remarks on the multiplier operators associated with a cylindrical distance function abstract in this note we consider l p and maximal l p estimates for the generalized riesz means which are associated with the cylindrical distance function max d 1 d 1 r d r we prove these estimates up to the currently known range of the spherical bochner riesz and its maximal operators this is done by establishing implications between the corresponding estimates for the spherical bochner riesz and the cylindrical multiplier operators

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 theorems for a surface with negative curvature we prove a bilinear restriction theorem for a surface of negative curvature this is the analogue of the results of t wolff 19 and t tao 14 15 for cones and paraboloids as a consequence we obtain an almost sharp linear restriction theorem

the l2 boundedness of pseudodifferential operators on donne une nouvelle demonstration du theoreme de calderon vaillancourt on obtient la l 2 continuite de a x d si le symbole a x satisfait a certaines conditions

an endpoint space time estimate for the schrodinger equation abstract we obtain endpoint estimates for the schrodinger operator f e i t f in l x q r n l t r r with initial data f in the homogeneous sobolev space h s r n the exponents and regularity index satisfy n 1 q 1 r n 2 and s n 2 n q 2 r for n 2 we prove the estimates in the range q 16 5 and for n 3 in the range q 2 4 n 1

multiplicity estimates on group varieties tions the field k will be the field q of algebraic numbers we remark at this point that the results remain true if we take instead of the field of complex numbers c its p adic analogue c for some fixed prime p and for k a corresponding subfield kp of cp we restrict ourselves to the complex case in order to avoid some minor complications appearing in the p adic domain these complications always arise when analytic functions come in our functions are defined globally in the case of complex numbers but only locally in the case when we deal with the p adic domain these difficulties can be avoided by a purely algebraic approach if we took this approach the only condition on the ground field would be that it should be algebraically closed and of characteristic zero but we prefer to avoid such an approach in order to keep the text understandable also for those who are mainly interested in the applications in transcendence

determination of the algebraic relations among special γ values in positive characteristic we devise a new criterion for linear independence over function fields using this tool in the setting of dual t motives we find that all algebraic relations among special values of the geometric r function over f q t are explained by the standard functional equations

arithmetic of characteristic p special l values recently the second author has associated a finite fq t module h to the carlitz module over a finite extension of fq t this module is an analogue of the ideal class group of a number field in this paper we study the galois module structure of this module h for cyclotomic extensions of fq t we obtain function field analogues of some classical results on cyclotomic number fields such as the p adic class number formula and a theorem of mazur and wiles about the fitting ideal of ideal class groups we also relate the galois module h to anderson s module of circular units and give a negative answer to anderson s kummer vandiver type conjecture these results are based on a kind of equivariant class number formula which refines the second author s class number formula for the carlitz module

radial fourier multipliers of lp r2 abstract we give sufficient conditions in terms of differentiability and growth properties for a radial function to be an lp r2 fourier multiplier these conditions are in the nature of best possible

propagation of singularities and maximal functions in the plane in this work we mainly generalize bourgain s circular maximal function to include variable coefficient averages our techniques involve a combination of bourgain s basic ideas plus microlocal analysis in particular to see the role of curvature we rely heavily on methods used in studying propogation of singularities for hyperbolic differential equations we also show that forp 2 there is local smoothing inl p for solutions to the wave equation

local smoothing properties of dispersive equations is it possible for time evolution partial differential equations which are reversible and conservative to smooth locally the initial data for the linear wave equation for instance the answer is no however in 10 t kato found a local smoothing property of the korteweg de vries equation the solution of the initial value problem is locally one derivative smoother than the initial datum kato s proof uses in a curcial way the algebraic properties of the symbol for the korteweg de vries equation and the fact that the underlying spatial dimension is one actually judging from the way several integrations by parts and cancellations conspire to reveal a smoothing effect one would be inclined to believe this was a special property of the k dv equation this is not however the case in this paper we attempt to describe a general local smoothing effect for dispersive equations and systems the smoothing effect is due to the dispersive nature of the linear part of the equation all the physically significant dispersive equations and systems known to us have linear parts displaying this local smoothing property to mention only a few the k dv benjamin ono intermediate long wave various boussinesq and schrodinger equations are included we study thus equations and systems of the form

