 maps of non autonomous hamiltonian flows on r z d 2 the periodic flow lines define braid conjugacy classes up to full twists we examine the dynamics relative to such braid classes and define a new invariant for such classes the braid floer homology this refinement of floer homology originally used for the arnol d conjecture yields a morse type forcing theory for periodic points of area preserving diffeomorphisms of the 2 disc based on braiding contributions of this paper include 1 a monotonicity lemma for the behavior of the nonlinear cauchy riemann equations with respect to algebraic lengths of braids 2 establishment of the topological invariance of the resulting braid floer homology 3 a shift theorem describing the effect of twisting braids in terms of shifting the braid floer homology 4 computation of examples and 5 a forcing theorem for the dynamics of hamiltonian disc maps based on braid floer homology

riesz transform characterizations of musielak orlicz hardy spaces for any j 1 n f in mathcal s mathbb r n and x in mathbb r n the j th riesz transform r j f of f is usually defined by displaystyle r j f x lim epsilon rightarrow 0 c n int y in mathbb r n vert y vert epsilon frac y j vert y vert n 1 f x y dy where displaystyle c n frac gamma n 1 2 pi n 1 2 and gamma denotes the gamma function

generalized orlicz spaces and related pde abstract we prove the boundedness of the maximal operator in generalized orlicz spaces defined on subsets of r n the proof is based on an extension result for functions we study generalized sobolev orlicz spaces and establish density of smooth functions and the poincare inequality as applications we establish the existence of solutions of the laplace equation with zero and non zero right hand side further we systematize assumptions for functions and prove several basic tools needed for the study of differential equations of generalized orlicz growth

nonlinear parabolic problems in musielak orlicz spaces abstract our studies are directed to the existence of weak solutions to a parabolic problem containing a multi valued term the problem is formulated in the language of maximal monotone graphs we assume that the growth and coercivity conditions of a nonlinear term are prescribed by means of a time and space dependent n function this results in the formulation of the problem in generalized musielak orlicz spaces we are using density arguments hence an important step of the proof is a uniform boundedness of appropriate convolution operators in musielak orlicz spaces for this purpose we shall need to assume a kind of logarithmic holder regularity with respect to t and x

stirling matrix via pascal matrix abstract the pascal type matrices obtained from the stirling numbers of the first kind s n k and of the second kind s n k are studied respectively it is shown that these matrices can be factorized by the pascal matrices also the ldu factorization of a vandermonde matrix of the form v n x x 1 x n 1 for any real number x is obtained furthermore some well known combinatorial identities are obtained from the matrix representation of the stirling numbers and these matrices are generalized in one or two variables

periodic approximations of irrational pseudo rotations using pseudoholomorphic curves we prove that every c 1 smooth area preserving dieomorphism of the closed 2 disk having not more than one periodic point is the uniform limit of periodic c 1 smooth dieomorphisms in particular every smooth irrational pseudo rotation can be c 0 approximated by integrable systems this partially answers a long standing question of a katok regarding zero entropy hamiltonian systems in low dimensions our approach uses pseudoholomorphic curve techniques from symplectic geometry

rotation vectors and fixed points of area preserving surface diffeomorphisms we consider the homological rotation vectors for area preserving diffeomorphisms of compact surfaces which are homotopic to the identity there are two main results the first is that if 0 is in the interior of the convex hull of the rotation vectors for such a diffeomorphism then f has a fixed point of positive index the second result asserts that if f has a vanishing mean rotation vector then f has a fixed point of positive index there are several applications including a new proof of the arnold conjecture for area preserving diffeomorphisms of compact surfaces in this paper we investigate area preserving diffeomorphisms of compact surfaces we are concerned with rotation vectors defined in terms of homology which can be associated to the points of the surface there are two main results the first is that if f is an area preserving diffeomorphism of a compact surface which is homotopic to the identity and 0 is in the interior of the convex hull of the rotation vectors of f then f has a fixed point of positive index the second result concerns the mean rotation vector of f it asserts that if this vector vanishes then f has a fixed point of positive index a special case of this second result when the surface in question is a surface of genus zero was proved in f5 in 6 we give several applications including a new proof of the arnold conjecture for area preserving diffeomorphisms of compact surfaces

