高等数学.pdf

Annals of Mathematics 156 2002 519 563 A uniform estimate By Christoph Thiele 1 Introduction In this article we study bilinear operators whose input consists of two functions on the real line and whose output is again a function on the real line We shall be concerned with a priori estimates for these operators For simplicity we initially require the two input functions to be Schwartz functions and allow the output function to be a tempered distribution By abstract generalities such a bilinear operation B can be formally written as B f1 f2 x f1 1 f2 2 e 2 ix 3M 1 2 3 d 1d 2d 3 for some tempered distribution M In principle it is not relevant that we have written this formula in terms of the Fourier transforms f of f but it makes the following discussion a bit simpler Under the natural assumption that the operation B should be invariant under simultaneous translation of the two input functions and the output function similar abstract generalities show that M lives on the hyperplane 1 2 3 0 Hence we fi nd another tempered distribution m such that 1 B f1 f2 x f1 1 f2 2 e2 ix 1 2 m 1 2 d 1d 2 Such an operation is called a bilinear multiplier operator with multiplier m Observe that if m is a constant equal to 1 then by the Fourier inversion formula this bilinear operation is just the pointwise product f1 x f2 x To obtain a priori estimates for this operation one has to make regularity assumptions on m Classically one requires m to be a measurable function with some smoothness outside the origin of the type 2 1 1 1 2 m 1 2 C dist 1 2 0 1 2 for all 1and 2up to some order which we do not specify here Such operators have been studied extensively and include as a main example the so called 520CHRISTOPH THIELE paraproducts A general reference is 11 Multiplier operators with regularity 2 satisfy estimates of the type 3 B f1 f2 p C f1 p1 f2 p2 for some constant C where f pdenotes the Lp norm of f and makes sense for tempered distributions if allowed to be infi nite We shall be more specifi c about the exact values of p p1 p2later but remark that the main question we are interested in here is whether there are any combinations p p1 p2for which such an a priori estimate holds The work by M Lacey and the author on the bilinear Hilbert transform has renewed interest in the case where m is singular on a larger set specifi cally on a one dimensional subspace R2 Thus one studies multipliers of the form 4 1 1 2 2 m 1 2 C dist 1 2 in this formulation introduced by Gilbert and Nahmod 2 Observe that such multipliers allow for a translation symmetry along in a nontrivial way unlike multipliers of the form 2 which have to be constant in order to have such a symmetry Such a symmetry is called a modulation symmetry of the bilinear operation For simplicity we shall consider only the case when m is constant on each of the two half planes separated by It turns out that one has to distinguish two cases concerning the sub space There is the degenerate case in which is determined by one of the three equations 1 0 2 0 and 1 2 Thus m is a function of either one of the coordinates only or 1 2 and the bilinear operator splits into a combination of some linear operator H and a pointwise product such as f1H f2 H f1 f2 H f1f2 In this case the discussion of Lp estimates is trivial by the linear theory and H older s inequality there are many combinations p p1 p2such that 3 holds The generic case is when is not of the above form In this case M Lacey and the author have shown in 6 and 8 that estimates of the form 3 hold The argument uses some subtle localized Fourier transform methods Now suppose we are given a tuple p p1 p2such that an estimate 3 holds for all both in the degenerate and the generic case There are many such tuples A natural question is whether one can choose the constant C inde pendently of the choice of say the multiplier is always normalized to take values 1 and 1 The diffi culty clearly arises when is generic but close to the degenerate case This question has not been answered in the work by Lacey and the author because their proof in the generic case does not transform in A UNIFORM ESTIMATE521 the limit into a proof in the degenerate case The use of H older s inequality in the trivial proof of the degenerate case cannot be combined easily with the Fourier methods used in the generic case The purpose of this paper is to address this question of uniform estimates in its simplest form and to prove uniform weak type estimates in one partic