bilinear restriction estimates for surfaces with curvatures of different signs recently the sharp l 2 bilinear adjoint restriction estimates for the cone and the paraboloid were established by wolff and tao respectively their results rely on the fact that for the cone and the paraboloid the nonzero principal curvatures have the same sign we generalize those bilinear restriction estimates to surfaces with curvatures of different signs

a local smoothing estimate for the schrodinger equation we show that the schrodinger operator eit is bounded from to for all a 2n 1 2 1 q 2 q and q2 4 n 1 this is almost sharp with respect to the sobolev index we also show that the schrodinger maximal operator sup0 s0 if and only if it is bounded from to when s 2s0 a corollary is that sup0 3 4

restriction estimates for some surfaces with vanishing curvatures abstract we obtain bilinear restriction estimates for surfaces with vanishing curvatures as application we also prove new linear restriction estimates for some class of conic surfaces

transference of local to global l2 maximal estimates for dispersive partial differential equations abstract in this paper we give an elementary proof for transference of local to global maximal estimates for dispersive pdes this is done by transferring local l 2 estimates for certain oscillatory integrals with rough phase functions to the corresponding global estimates the elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context and instead is based on scalings and classical oscillatory integral estimates

improved restriction estimate for hyperbolic surfaces in r3 abstract recently l guth improved the restriction estimate for the surfaces with strictly positive gaussian curvature in r 3 in this paper we extend his restriction estimate to the quadratic surfaces with strictly negative gaussian curvature in r 3

multi scale bilinear restriction estimates for general phases we prove adjoint bilinear restriction estimates for general phases at different scales in the full non endpoint mixed norm range and give bounds with a sharp and explicit dependence on the phases these estimates have applications to high low frequency interactions for solutions to partial differential equations as well as to the linear restriction problem for surfaces with degenerate curvature as a consequence we obtain new bilinear restriction estimates for elliptic phases and wave klein gordon interactions in the full bilinear range and give a refined strichartz inequality for the klein gordon equation in addition we extend these bilinear estimates to hold in adapted function spaces by using a transference type principle which holds for vector valued waves

parallel algorithms for solving systems of nonlinear equations abstract in this paper parallel algorithms are proposed for solving both systems of nonlinear algebraic equations and unconstrained optimization problems theoretical results for their convergence and orders of convergence are also presented the proposed algorithms are analytically compared in terms of their time efficiencies and speed up ratios

from harmonic analysis to arithmetic combinatorics arithmetic combinatorics or additive combinatorics is a fast developing area of research combining elements of number theory combinatorics harmonic analysis and ergodic theory its arguably best known result and the one that brought it to global prominence is the proof by ben green and terence tao of the long standing conjecture that primes contain arbitrarily long arithmetic progressions there are many accounts and expositions of the green tao theorem including the articles by kra 119 and tao 182 in the bulletin the purpose of the present article is to survey a broader highly interconnected network of questions and results built over the decades and spanning several areas of mathematics of which the green tao theorem is a famous descendant an old geometric problem lies at the heart of key conjectures in harmonic analysis a major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles an unexpected argument points harmonic analysts towards additive number theory with consequences that could have hardly been anticipated we will not try to give a comprehensive survey of harmonic analysis combinatorics or additive number theory we will not even be able to do full justice to our specific areas of focus instead referring the reader to the more complete expositions and surveys listed in section 7 our goal here is to emphasize the connections between these areas we will thus concentrate on relatively few problems chosen as much for their importance to their fields as for their links to each other the article is written from the point of view of an analyst who in the course of her work was gradually introduced to the questions discussed here and found them fascinating we hope that the reader will enjoy a taste of this experience

refined and microlocal kakeya nikodym bounds of eigenfunctions in higher dimensions we prove a kakeya nikodym bound on eigenfunctions and quasimodes which sharpens a result of the authors blair and sogge in anal pde 8 747 764 2015 and extends it to higher dimensions as in the prior work the key intermediate step is to prove a microlocal version of these estimates which involves a phase space decomposition of these modes that is essentially invariant under the bicharacteristic geodesic flow in a companion paper blair and sogge in j differ geom 2015 it will be seen that these sharpened estimates yield improved l q m bounds on eigenfunctions in the presence of nonpositive curvature when 2 q frac 2 d 1 d 1