harmonic analysis on sl 2 r and smoothness of the density of states in the one dimensional anderson model we consider infinite jacobi matrices with ones off diagonal and independent identically distributed random variables with distributionf v dv on diagonal iff has compact support and lies in some sobolev spacel 1 then we prove that the integrated density of states k e isc ine

bochner riesz means associated with conical surfaces abstract let d 1 and d 2 be two nonnegative integers greater than 2 we study the fourier multiplier t associated with a conical surface s r d 1 r d 2 defined by t f 1 2 2 f r d 1 r d 2 where is a smooth function defined on r that is supported in 1 2 2

estimations des solutions de l equation des ondes sur certaines varietes coniques on prouve des estimations l p pour les solutions de l equation des ondes analogues aux estimations de r strichartz sur certaines varietes coniques

algebraic relations among periods and logarithms of rank 2 drinfeld modules for any rank 2 drinfeld module rho defined over an algebraic function field we consider its period matrix p rho which is analogous to the period matrix of an elliptic curve defined over a number field suppose that the characteristic of the finite field bbb f q is odd and that rho does not have complex multiplication we show that the transcendence degree of the field generated by the entries of p rho over bbb f q theta is 4 as a consequence we show also the algebraic independence of drinfeld logarithms of algebraic functions which are linearly independent over bbb f q theta

sharp endpoint l p lp estimates for schrodinger groups let l be a non negative self adjoint operator acting on l 2 x where x is a space of homogeneous type with a dimension n suppose that the heat operator e tl satisfies the generalized gaussian p 0 p 0 estimates of order m for some 1 le p 0 2 in this paper we prove sharp endpoint l p sobolev bound for the schrodinger group e itl that is for every p in p 0 p 0 there exists a constant c c n p 0 independent of t such that begin aligned left i l s e itl f right p le c 1 t s vert f vert p quad t in mathbb r quad s ge n big 1 over 2 1 over p big end aligned as a consequence the above estimate holds for all 1 p infty when the heat kernel of l satisfies a gaussian upper bound this extends classical results due to feffermann and stein and miyachi for the laplacian on the euclidean spaces mathbb r n we also give an application to obtain an endpoint estimate for l p boundedness of the riesz means of the solutions of the schrodinger equations

singularities in spectra of disordered systems abstract this paper starts with a review of work on singularities in the spectral density of random harmonic chains detailed descriptions of singular behavior near the maximal frequency near special frequencies and near island frequencies are discussed next the site disordered anderson model is considered here the discussion is limited to the singularity in the density of states near the low energy band edge for systems with arbitrary disorder in dimensions d 1 2 3 in a specific example it is shown that renormalization improves the prediction for the band tail quantitatively

on the convergence of riesz means on compact manifolds the purpose of this paper is to study the lp mapping properties of the riesz means of eigenfunction expansions associated to elliptic differential operators on compact manifolds in order to be more specific let us first introduce some notation let m be a compact connected c manifold of dimension n 2 and p an elliptic differential operator on m with c00 coefficients we shall assume that p is self adjoint with respect to some positive c00 density dx kept fixed throughout that is pu v u pv where as usual

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

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

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

zero estimates on group varieties ii the aim of this article is to give an account of a class of estimates that have in recent years been found useful in the theory of transcendental numbers as our space is limited and the estimates are quite technical in nature wc can provide no hints of proofs and we prefer to illustrate the results with examples rather than supply full statements we also mention some of the applications one basic method in transcendence theory can be described in general terms as follows suppose it is desired to establish a certain proposition usually it will be possible to associate with the problem a collection of meromorphic functions ft fnm such a way that assuming the falsity of the proposition imposes strong arithmetical conditions on the functions these enable us to construct a polynomial p p x1 xn such that the function f p f1 fn is not identically zero but has a large number of zeroes by means of analytic interpolation arguments it can then be deduced that takes many small values and by using once more the arithmetical conditions we conclude that has even more zeroes than initially arranged but often merely knowing upper bounds for the degree of p leads to upper bounds for the number of zeroes of and if we are lucky these yield the contradiction that proves the original proposition in this article we shall limit attention to the last stage of this argument as it is here that the most interesting recent progress has taken place a simple example will suffice to illustrate the sort of estimates involved to show that w and e cannot both be algebraic for w 0 suppose they are then we end up with a non zero polynomial p p x y of degree at most l 1 in x and of degree at most jf 1 in y such that the function