ularly easy instance of p p1 p2 The innovation is to be able to combine the Fourier methods and H older s inequality inside the argument of 6 and 8 Roughly speaking there is a telescoping series a trick known in the para product community such that the summands are Fourier localized pieces while the sum in closed form can be estimated by H older s inequality The argument then consists of splitting the bilinear operator into pieces some of which are estimated by H older s inequality others by the localized Fourier methods Since the decision which pieces are estimated by either of the meth ods is done late and recursively in the argument one has to do a very careful decomposition so as to preserve the ability to recombine pieces into telescoping series The care that has to be taken for this is the reason for the argument having to be so technical We now give a more detailed description of the main theorem and then give a historical motivation for studying these uniform estimates By dualizing we pass to trilinear forms instead of bilinear operators and we reformulate things without the Fourier transform The Hilbert form with parameter 1 2 3 R3 x x x x R is the continuous trilinear form defi ned on S R S R S R by f1 f2 f3 p v 3 1 f x t dt t dx We assume without restriction that is a unit vector perpendicular to 1 1 1 Let 1 p1 p2 p3 We are interested in an a priori estimate of the form 5 f1 f2 f3 C f1 p1 f2 p2 f3 p3 for all Schwartz functions f1 f2 f3 We exclude exponents p 1 because for such exponents an a priori estimate of the form 5 is easily seen to be false It has been shown in 6 and 8 see also the research announcements 7 and 9 that if for all 1 2 3 with which we will refer to as the nondegenerate case the Hilbert form satisfi es 5 if 6 1 p1 1 p2 1 p3 1 522CHRISTOPH THIELE and this homogeneity condition is easily seen to be necessary If 2 3 then trivially satisfi es 5 if in addition 7 p1 This leads to the natural conjecture that for fi xed exponents p1 p2 p3 with p1 the constant C in 5 can be chosen independently of as 2 approaches 3 Such uniform estimates can not be read from 6 and 8 these arguments give only estimates polynomially growing in 2 3 1 Defi ne a dual bilinear operator H S R S R S R to by H f1 f2 x f3 x dx f1 f2 f3 for all f1 f2 S R It is easy to see that H f1 f2 is actually a continuous function Then 5 is equivalent to 8 H f1 f2 p3 C f1 p1 f2 p2 for all f1 f2 S R where p3 denotes the conjugate exponent to p3 The purpose of this paper is to prove the following theorem which gives partial progress towards the above conjecture Theorem 1 1 There is a constant C such that for all unit vectors R3 perpendicular to with 2 3 0 9 x H f1 f2 x C f1 2 f2 2 1 The conclusion of this theorem is a weak type estimate it shows that L2 L2is mapped into the Lorentz space L1 rather than into the desired L1 If one could vary the exponents p1and p2 which in this theorem are both equal to 2 in an open set ofR2 then one could use Marcinkiewicz interpolation to obtain strong type estimates of the form 8 In a desire to keep this paper short we have only attempted to discuss the case p1 p2 2 We give a few further remarks on this theorem Remark 1 It is no restriction to assume 1 0 Remark 2 The known proof for the nondegenerate case yields uniform bounds as long as remains in a compact set away from the degenerate case Hence it suffi ces to prove the theorem for 2 3 0 The main issue in proving this theorem is to combine the Fourier methods which are needed in the nondegenerate case with the simple argument of H older s inequality which works in the degenerate case We follow largely the A UNIFORM ESTIMATE523 proof in the nondegenerate case with the decomposition of the Hilbert form adapted to the parameter and then extract large pieces of the Hilbert form which can be estimated by H older s inequality This paper is essentially self contained however a knowledge of 6 and 8 may help understanding We conclude with some fi nal remarks on the historical motivation of this article Calder on s fi rst commutator is the expression C1 f x p v f t A x A t x t 2 dt where A is some Lipschitz function It is well known 1 that C1is bounded on L2 R One of the previously unsuccessful approaches to this result has been the formal calculation C1 f 1 0 p v f t A t x t 1 x t dt d Inside the integration we fi nd bilinear operators in f and A of the type H and one desires to prove that these operators map L2 L into L2with a bound that depends on such that