the conley zehnder index and the saddle center equilibrium abstract we consider hamiltonian systems with two degrees of freedom we suppose the existence of a saddle center equilibrium in a strictly convex component s of its energy level moser s normal form for such equilibriums and a theorem of hofer wysocki and zehnder are used to establish the existence of a periodic orbit in s with several topological properties we also prove the explosion of the conley zehnder index of any periodic orbit that passes close to the saddle center equilibrium

finite energy foliations of tight three spheres and hamiltonian dynamics surfaces of sections are a classical tool in the study of 3 dimensional dynamical systems their use goes back to the work of poincare and birkhoff in the present paper we give a natural generalization of this concept by constructing a system of transversal sections in the complement of finitely many distinguished periodic solutions such a system is established for nondegenerate reeb flows on the tight 3 sphere by means of pseudoholomorphic curves the applications cover the nondegenerate geodesic flows on t1s2 rp3 via its double covering s3 and also nondegenerate hamiltonian systems in ir4 restricted to sphere like energy surfaces of contact type

existence of closed geodesics on finsler n spheres abstract in this paper we prove that on every finsler n sphere s n f with reversibility satisfying f 2 1 2 g 0 and l s n f 1 1 there always exist at least n prime closed geodesics without self intersections where g 0 is the standard riemannian metric on s n with constant curvature 1 and l s n f is the length of a shortest geodesic loop on s n f we also study the stability of these closed geodesics

on a generic property of geodesic flows in this paper we are concerned with a sufficient condition for a riemannian metric on a compact simply connected manifold to have infinitely many geometrically distinct closed geodesics 1969 gromoll meyer proved in gm that for every riemannian metric on a compact simply connected manifold m there are infinitely many geometrically distinct closed geodesics if the sequence bi m f i of betti numbers of the free loop space m is unbounded for some field f using the theory of minimal models vigue poirrier sullivan proved that the rational cohomology algebra h m i q of m is generated by a single element if and only if the sequence bi m i q i of betti numbers of the free loop space m of m is bounded it is a conjecture that the same statement holds for all fields of prime characteristic partial results are due to mccleary ziller mz and halperin vigue poirrier hv now we turn to the manifolds for which the hypothesis of gromoll meyer s theorem does not hold for example spheres and projective spaces then stability properties of the closed geodesics become important a closed geodesic is hyperbolic if the linearized poincare map has no eigenvalue of norm 1 from the bumpy metrics theorem due to abraham ab and anosov an2 and from a pertubation result due to klingenberg takens kt one can conclude a c4 generic metric on a compact manifold has either a non hyperbolic closed geodesic of twist type or all closed geodesics are hyperbolic in the first case there are infinitely many geometrically distinct closed geodesics in every tubular neighborhood of the closed geodesic of twist type due to a theorem by moser mo in the second case there are infinitely many closed geodesics if m is simply connected due to results by hingston hi and the author ra1 hence it follows from these results that a c4 generic metric on a compact simply connected manifold has infinitely many geometrically distinct closed geodesics we remark that it is an open question whether there is a riemannian metric on a simply connected compact manifold all of whose closed geodesics are hyperbolic in ra2 we show that the examples of metrics on the 2

the second closed geodesic on finsler spheres of dimension n 2 we show the existence of at least two geometrically distinct closed geodesics on an n dimensional sphere with a bumpy and non reversible finsler metric for n 2 2000 msc classification 53c22 53c60 58e10

new orlicz hardy spaces associated with divergence form elliptic operators abstract let l be the divergence form elliptic operator with complex bounded measurable coefficients the positive concave function on 0 of strictly critical lower type p 0 1 and t t 1 1 t 1 for t 0 in this paper the authors study the orlicz hardy space h l r n and its dual space bmo l r n where l denotes the adjoint operator of l in l 2 r n several characterizations of h l r n including the molecular characterization the lusin area function characterization and the maximal function characterization are established the carleson measure characterization and the john nirenberg inequality for the space bmo l r n are also given as applications the authors show that the riesz transform l 1 2 and the littlewood paley g function g l map h l r n continuously into l the authors further show that the riesz transform l 1 2 maps h l r n into the classical orlicz hardy space h r n for p n n 1 1 and the corresponding fractional integral l for certain 0 maps h l r n continuously into h l r n where is determined by and and satisfies the same property as all these results are new even when t t p for all t 0 and p 0 1