a concept of nonlinear block diagonal dominance we introduce a concept of block diagonal dominance for nonlinear functions and discuss the relations to strictly diagonally dominant functions and m functions some sufficient conditions for the new kind of functions are given the global convergence of block asynchronous sor methods for finding zeros of block diagonal dominant nonlinear functions is proved

the optimal trilinear restriction estimate for a class of hypersurfaces with curvature abstract in 4 nearly optimal l 1 trilinear restriction estimates in r n 1 are established under transversality assumptions only in this paper we show that the curvature improves the range of exponents by establishing l p estimates for any p 2 n 4 3 n 2 in the case of double conic surfaces the exponent 2 n 4 3 n 2 is shown to be the universal threshold for the trilinear estimate

bilinear identities involving the k plane transform and fourier extension operators we prove certain l2 n bilinear estimates for fourier extension operators associated to spheres and hyperboloids under the action of the k plane transform as the estimates are l2 based they follow from bilinear identities in particular these are the analogues of a known identity for paraboloids and may be seen as higher dimensional versions of the classical l2 2 bilinear identity for fourier extension operators associated to curves in 2

on the cone multiplier in r3 abstract we prove the sharp l 3 bounds for the cone multiplier in r 3 and the associated square function which is known as mockenhaupt s square function

sharp null form estimates for the wave equation null form estimates from dot h alpha 1 times dot h alpha 2 to l q t l r x for the wave equation in bbb r n 1 are studied for n ge4 we obtain the sharp null form estimates except for the endpoints for n 2 3 we obtain the estimates under the additional assumption 4 n q when 2 r

restriction of the fourier transform to a quadratic surface in n we obtain estimates for the restriction of the fourier transform to a certain k dimensional quadratic submanifold of open image in new window n

on the multilinear restriction and kakeya conjectures we prove d linear analogues of the classical restriction and kakeya conjectures in rd our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians closely related to heat flow we conclude by giving some applications to the corresponding variable coefficient problems and the so called joints problem as well as presenting some n linear analogues for n d

nonlinear profile decomposition for the h 12 h 12 rd energy subcritical wave equation abstract we consider the nonlinear wave equation u t t u u u 4 d 1 0 in dimensions d 2 this equation is critical in h 1 2 h 1 2 r d and subcritical with respect to the energy we prove the nonlinear profile decomposition the proof must address the lack of compactness due to the lorentz symmetry

parallel variable metric algorithms for unconstrained optimization we classify straeter s ideas for parallel unconstrained optimization and apply them to huang s class of updating formulas straeter s rank one updating formula appears to be the only parallel extension within huang s class with the property of quadratic termination we also develop a parallel extension of broyden s 1965 rank one updating formula and prove quadratic termination finally we present numerical results obtained by testing the algorithms on several standard example problems

a parallel subgradient projections method for the convex feasibility problem abstract an iterative method to solve the convex feasibility problem for a finite family of convex sets is presented the algorithm consists in the application of a generalization of an acceleration procedure introduced by de pierro in a parallel version of the subgradient projections method proposed by censor and lent the generated sequence is shown to converge for any starting point some numerical results are presented

strichartz estimates via the schrodinger maximal operator we consider the schrodinger operator e it delta acting on initial data f in dot h s we show that an affirmative answer to a question of carleson concerning the sharp range of s for which lim t to 0 e it delta f x f x a e x in mathbb r n would imply an affirmative answer to a question of planchon concerning the sharp range of q and r for which e it delta is bounded in l x q mathbb r n l r t mathbb r when n 2 we unconditionally improve the range for which the mixed norm estimates hold

