Open question: boundedness of the trilinear Hilbert transform « What’s new

Open question: boundedness of the trilinear Hilbert transform
Thursday, May 10th, 2007 inquestion
Thisis a well-known problem in multilinear harmonic analysis; it isfascinating to me because it lies barely beyond the reach of the besttechnology we have for these problems (namely, multiscaletime-frequency analysis), and because the most recent developments inquadratic Fourier analysis seem likely to shed some light on thisproblem.
Recall that theHilbert transform is defined on test functions(up to irrelevant constants) as

where the integral is evaluated in the principal value sense (removing the regionto ensure integrability, and then taking the limit as.)
One of the basic results in (linear) harmonic analysis is that the Hilbert transform is bounded onfor every, thus for each such p there exists a finite constantsuch that

One can view boundedness result (which is of importance in complexanalysis and one-dimensional Fourier analysis, while also providing amodel case of the more general Calderón-Zygmundtheory of singular integral operators) as an assertion that the Hilberttransform is “not much larger than” the identity operator. And indeedthe two operators are very similar; both are invariant undertranslations and dilations, and on the Fourier side, the Hilberttransform barely changes the magnitude of the Fourier transform at all:

In fact, one can show the only reasonable (e.g.-bounded)operators which are invariant under translations and dilations are justthe linear combinations of the Hilbert transform and the identityoperator. (A useful heuristic in this area is to view the singularkernelas being of similar “strength” to theDirac delta function- for instance, they have same scale-invariance properties.)
Note that the Hilbert transform is formally a convolution of f with the kernel. This kernel is almost, but not quite, absolutely integrable - the integral ofdiverges logarithmically both at zero and at infinity. If the kernel was absolutely integrable, then the aboveboundedness result would be a simple consequence ofYoung’s inequality (orMinkowski’s inequality);the difficulty is thus “just” one of avoiding a logarithmic divergence.To put it another way, if one dyadically decomposes the Hilberttransform into pieces localised at different scales (e.g. restrictingto an “annulus”),then it is a triviality to establish boundedness of each component; thedifficulty is ensuring that there is enough cancellation ororthogonality that one can sum over the (logarithmically infinitenumber of) scales and still recover boundedness.
There are a number of ways to establish boundedness of the Hilberttransform. One way is to decompose all functions involved intowavelets- functions which are localised in space and scale, and whosefrequencies stay at a fixed distance from the origin (relative to thescale). By using standard estimates concerning how a function can bedecomposed into wavelets, how the Hilbert transform acts on wavelets,and how wavelets can be used to reconstitute functions, one canestablish the desired boundedness. The use of wavelets to mediate theaction of the Hilbert transform fits well with the two symmetries ofthe Hilbert transform (translation and scaling), because the collectionof wavelets also obeys (discrete versions of) these symmetries. One canview the theory of such wavelets as a dyadic framework for Calderón-Zygmund theory.
Just as the Hilbert transform behaves like the identity, it was conjectured by Calderón (motivated by the study of the Cauchy integral on Lipschitz curves) that the bilinear Hilbert transform

would behave like the pointwise product operator(exhibiting again the analogy betweenand), in particular one should have theHölder-type inequality
(*)
wheneverand.(There is nothing special about the “2″ in the definition of thebilinear Hilbert transform; one can replace this constant by any otherconstant except for 0, 1, or infinity, though it is a delicate issue tomaintain good control on the constantin that case. Note that by setting g=1 and looking at the limiting casewe recover the linear Hilbert transform theory from the bilinear one,thus we expect the bilinear theory to be harder.) Again, this claim istrivial when localising to a single scale, as it can then be quickly deduced from Hölder’s inequality. The difficulty is then to combine all the scales together.
It took some time to realise that Calderón-Zygmundtheory, despite being incredibly effective in the linear setting, wasnot the right tool for the bilinear problem. One way to see the problemis to observe that the bilinear Hilbert transform B (or more precisely,the estimate (*)) enjoys one additional symmetry beyond the scaling andtranslation symmetries that the Hilbert transform H obeyed. Namely, onehas the modulation invariance