c genericity of positive topological entropy for geodesic flows on s2 we show that there is a c open and dense set of positively curved metrics on s2 whose geodesic flow has positive topological entropy and thus exhibits chaotic behavior the geodesic flow for each of these metrics possesses a horseshoe and it follows that these metrics have an exponential growth rate of hyperbolic closed geodesics the positive curvature hypothesis is required to ensure the existence of a global surface of section for the geodesic flow our proof uses a new and general topological criterion for a surface diffeomorphism to exhibit chaotic behavior very shortly after this manuscript was completed the authors learned about remarkable recent work by hofer wysocki and zehnder 14 15 on three dimensional reeb flows in the special case of geodesic flows on s2 they show that if the geodesic flow has no parabolic closed geodesics this holds for an open and c dense set of riemannian metrics on s2 then it possesses either a global surface of section or a heteroclinic orbit it then immediately follows from the proof of our main theorem that there is a c open and dense set of riemannian metrics on s2 whose geodesic flow has positive topological entropy this concludes a program to show that every orientable compact surface has a c open and dense set of riemannian metrics whose geodesic flow has positive topological entropy

regular and irregular geodesics on spherical harmonic surfaces abstract the behavior of geodesic curves on even seemingly simple surfaces can be surprisingly complex in this paper we use the hamiltonian formulation of the geodesic equations to analyze their integrability properties in particular we examine the behavior of geodesics on surfaces defined by the spherical harmonics using the morales ramis theorem and kovacic algorithm we are able to prove that the geodesic equations on all surfaces defined by the sectoral harmonics are not integrable and we use poincare sections to demonstrate the breakdown of regular motion

the predual and john nirenberg inequalities on generalized bmo martingale spaces in this paper we introduce the generalized bmo martingale spaces by stopping time sequences which enable us to characterize the dual spaces of martingale hardy lorentz spaces h s p q for 0 p 1 1 q moreover by duality we obtain a john nirenberg theorem for the generalized bmo martingale spaces when the stochastic basis is regular we also extend the boundedness of fractional integrals to martingale hardy lorentz spaces

existence of renormalized solutions to elliptic equation in musielak orlicz space abstract we prove existence of renormalized solutions to general nonlinear elliptic equation in musielak orlicz space avoiding growth restrictions namely we consider div a x u f l 1 on a lipschitz bounded domain in r n the growth of the monotone vector field a is controlled by a generalized nonhomogeneous and anisotropic n function m the approach does not require any particular type of growth condition of m or its conjugate m neither 2 nor 2 the condition we impose is log holder continuity of m which results in good approximation properties of the space the proof of the main results uses truncation ideas the young measures methods and monotonicity arguments

non integrability of a class of painleve iv equations as hamiltonian systems in this paper we will prove the rational non integrability of a class of hamiltonian systems associated with painleve iv equation by using morales ramis theory and kovacic s algorithm which to some extent also implies the non integrability of the fourth painleve equation itself

the linear algebra of the pascal matrix abstract pascal s triangle can be represented as a square matrix in two basically different ways as a lower triangular matrix pn or as a full symmetric matrix qn it has been found that the pn pnt is the cholesky factorization of qn pn can be factorized by special summation matrices it can be shown that the inverses of these matrices are the operators which perform the gaussian elimination steps for calculating cholesky s factorization by applying linear algebra we produce combinatorial identities and an existence theorem for diophantine equation systems finally an explicit formula for the sum of the kth powers is given