weighted restriction estimates using polynomial partitioning we use the polynomial partitioning method of guth j amer math soc 29 2016 371 413 to prove weighted fourier restriction estimates in r3 with exponents p that range between 3 and 3 25 depending on the weight as a corollary to our main theorem we obtain new non weighted local and global restriction estimates for compact c surfaces s r3 with strictly positive second fundamental form for example we establish the global restriction estimate ef lp r3 f lq s in the full conjectured range of exponents for p 3 25 up to the sharp line and the global restriction estimate ef lp f l2 s for p 3 and certain sets r3 of infinite lebesgue measure as a corollary to our main theorem we also obtain new results on the decay of spherical means of fourier transforms of positive compactly supported measures on r3 with finite dimensional energies

improved bound for the bilinear bochner riesz operator we study l p times l q rightarrow l r bounds for the bilinear bochner riesz operator mathcal b alpha alpha 0 in mathbb r d d ge 2 which is defined by open image in new window we make use of a decomposition which relates the estimates for mathcal b alpha to the square function estimates for the classical bochner riesz operators in consequence we significantly improve the previously known bounds

weak version of restriction estimates for spheres and paraboloids in finite fields abstract we study l p l r restriction estimates for algebraic varieties in d dimensional vector spaces over finite fields unlike the euclidean case if the dimension d is even then it is conjectured that the l 2 d 2 d 3 l 2 stein tomas restriction result can be improved to the l 2 d 4 d 4 l 2 estimate for both spheres and paraboloids in finite fields in this paper we show that the conjectured l p l 2 restriction estimate holds in the specific case when test functions under consideration are restricted to d coordinate functions or homogeneous functions of degree zero to deduce our result we use the connection between the restriction phenomena for our varieties in d dimensions and those for homogeneous varieties in d 1 dimensions

restriction and kakeya phenomena for finite fields the restriction and kakeya problems in euclidean space have received much attention in the last few decades and they are related to many problems in harmonic analysis partial differential equations pdes and number theory in this paper we initiate the study of these problems on finite fields in many cases the euclidean arguments carry over easily to the finite setting and are in fact somewhat cleaner but there are some new phenomena in the finite case which deserve closer study

a bilinear approach to the restriction and kakeya conjectures the purpose of this paper is to investigate bilinear variants of the restriction and kakeya conjectures to relate them to the standard formulations of these conjectures and to give applications of this bilinear approach to existing conjectures the methods used are based on several observations and results of bourgain see 2 6 together with some refinements by moyua vargas and vega 17 18 this paper is organized as follows in the first section we discuss bilinear restriction estimates and show how one can pass back and forth between these estimates and the standard restriction estimates we also generalize the 12 7 bilinear restriction estimate of 18 to higher dimensions in the second section we give analogues of the above results for the kakeya operator in particular we give a bilinear improvement to wolff s kakeya theorem in arbitrary dimension in the third section we give applications of these bilinear estimates in three dimensions for example we are able to improve the 42 11 exponent in wolff s restriction theorem to 34 9 we are also able to prove a sharp lp lq restriction theorem which improves on the classical l2 l4 tomas stein theorem and also give some concrete progress on a bilinear restriction conjecture of klainerman and machedon we also give a non bilinear approach to these estimates which gives weaker results but is more direct and probably has a wider range of application

uniform estimates for the x ray transform restricted to polynomial curves abstract we establish near optimal mixed norm estimates for the x ray transform restricted to polynomial curves with a weight that is a power of the affine arclength the bounds that we establish depend only on the spatial dimension and the degree of the polynomial some of our results are new even in the well curved case

linear adjoint restriction estimates for paraboloid we prove a class of modified paraboloid restriction estimates with a loss of angular derivatives for the full set of paraboloid restriction conjecture indices this result generalizes the paraboloid restriction estimate in radial case from shao rev mat iberoam 25 2009 1127 1168 as well as the result from miao et al proc ams 140 2012 2091 2102 as an application we show a local smoothing estimate for a solution of the linear schrodinger equation under the assumption that the initial datum has additional angular regularity

