conormal problem of higher order parabolic systems with time irregular coefficients the paper is a comprehensive study of lp and schauder estimates for higher order divergence type parabolic systems with discontinuous coefficients on a half space and cylindrical domains with the conormal derivative boundary conditions for the lp estimates we assume that the leading coefficients are only bounded and measurable in the t variable and have vanishing mean oscillations vmox with respect to x we also prove the schauder estimates in two situations the coefficients are holder continuous only in the x variable the coefficients are holder continuous in the t variable as well on the lateral boundary

on the l p solvability of higher order parabolic and elliptic systems with bmo coefficients we prove the solvability in sobolev spaces for both divergence and non divergence form higher order parabolic and elliptic systems in the whole space on a half space and on a bounded domain the leading coefficients are assumed to be merely measurable only in the time variable and have small mean oscillations with respect to the spatial variables in small balls or cylinders for the proof we develop a set of new techniques to produce mean oscillation estimates for systems on a half space

upper bounds of limit cycles in abel differential equations with invariant curves abstract new criteria are established for upper bounds on the number of limit cycles of periodic abel differential equations having two periodic invariant curves one of them bounded the criteria are applied to obtain upper bounds of either zero or one limit cycle for planar differential systems

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

functional calculus for the ornstein uhlenbeck operator let be the gauss measure on rd and l the ornstein uhlenbeck operator which is self adjoint in l2 for every p in 1 p 2 set p arc sin 2 p 1 and consider the sector s p z c arg z p the main result of this paper is that if m is a bounded holomorphic function on s p whose boundary values on s p satisfy suitable hormander type conditions then the spectral operator m l extends to a bounded operator on lp and hence on lq for all q such that 1 q 1 2 1 p 1 2 the result is sharp in the sense that l does not admit a bounded holomorphic functional calculus in a sector smaller than s p

harmonic analysis and ultracontractivity let tt l 0 be a symmetric contraction semigroup on the spaces u m 1 0 is ultracontractive i e that ti t o satisfies the condi tion r oo t l f x for all f in ll m andall t in r if and only if the infinitesimal generator s has sobolev embedding properties namely w af q 0 tattf is bounded let be a positive possibly unbounded but with dense domain operator on l2 m where a7 is a ex finite measure space let 7 be the spectral resolution of the identity for which oo whenever 1 0 with the above properties will be called a symmetric contraction semigroup a symmetric contraction semigroup tt t o for which tt maps lx m into l m for every positive t is said to be ultracontractive the notion of ul tracontractivity appeared compatively recently in work of e b davies 4 and davies and b simon 6 in connection with logarithmic sobolev inequalities

bilinear embedding for real elliptic differential operators in divergence form with potentials we present a simple bellman function proof of a bilinear estimate for elliptic operators in divergence form with real coefficients and with nonnegative potentials the constants are dimension free the p range of applicability of this estimate is 1 for any real accretive nonsymmetric matrix a of coefficients

lp estimates on functions of the laplace operator on etudie la continuite l p de certaines fonctions du laplacien sur une variete de riemann complete a geometrie bornee

sharp lp estimates for discrete second order riesz transforms we show that multipliers of second order riesz transforms on products of discrete abelian groups enjoy the lp estimate p 1 where p max p q and p and q are conjugate exponents this estimate is sharp if one considers all multipliers of the form i iriri with i 1 and infinite groups in the real valued case we obtain better sharp estimates for some specific multipliers such as i iriri with 0 i 1 these are the first known precise lp estimates for discrete calderon zygmund operators

pointwise convergence for semigroups in vector valued l p spaces suppose that tt t 0 is a symmetric diffusion semigroup on l2 x and denote by widetilde t t t geq0 its tensor product extension to the bochner space l p x mathcal b where mathcal b belongs to a certain broad class of umd spaces we prove a vector valued version of the hopf dunford schwartz ergodic theorem and show that this extends to a maximal theorem for analytic continuations of widetilde t t t geq 0 on l p x mathcal b as an application we show that such continuations exhibit pointwise convergence

parabolic equations with vmo coefficients in sobolev spaces with mixed norms an lq lp theory of divergence and non divergence form parabolic equations is presented the main coefficients are supposed to belong to the class vmox which in particular contains all measurable functions depending only on t the method of proving simplifies the methods previously used in the case p q