inequalities for hilbert operator and its extensions the probabilistic approach we present a probabilistic study of the hilbert operator tf x 1 0f y dyx y x 0 tf x 1 0 f y dyx y x 0 defined on integrable functions ff on the positive halfline using appropriate novel estimates for orthogonal martingales satisfying the differential subordination we establish sharp moment weak type and inequalities for tt we also show similar estimates for higher dimensional analogues of the hilbert operator and by the further careful modification of martingale methods we obtain related sharp localized inequalities for hilbert and riesz transforms

a proof of two conjectures of deveci and karaduman abstract in a recent article in this journal deveci and karaduman 1 proposed several conjectures regarding the order of the cyclic groups generated by the generalized pascal s matrices in this note we give a proof of two of these conjectures

geodesics on s2 and periodic points of annulus homeomorphisms we show that an area preserving homeomorphism of the open or closed annulus which has at least one periodic point must in fact have infinitely many interior periodic points a consequence is the theorem that every smooth riemannian metric ons2 with positive gaussian curvature has infinitely many distinct closed geodesics

new martingale inequalities and applications to fourier analysis abstract let f p be a probability space and 0 0 be a musielak orlicz function in this article the authors prove that the doob maximal operator is bounded on the musielak orlicz space l using this and extrapolation method the authors then establish a fefferman stein vector valued doob maximal inequality on l as applications the authors obtain the dual version of the doob maximal inequality and the stein inequality for l which are new even in weighted orlicz spaces the authors then establish the atomic characterizations of martingale musielak orlicz hardy spaces h s p q h s and h m from these atomic characterizations the authors further deduce some martingale inequalities between different martingale musielak orlicz hardy spaces which essentially improve the corresponding results in orlicz space case and are also new even in weighted orlicz spaces by establishing the davis decomposition on h s and h m the authors obtain the burkholder davis gundy inequality associated with musielak orlicz functions finally using the previous martingale inequalities the authors prove that the maximal fejer operator is bounded from h 0 1 to l 0 1 which further implies some convergence results of the fejer means these results are new even for the weighted hardy spaces

generalizations of the poincare birkhoff theorem in this article we prove some generalizations of the classical poincar birkhoff theorem on area preserving homeomorphisms of the annulus which satisfy a boundary twist condition the work of g d birkhoff on this theorem and its applications can be found in b1 b2 and chapter v of b3 a more modern treatment can be found in b n we prove a theorem for the open annulus a s1 x 0 1 since as we will see the theorem for the closed annulus can easily be obtained from this result for the open annulus however it is not immediately obvious what should be the analogue of the twist condition it turns out that the most general hypothesis and the most natural from the point of view of our proof involves the notion of positively and negatively returning disks for some lift of f to the covering space a r x 0 1 more precisely if f a a is a lift of f a a we will say that the e is a positively returning disk for f if there is an open disk u c a such that f u n u 0 and fp u n u k 0 forsome n k 0 here u k denotes the set x k t i x t e u thus u is disjoint from its image but under iteration by f returns to a positive translate of itself a negatively returning disk is defined similarly but with k 0 such that fl u n u 0 in section 2 we prove the following

non hyperbolic closed geodesics on positively curved finsler spheres abstract in this paper we prove that for every finsler n dimensional sphere s n f n 3 with reversibility and flag curvature k satisfying 1 2 k 1 there exist at least three distinct closed geodesics and at least two of them are elliptic if the number of prime closed geodesics is finite when n 6 these three distinct closed geodesics are non hyperbolic

geometry and dynamics of planar linkages in this article we present an approach to describing the geometry and curvature of the configuration spaces of a class of simple idealized planar linkages this is based on determining the curvature of such configuration spaces canonically embedded into euclidean space and then the behaviour of the dynamics of such linkages can be understood via the associated geodesic flow our objective is to present a method which in principle can be applied to many different examples

a dynamical characterization of universally tight lens spaces we give necessary and sufficient conditions for a closed connected co orientable contact 3 manifold m to be a standard lens space based on assumptions on the reeb flow associated to a defining contact form our methods also provide rational global surfaces of section for non degenerate reeb flows on l p q std with prescribed binding orbits

real interpolation for variable martingale hardy spaces abstract in this paper we consider the real interpolation for variable martingale hardy spaces let 0 q and 0 1 then h p s h s q h p 1 q s where the variable exponent p is a measurable function defined on a probability space the variable exponent does not need any restriction