determination of algebraic relations among special zeta values in positive characteristic abstract as an analogue to special values at positive integers of the riemann zeta function we consider carlitz zeta values c n at positive integers n by constructing t motives after papanikolas we prove that the only algebraic relations among these characteristic p zeta values are those coming from the euler carlitz relations and the frobenius pth power relations

special energies and special frequencies special frequencies have been asserted to be zeros of the density of frequencies corresponding to a random chain of coupled oscillators our investigation includes both this model and the random one dimensional schrodinger operator describing an alloy or its discrete analogue using the phase method we exactly determine a bilateral lifsic asymptotic of the integrated density of statesk e at special energiese s which is not only of the classical type exp c e e s 1 2 but also exp c e e s is a typical behaviour in addition other asymptotics occur e g e e s c which show thatk e need not bek

long time existence for small amplitude semilinear wave equations we prove that a certain class of semilinear wave equations has global solutions if the initial data is small these existence results rely on mixed norm angular radial space time estimates more specifically we consider power nonlinearities u f p u where f p u u p and 2 t denotes the d alembertian on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i in 1979 john investigated this equation for small initial data in dimension n 3 he proved that if p 1 2 then there is a global solution whereas for p n dimensions and the critical power p c above which global solutions exist should be the positive root of n 1 p 2 c n 1 p c 2 0 sideris proved that for all n the solutions can blow up if p p c recently zhou proved that for n 4 one has global solutions if p p c 2 here we prove that for n 8 one has global solutions if p p c furthermore we proved that for all n one has global solutions if p p c and the initial data are spherically symmetric in the radial case we use l p estimates derived from the explicit form of the fundamental solution using inequalities from classical analysis in the nonradial case we write the fundamental solution in polar coordinates and use fourier integral and special function estimates to handle the angular part

linear independence of gamma values in positive characteristic we investigate the arithmetic nature of special values of thakur s func tion field gamma function at rational points our main result is that all linear dependence relations over the field of algebraic functions are consequences of the anderson deligne thakur bracket relations

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

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

log holder continuity of the integrated density of states for stochastic jacobi matrices we consider the integrated density of states k e of a general operator on l2 v of the formh h0 v where h 0 u n sum limits left i right 1 u n i and vu n v n u n wherev is a general bounded ergodic stationary process on v we show that k e k e c log e e 1 when e e 1 2 the key is a thouless formula for the strip

tannakian duality for anderson drinfeld motives and algebraic independence of carlitz logarithms we develop a theory of tannakian galois groups for t motives and relate this to the theory of frobenius semilinear difference equations we show that the transcendence degree of the period matrix associated to a given t motive is equal to the dimension of its galois group using this result we prove that carlitz logarithms of algebraic functions that are linearly independent over the rational function field are algebraically independent

transcendence and drinfeld modules it has been ten years since v g drinfeld introduced in 3 the concept of elliptic a module which we now call drinfeld a module these modules are analogues of elliptic curves for global function field k and we have witnessed a beautiful algebraic theory gradually taking shape in 13 we have asked a few natural t ranscendence questions conjectures concerning these modules having in mind the analogue of the classical work of siegel schneider about elliptic integrals and based on evidences provided by the old papers of carlitz and wade cf 1 11 here in this article we will prove those conjectures we will show that indeed there is an interesting transcendence theory for each algebraic curve over a finite field fol lowing drinfeld we let m be a lattice in i e a finitely generated discrete a module in k and form