limit cycles for rigid cubic systems we consider planar cubic systems with a unique rest point of center focus type and constant angular velocity for such systems we obtain an affine classification in three families and for two of them their corresponding phase portraits on the poincare sphere we also prove that for two of these families there is uniqueness of limit cycle with respect the third family we give the bifurcation diagram and phase portraits on the poincare sphere of a one parameter sub family exhibiting at least two limit cycles

bellman function and linear dimension free estimates in a theorem of bakry by using an explicit bellman function we prove a bilinear embedding theorem for the laplacian associated with a weighted riemannian manifold m having the bakry emery curvature bounded from below the embedding acting on the cartesian product of lp m and lq t m 1 p 1 q 1 involves estimates which are independent of the dimension of the manifold and linear in p as a consequence we obtain linear dimension free estimates of the lp norms of the corresponding shifted riesz transform all our proofs are analytic

radial fourier multipliers in high dimensions given a fixed p 2 we prove a simple and effective characterization of all radial multipliers of mathcal f l p left mathbb r d right provided that the dimension d is sufficiently large the method also yields new l q space time regularity results for solutions of the wave equation in high dimensions

the conormal derivative problem for elliptic equations of variational type on considere des solutions du probleme qu 0 dans nu 0 sur ou est un domaine borne dans r n et q et n sont definis par qu x div a x u x du x b x u x du x x nu x a x u x du x x x x

lpestimates for some ultraparabolic operators with discontinuous coefficients we consider a class of ultraparabolic operators of the kind formula z x t rn 1 where the principal part is uniformly elliptic on rq q n and the constant matrixbis upper triangular and such that the operator obtained by freezing the coefficientsaijat any pointz0 rn 1is hypoelliptic we prove local lp estimates for the derivatives xixju i j 1 q of a solution to the equationlu f under the assumption that the coefficientsaijbelong to the space vmo vanishing mean oscillation with respect to a suitable metric related tob

functional calculus for generators of symmetric contraction semigroups we prove that every generator of a symmetric contraction semigroup on a finite measure space admits for 1 p a hormander type holomorphic functional calculus on lp in the sector of angle p arcsin 1 2 p the obtained angle is optimal

cubic systems and abel equations abstract we investigate cubic systems which can be transformed to an equation of abel form the conditions for the origin to be a centre and in particular an isochronous centre are obtained the maximum number of limit cycles which can bifurcate from a fine focus is determined and some information is obtained about the global phase portrait

bellman functions and dimension free lp estimates for the riesz transforms in bessel settings abstract in this article we prove dimension free l p boundedness of riesz transforms associated with a bessel differential operator we obtain explicit estimates of the l p norms for the bessel riesz transforms in terms of p establishing a linear behavior with respect to p we use the bellman function technique to prove a bilinear dimension free inequality involving poisson semigroups defined through this bessel operator

admissible decomposition for spectral multipliers on gaussian l p l p this paper concerns harmonic analysis of the ornstein uhlenbeck operator l on the euclidean space we examine the method of decomposing a spectral multiplier phi l into three parts according to the notion of admissibility which quantifies the doubling behaviour of the underlying gaussian measure gamma we prove that the above mentioned admissible decomposition is bounded in l p gamma for 1 p le 2 in a certain sense involving the gaussian conical square function the proof relates admissibility with e nelson s hypercontractivity theorem in a novel way

a new uniqueness criterion for the number of periodic orbits of abel equations abstract a solution of the abel equation x a t x 3 b t x 2 such that x 0 x 1 is called a periodic orbit of the equation our main result proves that if there exist two real numbers a and b such that the function a a t b b t is not identically zero and does not change sign in 0 1 then the abel differential equation has at most one non zero periodic orbit furthermore when this periodic orbit exists it is hyperbolic this result extends the known criteria about the abel equation that only refer to the cases where either a t 0 or b t 0 does not change sign we apply this new criterion to study the number of periodic solutions of two simple cases of abel equations the one where the functions a t and b t are 1 periodic trigonometric polynomials of degree one and the case where these two functions are polynomials with three monomials finally we give an upper bound for the number of isolated periodic orbits of the general abel equation x a t x 3 b t x 2 c t x when a t b t and c t satisfy adequate conditions

harmonic analysis on semigroups wherever t 0 and 1 p so e m stein st 3 developed a littlewoodpaley theory for such semigroups with some additional hypotheses r r coifman r rochberg and g weiss crw pointed out that some of stein s estimates could be obtained more simply by the use of transference techniques which were developed by coifman and weiss cw our object is to push these transference methods to present an alternative and simpler approach to some of the principal results of stein s book we shall also use these results to prove estimates for the wave equation