the integral over is fi nite By duality one seeks to prove that the operator B f1 f2 x p v f1 x t f2 x 1 t dt t maps L2 L2 L1with a constant integrable in 0 1 Assume 1 2 Keeping track of the constants in 6 and 8 gives B f1 f2 p C m f1 p1 f2 p2 for some m 0 provided 1 p 1 p1 1 p2and p1 p2 p remains in a fi xed small neighborhood of 2 2 1 Interpolation with Theorem 1 1 which states uniform weak type for p1 p2 p 2 2 1 gives the weak type estimate x B f1 f2 x C m f1 p p1 f2 p p2 p for some m 0 provided 1 p 1 p1 1 p2and the distance from p1 p2 p to 2 2 1 is less than Thus x B f1 f2 x C f1 p p1 f2 p p2 p provided p 1 p1 1 p2and the distance from p1 p2 p to 2 2 1 is less than log a 1 Thus interpolation as in 5 gives the strong type estimate 524CHRISTOPH THIELE B f1 f2 1 C log m f1 2 f2 2 for some m 0 In particular the dependence on is locally integrable near 0 A symmetric argument deals with in the interval 1 2 1 Thus we are able to complete the above approach to Calder on s commutator and reprove boundedness of the commutator in L2 The result of this paper is part of the author s senior thesis Habilitation 1998 at the Christian Albrechts Universit at Kiel 14 However by a faulty in terpolation argument it is claimed there that one can deduce from Theorem 1 1 strong type L2 L2 L1instead of weak type I am indebted to Xiaochun Li for pointing out this mistake Since then Grafakos Li 3 and Li 10 have ex tended the results of this paper to obtain weak type estimates for more tuples of exponents so that they can in particular deduce by a correct interpolation argument the strong type estimate claimed in 14 It should also be noted that between submission of the current article and its appearance further work following up on the results of this paper has been circulated mostly in form of preprints Muscalu 12 and Grafakos Li 4 have discussed bilinear mul tipliers with singularities along curved manifolds using techniques of uniform estimates Mucalu Tao and the author 13 have proved uniform estimates for multilinear generalizations of the bilinear operators discussed here A large part of the work presented here has been done during delightful stays at Berkeley MSRI and Princeton IAS and University I am grateful to M Christ C Feff erman M Lacey and D M uller for many stimulating discussions concerning this subject I am also grateful to C Muscalu and T Tao for carefully reading this manuscript and sharing their views on it which has lead to a deeper understanding of the matter Finally I would like to thank the referee for carefully reading the paper and passing on many useful comments 2 The decomposition of the Hilbert forms In this paper C and m denote universal constants whose value may change from line to line Defi ne the following isometries of Lp R Lyf x f x y U f x f x ei x Dp f x 1 pf 1x Defi ne the Fourier transform f of a function f onRdby f f x e i xdx If F S R2 G S R2 and A is a regular linear transformation ofR2 then A UNIFORM ESTIMATE525 an easy application of Plancherel s theorem shows R2 F Ax G x dx C R2 F G AT d Applying this with F x1 x2 f1 x1 f2 x2 and G x1 x2 f3 x1 p v 1 x2 gives f1 f2 f3 10 f1 x 1 3 t f2 x 2 3 t f3 x p v 1 t dtdx C f1 1 f2 2 f3 1 2 sign 1 3 1 2 3 2 d 1d 2 C 1 f1 1 f2 2 f3 3 sign d Here d denotes the Lebesgue measure on the hyperplane defi ned by 0 By a trivial decomposition of sign and by symmetry it suffi ces to consider instead of the trilinear form 1 f1 1 f2 2 f3 3 1R d Defi ne 2 100and N 100 By the remarks after Theorem 1 1 we can assume 0 2 3 N 100and 1 1 and 1 1 This discussion yields in particular that vanishes outside the half plane 0 Hence averaging gives for all with 0 c 1R R R 3 1 D N kL L D 2 N 1 dkd The estimate on the support of and the fact that maps into 0 1 imply that c is bounded above uniformly in To see that c is also bounded below uniformly in we discuss the preimage of 1 under A suffi cient condition on for 1 is 2N 1 for each This is in particular satisfi ed if with 2 2 N 1 c 0 for some universal c Hence it suffi ces to prove uniform bounds on the trilinear form defi ned by f1 f2 f3 R3 3 1 f k x dkd dx C 0 R R 3 1 D 2 N k 1L 2N 1 1 N f dkd d Here we have set k D1 2 1 Nk 1U 2N 1 1 N 3 The truncated trilinear form Let be an integer such that 12 N 2 3 2 3 1 Then E f1 f2 f3 C for a universal constant C Here Mpdenotes the maximal operator Mpf x sup x I 1 I I f x pdx 1 p We postpone the proof of Lemma 3 1 and prove that Lemma 3 1 implies The orem 1 1 Defi ne the operator H mapping S R S