bochner riesz profile of anharmonic oscillator l d2dx2 x we investigate spectral multipliers bochner riesz means and the convergence of eigenfunction expansion corresponding to the schrodinger operator with anharmonic potential l d2dx2 x we show that the bochner riesz profile of the operator l completely coincides with such profile of the harmonic oscillator h d2dx2 x2 it is especially surprising because the bochner riesz profile of the one dimensional standard laplace operator is known to be essentially different and the case of operators h and l resembles more the profile of multidimensional laplace operators another surprising element of the main obtained result is the fact that the proof is not based on restriction type estimates and instead an entirely new perspective has to be developed to obtain the critical exponent for bochner riesz means convergence

lifshitz singularities in the total and the wavenumber dependent spectral density of random harmonic chains abstract lifshitz has shown that the density of states of random systems becomes exponentially small near the band edge we perform a detailed computation in a one dimensional system and calculate the periodic prefactor of this essential singularity the analysis is extended to the wavenumber dependent spectral density the periodic amplitude is derived for the exactly soluble cases where the distribution of masses is exponential for binary distributions the leading behavior in the concentration of light masses is also derived

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

the weak type l1 convergence of eigenfunction expansions for pseudodifferential operators soit m une variete compacte connexe lisse sans bord de dimension n 2 soit p d un operateur pseudodifferentiel elliptique d ordre 1 sur m qui est auto adjoint par rapport a une densite positive lisse dx on demontre la convergence faible de type l 1 pour les developpements en fonctions propres

on the regularity of inverses of singular integral operators sur r d on considere un operateur de convolution sf f k ou k est une distribution homogene de degre d on suppose que s s etend a un operateur borne et inversible sur l 2 r d l inverse de s est donne par convolution avec une distribution l aussi homogene de degre d on montre qu une regularite possedee par k est partagee par l

on radial weights for the spherical summation operator abstract in this paper we give a sufficient condition for radial weights such that the spherical summation operator is bounded on l 2 r n x dx n 2 when applied to power weights x x x these conditions are also necessary moreover in the general case we shall show that our condition is almost necessary furthermore we give an example which shows that the conjecture in anderson proc amer math soc 83 no 2 1981 269 275 fails to hold

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

exact electronic spectra and inverse localization lengths in one dimensional random systems i random alloy liquid metal and liquid alloy analytic continuations into the complex energy plane of dyson schmidt type of equations for the calculation of the density of states are constructed for a random alloy model a liquid metal and for a liquid alloy in all these models the characteristic function follows from the solution of this equation its imaginary part yields the accumulated density of states and its real part is a measure for the inverse of the localization length of the eigenfunctions the equations have been solved exactly for some distributions of the random variables in the random alloy case the strengths of the delta potentials have an exponential distribution they may also have finite exponentially distributed values with probability 0 p 1 and be infinite with probability q 1 p in the liquid metal the liquid particles are assumed to behave like hard rods this implies an exponential distribution of the distances between the particles the common electronic potential may be arbitrary but is assumed to vanish outside the rods in the one dimensional liquid alloy there is apart from positional randomness of the liquid particles a distribution of the strengths of the electronic delta potentials for cauchy distributions an argument of lloyd is extended to obtain the characteristic function from the one in the model with equal strengths for the case of a liquid of point particles a three parameter class of distributions of the strengths is shown to yield a solution in the form of known functions of the equation mentioned above for several cases numerical calculations of the density of states and the inverse localization length of the eigenfunctions are presented and discussed new results are found exponential decay of the density of states near special energies in the random alloy and liquid metal divergence of the density of states at certain energies with non classical exponent 13 in the random alloy if the average of the potential strengths vanishes exponentially small broadening of the bound state levels for low concentrations of the liquid particles peak in the localization length at the bound state energies which becomes exponentially narrow for low concentrations different exponent in the decay of the inverse localization length at large energies for delta potentials and square well potentials further an expression for the grand potential is given involving a sum over the characteristic function at certain points and divergence of the zero point energy is found for cauchy distributions of the delta potential strengths