for any frequency, whereis the linear plane wave of frequency,which leads to a modulation symmetry for the estimate (*). Thissymmetry - which has no non-trivial analogue in the linear Hilberttransform - is a consequence of the algebraic identity

which can in turn be viewed as an assertion that linear functions have a vanishing second derivative.
It is a general principle that if one wants toestablish a delicate estimate which is invariant under some non-compactgroup of symmetries, then the proof of that estimate shouldalso be largely invariant under that symmetry (or, if it doeseventually decide to break the symmetry (e.g. by performing anormalisation), it should do so in a way that will yield some tangibleprofit). Calderón-Zygmund theory gives the frequency origina preferred role (for instance, all wavelets have mean zero, i.e. theirFourier transforms vanish at the frequency origin), and so is not theappropriate tool for any modulation-invariant problem.
The conjecture of Calderón was finally verified in a breakthroughpair ofpapers by Lacey and Thiele, first in the “easy” region(in which all functions are locally inand so local Fourier analytic methods are particularly tractable) and then in the significantly larger region where. (Extending the latter result toor beyond remains open, and can be viewed as a toy version of thetrilinear Hilbert transform question discussed below.) The key idea(dating back to Fefferman) was to replace the wavelet decomposition bya more general wave packet decomposition - wave packets beingfunctions which are well localised in position, scale, and frequency,but are more general than wavelets in that their frequencies do notneed to hover near the origin; in particular, the wave packet frameworkenjoys the same symmetries as the estimate that one is seeking toprove. (As such, wave packets are a highly overdetermined basis, incontrast to the exact bases that wavelets offers, but this turns out tonot be a problem, provided that one focuses more on decomposing the operator Brather than the individual functions f,g.) Once the wave packets areused to mediate the action of the bilinear Hilbert transform B, Laceyand Thiele then used a carefully chosen combinatorial algorithm toorganise these packets into “trees” concentrated in mostly disjointregions of phase space, applying (modulated) Calderón-Zygmundtheory to each tree, and then using orthogonality methods to sum thecontributions of the trees together. (The same methodalso leads to the simplest proof known ofCarleson’s celebrated theorem on convergence of Fourier series.)
Since the Lacey-Thiele breakthrough, there has been a flurry ofother papers (including some that I was involved in) extending thetime-frequency method to many other types of operators; all of thesehad the characteristic that these operators were invariant (or“morally” invariant) under translation, dilation, and some sort ofmodulation; this includes a number of operators of interest to ergodictheory and to nonlinear scattering theory. However, in this post I wantto instead discuss an operator which does not lie in this class, namelythe trilinear Hilbert transform

Again, since we expectto behave like, we expect the trilinear Hilbert transform to obey a Hölder-type inequality
(**)
wheneverand. This conjecture is currently unknown for any exponents p,q,r - even thecase p=q=r=4, which is the “easiest” case by symmetry, duality andinterpolation arguments. The main new difficulty is that in addition tothe three existing invariances of translation, scaling, and modulation(actually, modulation is now a two-parameter invariance), one now alsohas a quadratic modulation invariance

for any “quadratic frequency”, whereis the quadratic plane wave of frequency,which leads to a quadratic modulation symmetry for the estimate (**).This symmetry is a consequence of the algebraic identity