R S R by 14 H f1 f2 f3 f1 f2 f3 The estimate of Theorem 1 1 is equivalent to the estimate 15 x H f1 f2 x C 1 f1 2 f2 2 for all f1 f2 S R and 0 By homogeneity and linearity it suffi ces to prove 16 x H f1 f2 x C C for all f1 f2with f1 2 f2 2 1 Fix such f1and f2 and defi ne E 2 1 x M2f x 1 It suffi ces to prove the weak type estimate 16 for HEand H HEseparately where HEis the dual of Eas in 14 Consider the set F x HE f1 f2 x 1 528CHRISTOPH THIELE Let f3be a Schwartz function with f3 1 and f3 1F 2 min 1 HE f2 f3 1 2 We can assume that f3 2 1 because there is an immediate bound on F otherwise Hence M2 f3 f3 2 is bounded by 1 and we can apply Lemma 3 1 to conclude F E f1 f2 1F f3 2 E f1 f2 f3 f3 2 C C F 1 2 C This proves a uniform bound on F To complete the proof of Theorem 1 1 it now suffi ces to prove H HE f1 f2 L1 Ec C which in turn follows from the a priori estimate E f1 f2 f3 C for all Schwartz functions f3supported in Ecwith f3 1 First observe that since ktakes values between 0 and 1 1 1 k 2 k 3 k 1 1 k 1 2 k 1 3 k 17 1Ek D1 Nk 2 2 1Ek D1 Nk 2 Accordingly E f1 f2 f3 can be estimated by two terms the fi rst one being 18 R Ek R Lx0D1 Nk 2 x R 3 1 f k x d dxdx0dk By H older s inequality the inner integral is bounded by 2 1 R f k x 2d 1 2 sup f3 3 k x A calculation using Plancherel s theorem and the symmetry of show f k x L2 D 2 N k 1L xf U 2N 1 1 N 0 L2 C D 2 N k 1L xf 2 C f LxD2 2 1 Nk 1 2 For 1 2 we estimate this by C 1 2 N k 1dist x Ec sup y Ec M2f y C 1 N kdist x Ec A UNIFORM ESTIMATE529 For 3 we calculate similarly f3 3 k x L f3LxD1 2 1 3 Nk 1 1 C 1 N kdist x Ec 3 In the last inequality we have used the fact that f3is supported in Ec Hence we can estimate 18 by C R Ek R Lx0D1 Nk 2 x 1 N kdist x Ec 1 dxdx0dk If we do the x integration separately on the region 2 x x0 dist x0 Ec and on the complement of this region we see easily that this is bounded by C R Ek 1 N kdist x0 Ec 1 dx0dk C E Nk dist x0 Ec 1 N kdist x0 Ec 1 dkdx0 C E dx0 C The modifi cations to the last three lines which are needed to estimate the term corresponding to the second summand in 17 are left to the reader This completes the reduction of Theorem 1 1 to Lemma 3 1 4 The geometric picture and making discrete the trilinear forms We write E f1 f2 f3 0 1 2 k0 0 f1 f2 f3 dk0d 0 with k0 0 f1 f2 f3 k l Z R 3 1 k k0 x f k k0 l 0 x dx Obviously it suffi ces to prove Lemma 3 1 for k0 0instead of E For simplicity of notation we will only consider the case k0 0 0 however it will be clear that the estimates we obtain hold for arbitrary k0 0 0 1 Let be a nonnegative Schwartz function such that is supported in 1 1 and 0 1 Defi ne for any k n Z Ik n Nkn Nk n 1 We call such an interval Ik n an N adic interval Defi ne 1 k n 1Ik n D1 Nk 2 and k n 1Ik n D1 Nk 2 530CHRISTOPH THIELE for 2 3 Then we can write 19 0 0 f1 f2 f3 k l Z R 3 1 n Z k n x k x f k l x dx We introduce intervals k lsuch that the support of D 1 k lis con tained in k l More precisely we choose for each k l Z 1 2 3 a compact interval k l such that the following fi ve properties hold 1 The center c k l satisfi es 20 c k l N k l 1 2N k 2 The length k l satisfi es 21 N k 1 4 k l N k 1 6 3 For all k l 2 k l 3 k l 4 If two intervals k land k l are not disjoint then one is contained in the other 5 If k lstrictly contains k l then k lcontains 0 1 k l for all 1 2 3 Since 2 and 3 are very large the third property obviously does not confl ict with the fi rst two properties Hence it is obvious that one can choose intervals which satisfy the fi rst three properties The method to choose the intervals such that in addition properties 4 and 5 are satisfi ed is described in 6 One fi rst chooses intervals satisfying the fi rst three properties then one enlarges these intervals a little bit so that they in addition satisfy properties 4 and 5 We do not work out the details here Observe that the distance between 1 k land 2 k lis approximately 1N k which is large compared to the size of the intervals In particular the two inter vals are disjoint Since 1 0 we have 1 k l 2 k lin the sense that 1 0 and any interval J denotes the interval which has length A J and the same center as J Defi ne Tkto be the set of integers n such that k n T We will describe below an algorithm which selects certain trees of type 2 with the property that if P T and P P such that IP IP ITand T 2 P then P T Observe that if T is such a tree then 1 k 0 implies Tk Tk We are aiming to use Littlewood Paley theory near the frequencies T hence we defi ne k T an