characterizations of hankel multipliers we give characterizations of radial fourier multipliers as acting on radial l p functions 1 p 2d d 1 in terms of lebesgue space norms for fourier localized pieces of the convolution kernel this is a special case of corresponding results for general hankel multipliers besides l p l q bounds we also characte rize weak type inequalities and intermediate inequalities involving lorentz spaces applications include results on interpolation of multiplier spaces

plancherel type estimates and sharp spectral multipliers abstract we study general spectral multiplier theorems for self adjoint positive definite operators on l2 x where x is any open subset of a space of homogeneous type we show that the sharp hormander type spectral multiplier theorems follow from the appropriate estimates of the l2 norm of the kernel of spectral multipliers and the gaussian bounds for the corresponding heat kernel the sharp hormander type spectral multiplier theorems are motivated and connected with sharp estimates for the critical exponent for the riesz means summability which we also study here we discuss several examples which include sharp spectral multiplier theorems for a class of scattering operators on r3 and new spectral multiplier theorems for the laguerre and hermite expansions

functional calculus for some perturbations of the ornstein uhlenbeck operator in a recent paper garcia cuerva et al have shown that for every p in 1 the symmetric finite dimensional ornstein uhlenbeck operator mathcal l ou frac 1 2 delta x cdot nabla has a bounded holomorphic functional calculus on l p in the sector of angle phi p arcsin 1 2 p we prove a similar result for some perturbations of the ornstein uhlenbeck operator

holomorphy of spectral multipliers of the ornstein uhlenbeck operator let be the gauss measure on rd and l the ornstein uhlenbeck operator for every p in 1 2 set p arcsin 2 p 1 and consider the sector s p z c argz 0 m tl lp 0 m tl l1 if and only if m extends to a bounded holomorphic function on the right half plane and its boundary value m i on the imaginary axis is the euclidean fourier transform of a finite borel measure on the real line we prove similar results for uniform spectral multipliers of second order elliptic differential operators in divergence form on rd belonging to a wide class which contains l from these results we deduce that operators in this class do not admit an h functional calculus in sectors smaller than s p

limit cycles of abel equations of the first kind abstract consider the scalar differential equation x i 0 m a i t x n i where a i t are t periodic analytic functions and 1 n i n for any polynomial q x x n 0 i 1 m i x n i the equation can be written as x a 0 q x r t x let w be the wronskian of q and r with respect to x and q w the previous polynomials after removing multiplicity of roots and solutions of the differential equation we prove that if the vector field defined by the differential equation is transversal at every point of q x 0 or w t x 0 then the number of limit cycles isolated periodic solutions in the set of periodic solutions of the differential equation is at most 3 n 1

2 solvability of the dirichlet problem for nondivergence elliptic equations with vmo coefficients we prove a well posedness result in the class w 2 p w 0 1 p for the dirichlet problem lu f a e in u 0 on we assume the coefficients of the elliptic nondivergence form equation that we study are in vmo l

the conormal derivative problem for equations of variational type in nonsmooth domains it is well known that elliptic boundary value problems in smooth domains have smooth solutions but if the domain is say c 1 the solutions need not be lipschitz recently korevaar has identified a class of lipschitz domains in which solutions of the capillary problem are lipschitz assuming the contact angle relates correctly to the geometry of the domain lipschitz bounds for more general boundary value problems in the same class of domains are proved applications to variational inequalities are also considered

functional calculus for the laguerre operator in this paper we study the boundedness of spectral multipliers associated to the multidimensional laguerre operator open image in new window it is well known that for special values of the analysis of the laguerre operator can be interpreted as the analysis of the ornstein uhlenbeck operator acting on polyradial functions exploiting this relation we prove that if m is a bounded holomorphic function in the sector open image in new window satisfying suitable hormander type conditions on the boundary then the spectral operator m open image in new window is bounded on lp with respect to the laguerre measure we also prove that holomorphy in the sector s is a necessary condition for multipliers whose norm is invariant under dilations

higher order elliptic and parabolic systems with variably partially bmo coefficients in regular and irregular domains abstract the solvability in sobolev spaces is proved for divergence form complex valued higher order parabolic systems in the whole space on a half space and on a reifenberg flat domain the leading coefficients are assumed to be merely measurable in one spacial direction and have small mean oscillations in the orthogonal directions on each small cylinder the directions in which the coefficients are only measurable vary depending on each cylinder the corresponding elliptic problem is also considered