restriction estimates using polynomial partitioning ii we improve the estimates in the restriction problem in dimension n geqslant 4 to do so we establish a weak version of a k linear restriction estimate for any k the exponents in this weak k linear estimate are sharp for all k and n

fractal strichartz estimate for the wave equation abstract we consider strichartz estimates for the wave equation with respect to general measures which satisfy certain growth conditions in r 3 1 we obtain the sharp estimate and in higher dimensions improve the previous results

logarithmic improvements in l p bounds for eigenfunctions at the critical exponent in the presence of nonpositive curvature we consider the problem of proving l p bounds for eigenfunctions of the laplacian in the high frequency limit in the presence of nonpositive curvature and more generally manifolds without conjugate points in particular we prove estimates at the critical exponent p c frac 2 d 1 d 1 where a spectrum of scenarios for phase space concentration must be ruled out our work establishes a gain of an inverse power of the logarithm of the frequency in the bounds relative to the classical l p bounds of the second author

mixed norm strichartz type estimates for hypersurfaces in three dimensions in their work ikromov and muller fourier restriction for hypersurfaces in three dimensions and newton polyhedra princeton university press princeton 2016 proved the full range l p l 2 fourier restriction estimates for a very general class of hypersurfaces in mathbb r 3 which includes the class of real analytic hypersurfaces in this article we partly extend their results to the mixed norm case where the coordinates are split in two directions one tangential and the other normal to the surface at a fixed given point in particular we resolve completely the adapted case and partly the non adapted case in the non adapted case the case when the linear height h text lin phi is below two is settled completely

parallel newton methods for the nonlinear complementary problem in this paper we discuss how the basic newton method for solving the nonlinear complementarity problem can be implemented in a parallel computation environment we propose some synchronized and asynchronous newton methods and establish their convergence

partially updated switching method for systems of nonlinear equations abstract a hybrid method for solving systems of n nonlinear equations is given the method does not use derivative information and is especially attractive when good starting points are not available and the given system is expensive to evaluate it is shown that after a few steps each iteration requires 2 k 1 function evaluations where k 1 k n is chosen so as to have an efficient algorithm global convergence results are given and superlinear convergence is established some numerical results show the numerical performance of the proposed method

estimates for the x ray transform restricted to line complexes of kirillov type let there be given a piecewise continuous rectifiable curve r r n let g 1 n m 1 n be the usual grassmannian bundle in r n define an n dimensional submanifold m r n of m 1 n as the set of all copies of g 1 n along the curve following krillov we know that a nice function f x can be recovered from its x ray transform r 1 n f on m r n if and only if the curve intersects almost every affine hyperplane

an endline bilinear cone restriction estimate for mixed norms we prove an l 2 times l 2 rightarrow l t ql x p bilinear fourier extension estimate for the cone when p q are on the critical line 1 q frac n 1 2 1 1 p this extends previous results by wolff tao and lee vargas

an asynchronous parallel newton method a parallel newton method is described for the minimization of a twice continuously differentiable uniformly convex functionf x the algorithm generates a sequence x j which converges superlinearly to the global minimizer off x

on the restriction of the fourier transform to a conical surface let r be the su rface of a circular cone in r3 we show that if 1 p 4 3 1 q 3 1 1 p and f e lp r3 then the fourier transform of f belongs to lq r da for a certain natural measure a on r following p tomas we also establish bounds for restrictions of fourier transforms to conic annuli at the endpoint p 4 3 with logarithmic growth of the bound as the thickness of the annulus tends to zero

sharp local well posedness results for the nonlinear wave equation this article is concerned with local well posedness of the cauchy problem for second order quasilinear hyperbolic equations with rough initial data the new results obtained here are sharp in low dimension

a sobolev estimate for the adjoint restriction operator in this note we consider the adjoint restriction estimate for hypersurface under additional regularity assumption we obtain the optimal h s l q estimates and their mixed norm generalizations as applications we prove some weighted strichartz estimates for the propagator varphi rightarrow e it delta alpha 2 varphi alpha 0