weighted estimates for the helmholtz equation and some applications we characterize the radial functionsvin rnfor which the a priori inequality u1 2 2 k uv1 2 2 c v 1 2 k2 u 2holds with constant independent ofk the condition is forvto have the x rays transform everywhere bounded we apply these estimates to the well posedness of evolution schrodinger equations with time dependent drift terms and to the restriction of the fourier transform to euclidean spheres

weighted norm inequalities for oscillatory integrals with finite type phases on the line abstract we obtain two weighted l 2 norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type the conditions imposed on the weights involve geometrically defined maximal functions and the inequalities are best possible in the sense that they imply the full l p r l q r mapping properties of the oscillatory integrals our results build on work of carbery soria vargas and the first author

the dynamics of some discrete population models in this paper we are interested in providing sufficient conditions guaranteeing the fact that all positive solutions of the discrete delay difference equation a ii 1 aa f a e converge as n 00 in e a is a constant real number with 0 0 sufficiently small thus formally equation e comes from x t h x t h yx t d x t r for small h if we set z f x ht then the preceding equation for t n and r h m becomes yh 1 1 h y n hd y n m which is of the form e where a 1 yh and f u hd u u 2 0 clearly h 0 is small if and only if a e 0 1 1069 h

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

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

on certain maximal functions and approach regions if f e l ir and u x y is the poisson integral of j a classical theorem of fatou 2 asserts that u has nontangential limits almost everywhere on r littlewood 5 and later zygmund 9 showed that this conclusion is false if nontangential approach is replaced by approach along translates of a curve which approaches the boundary tangentially in this paper we find a necessary and sufficient condition on an approach region for the associated maximal function to be appropriately bounded it then turns out that there are many approach regions which are not contained in any nontangential region but for which the conclusions of fatou s theorem remain true suppose q c ir has the property that whenever xi u e b and whenever x x y y it follows that x v e q then the maximal function associated to such a set is bounded if and only if the cross sectional measure at height y is bounded by a constant times y we also study regions where the cross sectional area is larger than y we introduce a modified maximal function which is again bounded this result leads to a simple proof of the boundedness of certain tangential maximal functions of poisson integrals of potentials studied in 161 the plan of this paper is as follows in section 1 we motivate our main ideas by discussing two particular examples the method of proof here is somewhat different from the proofs of the more general results in section 2 we obtain the necessary and sufficient conditions for fatou s theorem in section 3 we study more general approach regions and the appropriately modified maximal functions in section 4 we show how this maximal function controls the tangential boundary behavior of poisson integrals of potentials

on the laplace equation with dynamical boundary conditions of reactive diffusive type this paper deals with the laplace equation in a bounded regular domain of rn n 2 coupled with a dynamical boundary condition of reactive diffusive type in particular we study the problem u 0in 0 ut ku l uon 0 u 0 x u0 x on where u u t x t 0 x x denotes the laplacian operator with respect to the space variable while denotes the laplace beltrami operator on is the outward normal to and k and l are given real constants well posedness is proved for any given initial distribution u0 on together with the regularity of the solution moreover the fourier method is applied to represent it in term of the eigenfunctions of a related eigenvalue problem

radial weights and mixed norm inequalities for the disc multiplier in this paper we show that the l2 weighted inequalities with radial weights of the multiplier for the ball are controlled by the maximal function defined by averages on arbitrary rectangles this solves the case of radial weights of a problem proposed by e stein in 1978 it also gives an elegant and simple proof of the boundedness of the multiplier for the ball on the spaces of mixed norm lradp lang2 for 2n n 1 p 2n n 1 obtained by a cordoba and by g mockenhaupt additional work is presented in order to extend these mixed norm estimates to the end points 2n n 1 and 2n n 1 in the sharp form of restricted weak type inequalities

weighted estimates for conic fourier multipliers we prove a weighted inequality which controls conic fourier multiplier operators in terms of lacunary directional maximal operators by bounding the maximal operators this enables us to conclude that the multiplier operators are bounded on l p mathbb r 3 with 1 p infty

attractivity and global stability for linearizable difference equations consider the difference equation x n 1 f x n x n k n 0 1 where email protected 0 1 and the initial conditions are real numbers we use the linearization of this equation in the form x n email protected i 1 lkg ix n i n 0 1 where email protected 1 2 and the functions g i r k l r to investigate the asymptotic behavior of the solutions of the considered equation we illustrate our results with various examples