which can in turn be viewed as an assertion that quadratic functions have a vanishing third derivative.
It is because of this symmetry that time-frequencymethods based on Fefferman-Lacey-Thiele style wave packets seem to beineffective (though the failure is very slight; one can control entire“forests” of trees of wave packets, but when summing up all therelevant forests in the problem one unfortunately encounters alogarithmic divergence; also, it is known that if one ignores the signof the wave packet coefficients and only concentrates on the magnitude- which one can get away with for the bilinear Hilbert transform - thenthe associated trilinear expression is in fact divergent). Indeed, wavepackets are certainly not invariant under quadratic modulations. Onecan then hope to work with the next obvious generalisation of wavepackets, namely the “chirps” - quadratically modulated wave packets -but the combinatorics of organising these chirps into anythingresembling trees or forests seems to be very difficult. Also, recentwork in the additive combinatorial approach to Szemerédi’s theorem (aswell as in the ergodic theory approaches) suggests that these quadraticmodulations might not be the only obstruction, that other “2-stepnilpotent” modulations may also need to be somehow catered for. IndeedI suspect that some of the modern theory of Szemerédi’s theorem forprogressions of length 4 will have to be invoked in order to solve thetrilinear problem. (Again based on analogy with the literature onSzemerédi’s theorem, the problem of quartilinear and higher Hilberttransforms is likely to be significantly more difficult still, and thusnot worth studying at this stage.)
This problem may be too difficult to attack directly,and one might look at some easier model problems first. One that wasalready briefly mentioned above was to return to the bilinear Hilberttransform and try to establish an endpoint result at r=2/3. At thispoint there is again a logarithmic failure of the time-frequencymethod, and so one is forced to hunt for a different approach. Anotheris to look at the bilinear maximal operator

which is a bilinear variant of the Hardy-Littlewood maximaloperator, in much the same way that the bilinear Hilbert transform is avariant of the linear Hilbert transform. It was shownby Laceythat this operator obeys most of the bounds that the bilinear Hilberttransform does, but the argument is rather complicated, combining thetime-frequency analysis with some Fourier-analytic maximal inequalitiesof Bourgain.In particular, despite the “positive” (non-oscillatory) nature of themaximal operator, the only known proof of the boundedness of thisoperator is oscillatory. It is thus natural to seek a “positive” proofthat does not require as much use of oscillatory tools such as theFourier transform, in particular it is tempting to try an additivecombinatorial approach. Such an approach has had some success with aslightly easier operator in a similar spirit, in an unpublished paperof Demeter, Thiele, and myself. There is also apaper of Christ in which a different type of additive combinatorics (coming, in fact, fromwork on the Kakeya problem)was used to establish a non-trivial estimate for single-scale model ofvarious multilinear Hilbert transform or maximal operators. If theseoperators are understood better, then perhaps additive combinatoricscan be used to attack the trilinear maximal operator, and thence to thetrilinear Hilbert transform. (This trilinear maximal operator,incidentally, has some applications to pointwise convergence ofmultiple averages in ergodic theory.)
Another, rather different, approach would be to work in the “finite field model” in which the underlying fieldis replaced by a Cantor ringof formal Laurent series over a finite field F; in such “dyadic models”the analysis is known to be somewhat simpler (in large part because inthis non-Archimedean setting it now becomes possible to create wavepackets which are localised in both space and frequency). Nazarov hasan unpublished proof of the boundedness of the bilinear Hilberttransform in characteristic 3 settings based on a Bellman functionapproach; it may be that one could achieve something similar over thefield of 4 elements for (a suitably defined version of) the trilinearHilbert transform. This would at least give supporting evidence for theanalogous conjecture in, although it looks unlikely that a positive result in the dyadic setting would have a direct impact on the continuous one.
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May 20th, 2007 at 5:56 am
Gil Kalai
Whilequite far from my own research I find this problem and posting veryinteresting for the following reasons. First, shortly after theCalderon conjecture was proved I heard a talk about it by Chris Thieleat the HU colloquium which (to my surprise) I could follow and enjoy.(And even find there was some combinatorics.) Second, at a later timeRafi Coifman gave me the Lacey-Thiele theorem as an example howconcepts and ideas from applied mathematics interlace and influence thestudy of classical pure math problem. (This aspect of the (even wider)story is something I will be happy to learn more about.) Third, Ialways wondered if Carleson’s theorem could ever be “industrialized”.(This is a fancy way to the ask if we can hope for a proof that can(really) be presented at class.) I do not know if the Lacey-Thieleproof quite get there but it is certainly in this direction. And, theanalogy with Szemeredi theorem is intruiging.
Is there a quick reference/link for an easy proof for the boundedness of Hilbert transform, especially the proof using wavelets?
Regarding the analogy with Szemeredi’s theorem. Are wavelets orsomething similar useful there? Is there a way to think about the Rothcase (or its improved versions) as related to a large spanning class offunctions like wavelets?
I remember that various people (e.g. Eli Shamir) raised the ideathat appropriate notion of wavelets can be useful to improve knownresults and find further results in places were Fourier analysisapplies to combinatorics, (mainly Fourier analysis over the discretecube, or products of fixed graphs); but I am not aware of some nicedefinitions/applications. In particular, I do not know what waveletsare in these contexts.
May 20th, 2007 at 7:28 am
Terence Tao