simple closed geodesics on most alexandrov surfaces abstract we study the existence of simple closed geodesics on most in the sense of baire category alexandrov surfaces with curvature bounded below compact and without boundary we show that it depends on both the curvature bound and the topology of the surfaces

rotation sets for homeomorphisms and homology in this article we propose a definition of rotation sets for homeomorphisms of arbitrary compact manifolds this approach is based on taking the suspended flow and using ideas of schwartzmann on homology and winding cycles for flows our main application is to give a generalisation of a theorem of libre and mackay for tori to the context of surfaces of higher genus

bounded minimisers of double phase variational integrals bounded minimisers of the functional w mapsto int dw p a x dw q rm d x where 0 leqq a cdot in c 0 alpha and 1 p q are c 1 beta regular provided the sharp bound q leqq p alpha holds

non integrability criteria for hamiltonians in the case of lame normal variational equations abstract we consider complex analytic classical hamiltonian systems with two degrees of freedom and an invariant plane furthermore we assume that the normal variational equations nve are of lame type we describe the possible potentials giving rise to this kind of problems and non integrability criteria based on the differential galois approach to the ziglin theorem some examples are included

hardy lorentz spaces for b valued martingales abstract in this paper we study the hardy lorentz spaces for banach space valued martingales the dual spaces are characterized and several martingale inequalities are established the proofs mainly depend on the classical tool of atomic decompositions as usual these conclusions are closely related to the geometrical properties of the underlying banach spaces

computation of invariant tori by newton krylov methods in large scale dissipative systems abstract a method to compute invariant tori in high dimensional systems obtained as discretizations of pdes by continuation and newton krylov methods is described invariant tori are found as fixed points of a generalized poincare map so that the dimension of the system of equations to be solved is that of the original system due to the dissipative nature of the problems studied the convergence of the linear solvers is extremely fast the computation of periodic orbits inside the arnold s tongues is also considered thermal convection of a binary mixture of fluids in a rectangular cavity has been used to test the method

on a conjecture of deveci and karaduman abstract in a recent article in this journal deveci and karaduman 3 proposed several conjectures two of which have been proven by hiller 4 regarding the order of the cyclic group generated by the generalized pascal matrix in this paper we give a counterexample to one of these conjectures and present some related results

swinging atwood machine experimental and numerical results and a theoretical study a swinging atwood machine sam is built and some experimental results concerning its dynamic behaviour are presented experiments clearly show that pulleys play a role in the motion of the pendulum since they can rotate and have non negligible radii and masses equations of motion must therefore take into account the moment of inertia of the pulleys as well as the winding of the rope around them their influence is compared to previous studies a preliminary discussion of the role of dissipation is included the theoretical behaviour of the system with pulleys is illustrated numerically and the relevance of different parameters is highlighted finally the integrability of the dynamic system is studied the main result being that the machine with pulleys is non integrable the status of the results on integrability of the pulley less machine is also recalled

the cyclic groups via the pascal matrices and the generalized pascal matrices abstract in this paper given a positive integer m we consider the multiplicative order of upper and lower triangular matrices and symmetric matrices derived from pascal s triangle when read modulo m we also consider the corresponding problems when the binomial coefficients i j are replaced by x i j i j and x i j i j where x is an integer satisfying certain divisibility hypotheses we also offer some open conjectures on these orders

new real variable characterizations of musielak orlicz hardy spaces abstract let r n 0 0 be such that x is an orlicz function and t is a muckenhoupt a r n weight the musielak orlicz hardy space h r n is defined to be the space of all f s r n such that the grand maximal function f belongs to the musielak orlicz space l r n luong dang ky established its atomic characterization in this paper the authors establish some new real variable characterizations of h r n in terms of the vertical or the non tangential maximal functions or the littlewood paley g function or g function via first establishing a musielak orlicz fefferman stein vector valued inequality moreover the range of in the g function characterization of h r n coincides with the known best results when h r n is the classical hardy space h p r n with p 0 1 or its weighted variant