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

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 the Hilbert transform is defined on test functions

(up to irrelevant constants) as

where the integral is evaluated in the principal value sense (removing the region

to ensure integrability, and then taking the limit as

.)

One of the basic results in (linear) harmonic analysis is that the Hilbert transform is bounded on

for every

, thus for each such p there exists a finite constant

such 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 singularkernel

as being of similar “strength” to the Dirac 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 of

diverges logarithmically both at zero and at infinity. If the kernel was absolutely integrable, then the above

boundedness result would be a simple consequence of Young’s inequality (or Minkowski’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 into wavelets- 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 between

and

), in particular one should have the Hölder-type inequality

(*)

whenever

and

.(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 constant

in that case. Note that by setting g=1 and looking at the limiting case

we 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

, where

is 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 origin

a 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 breakthrough pair of papers by Lacey and Thiele, first in the “easy” region

(in which all functions are locally in

and so local Fourier analytic methods are particularly tractable) and then in the significantly larger region where

. (Extending the latter result to

or 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 method also leads to the simplest proof known of Carleson’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 expect

to behave like

, we expect the trilinear Hilbert transform to obey a Hölder-type inequality

(**)

whenever

and

. 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”

, where

is 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 shown by 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 paper of Demeter, Thiele, and myself. There is also a paper of Christ in which a different type of additive combinatorics (coming, in fact, from work 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 field

is replaced by a Cantor ring

of 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.