Dear Gil,
Good questions! I may need several responses to answer all of them.
Coifman actually has a nice principle connecting pure and appliedanalysis: given any operator T, there is a positive correlation betweenT being tractable to estimate in pure analysis, and T having anefficiently computable representation for applied analysis purposes.The point is that the representations which allow T to be computedquickly, tend to also be the representations which allow one to analyseT efficiently, and vice versa.
For instance, suppose one wanted to compute the discrete Hilberttransform Hf of a function f on an N point set (e.g. the cyclic groupZ/NZ). A naive computation of Hf would take O(N^2) computations, but byusing either the Fourier representation or the wavelet representation(and using the FFT or FWT) one cuts this down to O(N log N).
For the bilinear Hilbert transform B, discretised to Z/NZ, I don’t knowof a way to compute B(f,g) for given f,g which is any faster thanO(N^2) steps. However, given f, g, h, one can compute the inner productin something like O( N log N ) steps by using wave packets; roughly speaking, there is a representation of the form

where P ranges over all Heisenberg tiles (dyadic rectangles in phase space with area),is the spatial width of P, andare (-normalised)wave packets which are essentially localised in phase space to P. Thereare about O(N log N) of these tiles, and the above inner products canall be computed by an FFT-like algorithm in O(N log N) time. (The FFTperforms O(N log N) computations but only retains N of the numberscomputed as the Fourier transform. If instead one saves all of the O(Nlog N) numbers that one computes, one essentially obtains the wavepacket transform.) In accordance with Coifman’s principle, the aboverepresentation is precisely what Lacey and Thiele need to control thebilinear Hilbert transform properly.
No such efficient representation (better than O(N^2)) is known for thetrilinear Hilbert transform; finding a fast representation here may bethe key to this problem.
There is a “finite field”, “single scale” model which might be moretractable for this latter problem. Let G be a finite group (whose orderis coprime to 2 and 3). If one is given three functionsand one wants to compute (exactly) the expression

then by using the FFT one can achieve this in O( |G| log |G| ) computations. But what if one is given four functionsand wants to compute

I know of no way to compute this in fewer thancomputations, even for nice groups G such as.A faster algorithm to compute this expression exactly may be veryuseful to the trilinear Hilbert transform problem, by suggesting the“correct” way to represent that transform.
There is an analogous problem for the Gowers norms; the FFT allows one to compute thenorm of f in O( |G| log |G| ) steps, but the fastest I can compute thenorm in issteps (by using the recursive formula connecting each Gowers norm to its predecessor). Is there a faster way?
May 20th, 2007 at 7:48 am
Terence Tao

TheLacey-Thiele proof Carleson’s theorem can be presented in a graduateharmonic analysis class in a handful of lectures. It is “simple” moduloknowing all the modern machinery of harmonic analysis (Calderon-Zygmundtheory, Littlewood-Paley theory, interpolation theory, and theuncertainty principle). Things are a little simpler in a “finite field”model, in which the real line is replaced by the Walsh-Cantor group. See for instancemy lecture notes on these topics.
Wavelets appear in harmonic analysis due to the role of scale,which is largely absent in additive combinatorics, and in particular inSzemeredi’s theorem. In Szemeredi’s theorem there is essentially nodistinction made between an arithmetic progression of small step and anarithmetic progression of large step, thus there is no need to treatthese two scales separately (and indeed, since there are so many morelarge steps than small steps, the net contribution of the small stepsis negligible compared to the large). But in things like the Hilberttransforms, the presence of the 1/t factor in the integral causes thesmall scales to be elevated to be of “equal strength” to the largescales, and so they have to be treated separately. Wavelets are a goodway to separate the fine-scale and coarse-scale behaviours from eachother. Wave packets are generalisations of wavelets which also capturefrequency information, indeed the (overdetermined) wave packet basiscontains both the wavelet basis and the Fourier basis as subsets.
So, in summary, I would only expect wavelet-type tools to be usefulin “multiscale” situations in which one needs to treat fine-scale andcoarse-scale oscillations separately.
As to the “easy” proof of the boundedness of the Hilbert transformusing wavelets, this is a little trickier to answer; again, this resultis easy only modulo a certain amount of standard wavelet theory.Roughly speaking, the main tool is the Littlewood-Paley inequality forwavelets