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

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

integrable systems and number theory in finite characteristic the purpose of this paper is to give an overview of applications of the concepts and techniques of the theory of integrable systems to number theory in finite characteristic the applications include explicit class field theory and langlands conjectures for function fields effect of the geometry of the theta divisor on factorization of analogs of gauss sums special values of function field gamma zeta and l functions analogs of theorems of weil and stickelberger control of the intersection of the jacobian torsion with the theta divisor the techniques are the krichever drinfeld dictionaries and the theory of solitons akhiezer baker and tau functions developed in this context of arithmetic geometry by anderson 2001 elsevier science b v all rights reserved

alternating multizeta values in positive characteristic we introduce alternating multizeta values in positive characteristic which are generalizations of thakur multizeta values we establish their fundamental properties including non vanishing sum shuffle relations period interpretation and linear independence which is a direct sum result for these values

regularity of solutions to the schrodinger equation soit f dans l espace de schwartz s r n et soit s t f x u x t rn exp ix exp it 2 f d x r n t r f est la transformee de fourier de f on considere u solution de l equation de schrodinger u i u t on pose s f x sup 0 t 1 s t f x x r n on introduit des espaces de sobolev h s en posant h s f s f hd s s r ou f hd s rn 1 2 s f 2 d 1 2 on etudie pour quelles valeurs de s on a b s f x 2 dx 1 2 c b f hd s f s r n pour toutes les boucles b de r n

local smoothing estimates for some half wave operators in this paper we study the local smoothing estimates for some half wave operators associated to nondegenerate convex hypersurfaces

on the density of states of schrodinger operators with a random potential using very recent results on ergodic theorems for superadditive processes on rd the authors prove the existence of the density of states for a wide class of random schrodinger operators in particular new non asymptotic estimates on the density of states are obtained and examples are discussed

on the spectrum of schrodinger operators with a random potential we investigate the spectrum of schrodinger operatorsh of the type h qi f x xi i qi and i independent identically distributed random variables i d we establish a strong connection between the spectrum ofh and the spectra of deterministic periodic schrodinger operators from this we derive a condition for the existence of forbidden zones in the spectrum ofh for random one and three dimensional kronig penney potentials the spectrum is given explicitly

taylor coefficients of anderson thakur series and explicit formulae for each positive characteristic multiple zeta value defined by thakur in function field arithmetic world scientific publishing river edge 2004 the first and third authors in chang and mishiba invent math 2020 https doi org 10 1007 s00222 020 00988 1 constructed a t module together with an algebraic point such that a specified coordinate of the logarithmic vector of the algebraic point is a rational multiple of that multiple zeta value the objective of this paper is to use the taylor coefficients of anderson thakur series and t motivic carlitz multiple star polylogarithms to give explicit formulae for all of the coordinates of this logarithmic vector

improved bounds for stein s square functions supported in part by nrf grant 2009 0072531 korea micinn grant mtm2010 16518 spain erc grant 277778 europe and nsf grant 0652890 usa

singular behavior of the density of states and the lyapunov coefficient in binary random harmonic chains we study the integrated density of statesh 2 of a chain of harmonic oscillators with a binary random distribution of the masses we show in particular that there is a dense set of values of the squared frequency for which the differenceh 2 h 2 has a singularity of the type 2 multiplied by a periodic function of ln where the exponent and the period depend continuously on 2 in the region where 1 2 h is not differentiate on a dense set of points the same type of singularities is also present in the lyapunov coefficient

estimations l p de l equation des ondes sur les varietes a singularite conique dans cet article on se propose d abord d ameliorer le resultat de trans amer math soc 355 689 711 2003 plus precisement soit n une variete riemannienne compacte de dimension n 1 2 sans bord on montre que sur le cone c n l operateur des ondes frac sin t sqrt delta sqrt delta est borne sur lp c n avec la norme cpt pour frac 1 p frac 1 2 0 t 0 et apparentes

sharp resolvent estimates outside of the uniform boundedness range in this paper we are concerned with resolvent estimates for the laplacian delta in euclidean spaces uniform resolvent estimates for delta were shown by kenig et al duke math j 55 2 329 347 1987 who established rather a complete description of the lebesgue spaces allowing such estimates however the problem of obtaining sharp l p l q bounds depending on z has not been considered in a general framework which admits all possible p q in this paper we present a complete picture of sharp l p l q resolvent estimates which may depend on z we also obtain the sharp resolvent estimates for the fractional laplacians and a new result for the bochner riesz operators of negative index