on the laplace equation with non linear dynamical boundary conditions the main part of the paper deals with local existence and global existence versus blow up for solutions of the laplace equation in bounded domains with a non linear dynamical boundary condition more precisely we study the problem consisting in 1 the laplace equation in 0 infty times omega 2 a homogeneous dirichlet condition 0 infty times gamma 0 3 the dynamical boundary condition frac partial u partial nu u t m 2 u t u p 2 u on 0 infty times gamma 1 4 the initial condition u 0 x u 0 x on partial omega here omega is a regular and bounded domain in mathbb r n with n ge 1 and gamma 0 and gamma 1 endow a measurable partition of partial omega moreover m 1 2 le p r where r 2 n 1 n 2 when n ge 3 r infty when n 1 2 and u 0 in h 1 2 partial omega u 0 0 on gamma 0 the final part of the paper deals with a refinement of a global non existence result by levine park and serrin which is applied to the previous problem

a bilinear extension inequality in two dimensions abstract we provide sharp decay estimates for circular averages of a certain bilinear extension operator on l 2 s 1 l 2 s 1

subdyadic square functions and applications to weighted harmonic analysis abstract through the study of novel variants of the classical littlewood paley stein g functions we obtain pointwise estimates for broad classes of highly singular fourier multipliers on r d satisfying regularity hypotheses adapted to fine subdyadic scales in particular this allows us to efficiently bound such multipliers by geometrically defined maximal operators via general weighted l 2 inequalities in the spirit of a well known conjecture of stein our framework applies to solution operators for dispersive pde such as the time dependent free schrodinger equation and other highly oscillatory convolution operators that fall well beyond the scope of the calderon zygmund theory

a stein conjecture for the circle we discuss the manner in which one might expect directional maximal functions to control the fourier extension operator via l 2 weighted inequalities we prove a general inequality of this type for the extension operator restricted to circles in the plane

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

global stability two conjectures and maple abstract consider the second order difference equation u 1 0 u 0 0 and u n 1 f u n 1 u n for n 0 where either a f u v u p v u q v or b f u v p q v 1 u if 0 q p in case a or p 0 and q 0 in case b it has been conjectured see m r s kulenovic g ladas dynamics of second order rational difference equations chapman and hall crc press 2001 that lim n u n exists and equals l where l 0 and l f l l we prove this conjecture in case a and significantly extend the range of p and q for which it is known in case b in cases a and b these questions are equivalent to global stability of the fixed point l l of the planar map u v v f u v for as in case a we consider natural four dimensional extensions t of 3 and s of 2 for 0 q p we prove that l l l l is a global stable fixed point of t but we also describe precisely a range of parameters 0 q p for which s has at least three distinct fixed points in the positive orthant we describe section 3 some general principles underlying our arguments symbolic calculations using maple play a crucial role in our arguments in section 4

geometric stability conditions for higher order difference equations is a higher order or delay autonomous difference equation if the order m is a fixed positive integer larger than 1 the real valued function f is assumed continuous on r and x x g r are the initial values in 1ym 0 addition to the fact that discretizations of scalar differential equations result in equations of type 1 direct applications of difference equations of order greater than 1 date back to more than half a century ago and nowadays these equations are increasingly encountered in both social and w x w x natural sciences see 4 and 6 for some applications and additional references a solution x of f x x s x is a fixed point or equilibrium of eq 1 like its differential analogs 1 can be and often is translated into a first order vector equation the instability or the asymptotic stability of x may then be determined by linearizing the associated vector function

global dynamics of quadratic second order difference equation in the first quadrant we investigate the global behavior of a quadratic second order difference equation x n 1 ax n 2 bx n x n 1 cx n 1 2 dx n ex n 1 f n 0 1 with non negative parameters and initial conditions we find the global behavior for all ranges of parameters and determine the basins of attraction of all equilibrium points

restriction estimates for space curves with respect to general measures in this paper we consider adjoint restriction estimates for space curves with respect to general measures and obtain optimal estimates when the curves satisfy a finite type condition the argument here is new in that it doesn t rely on the offspring curve method which has been extensively used in the previous works our work was inspired by the recent argument due to bourgain and guth which was used to deduce linear restriction estimates from multilinear estimates for hypersurfaces

the global attractivity of difference equations of nonincreasing nonlinearities with applications abstract the global attractivity of the unique positive equilibrium of the equation x n 1 g x n k 1 x n k 2 x n k m n 0 1 2 is investigated where g is a nonincreasing continuous function in each of its arguments and the k are positive integers we are able to use our results to extract some important global attractivity criteria of the positive equilibria of the discrete clark model x n 1 x n f x n k and the rational recursive sequence x n 1 a bx n a x n 1 as well

on global asymptotic stability of nonlinear higher order difference equations in this paper we generalize the main theorem of liz and ferreiro e liz j b ferreiro a note on the global stability of generalized difference equations appl math lett 15 2002 655 659 and some other global stability results for nonautonomous higher order difference equations to the case when contraction type steps are incorporated together with the steps when the difference sequence can increase the relation to the q method for discretization of delay equations is discussed and some sufficient stability conditions for the numerical scheme are deduced several examples are presented to demonstrate sharpness and applications of the results