which asserts in particular that thenorm of a function f is controlled by the magnitude of the wavelet coefficients,but is insensitive to the phase. In the wavelet basis, the Hilberttransform is almost a diagonal operator; it alters the phase of thewavelet coefficients, and also mixes some nearby wavelet coefficientstogether, but does not significantly affect the magnitude of thesecoefficients. Making this precise, one can easily deduce theboundedness of the Hilbert transform (and many other operators) fromthe above inequality (plus some other useful tools, such as theFefferman-Stein maximal inequality). The proof of the Littlewood-Paleyinequality is not too difficult, but is basically a fancy version ofthe standard arguments used to establish boundedness of the Hilberttransform. Thus if boundedness of the Hilbert transform was your onlygoal, this would not be the most efficient way to establish it; but ifone was interested in treating systematically a large class ofoperators by a single theory, this is a good approach.
May 20th, 2007 at 10:08 pm
Gil Kalai
Thanks, Terry
You wrote: “In Szemeredi’s theorem there is essentially nodistinction made between an arithmetic progression of small step and anarithmetic progression of large step, thus there is no need to treatthese two scales separately (and indeed, since there are so many morelarge steps than small steps, the net contribution of the small stepsis negligible compared to the large).”
Is what you wrote a characteristic of Szemeredi’s theorem or ratherof its proofs? (We can talk just about Roth’s theorem or the analogousproblem for bounds for cap sets.) We will be happy to find a 3-term APof any step; For the analysis in the present proofs AP with small stepsare negligebale. Couldn’t it give an advantage to consider kernelswhich introduce delicate multi scale situations?
May 21st, 2007 at 8:40 am
Terence Tao

Dear Gil,
This is possible, but I think it is unlikely (at least if one uses“naive” notions of scale) because it doesn’t seem to be compatible withthe symmetries of the problem. For instance, in Z/pZ, the statement ofSzemeredi’s theorem is invariant under any dilation by an invertibleelement of the field Z/pZ. Thus, there is no reason to give small steps(i.e. steps r in an interval such as {1,…,R}) any more privileged rolethan steps in a dilated interval such as.This observation was used by Varnavides to show that once Szemeredi’stheorem is proven for a small interval, it automatically holds for alarge interval as well, and furthermore provides a positive densityfamily of arithmetic progressions in that larger interval.
That said, though, there is certainly a lot of scope for utilising adaptivenotions of scale, tailored to the set or function that one is seekingarithmetic progressions inside. Indeed, the known proofs of Roth andSzemeredi all do this in one way or another. For instance, to findarithmetic progressions of length 3 inside a set A in Z/pZ, one canfirst look for a large Fourier coefficient. If there is no such large Fourier coefficient, then we “win”; if instead we find a frequencywhere things are large, we can use that frequency to determine a“scale” (we can give each element x in Z/pZ a “norm”, defined as thedistance ofto the nearest integer). We can then pass to a smaller scale (i.e. to a Bohr set generated by),and this is the heart of the original “density-incremented” proof ofRoth’s theorem. Or we can then go and hunt for more types of scales tocapture as much of the global structure of A as one can, until we haveso much control that it becomes “obvious” that there are lots ofprogressions; this brings us closer to the more “ergodic” or“energy-incremented” proofs of Roth’s theorem, or to the hybridenergy+density increment proofs of Heath-Brown and Szemeredi.
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