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I have just uploaded to the arXiv my paper “Sharp bounds for multilinear curved Kakeya, restriction and oscillatory integral estimates away from the endpoint“, submitted to Mathematika. In this paper I return (after more than a decade’s absence) to one of my first research interests, namely the Kakeya and restriction family of conjectures. The starting point is the following “multilinear Kakeya estimate” first established in the non-endpoint case by Bennett, Carbery, and myself, and then in the endpoint case by Guth (with further proofs and extensions by Bourgain-Guth and Carbery-Valdimarsson:

Theorem 1 (Multilinear Kakeya estimate)Let be a radius. For each , let denote a finite family of infinite tubes in of radius . Assume the following axiom:

- (i) (Transversality) whenever is oriented in the direction of a unit vector for , we have
for some , where we use the usual Euclidean norm on the wedge product .

where are the usual Lebesgue norms with respect to Lebesgue measure, denotes the indicator function of , and denotes the cardinality of .

The original proof of this proceeded using a heat flow monotonicity method, which in my previous post I reinterpreted using a “virtual integration” concept on a fractional Cartesian product space. It turns out that this machinery is somewhat flexible, and can be used to establish some other estimates of this type. The first result of this paper is to extend the above theorem to the curved setting, in which one localises to a ball of radius (and sets to be small), but allows the tubes to be curved in a fashion. If one runs the heat flow monotonicity argument, one now picks up some additional error terms arising from the curvature, but as the spatial scale approaches zero, the tubes become increasingly linear, and as such the error terms end up being an integrable multiple of the main term, at which point one can conclude by Gronwall’s inequality (actually for technical reasons we use a bootstrap argument instead of Gronwall). A key point in this approach is that one obtains optimal bounds (not losing factors of or ), so long as one stays away from the endpoint case (which does not seem to be easily treatable by the heat flow methods). Previously, the paper of Bennett, Carbery, and myself was able to use an induction on scale argument to obtain a curved multilinear Kakeya estimate losing a factor of (after optimising the argument); later arguments of Bourgain-Guth and Carbery-Valdimarsson, based on algebraic topology methods, could also obtain a curved multilinear Kakeya estimate without such losses, but only in the algebraic case when the tubes were neighbourhoods of algebraic curves of bounded degree.

Perhaps more interestingly, we are also able to extend the heat flow monotonicity method to apply directly to the multilinear restriction problem, giving the following global multilinear restriction estimate:

Theorem 2 (Multilinear restriction theorem)Let be an exponent, and let be a parameter. Let be a sufficiently large natural number, depending only on . For , let be an open subset of , and let be a smooth function obeying the following axioms:for any , , extended by zero outside of , and denotes the extension operator

Local versions of such estimate, in which is replaced with for some , and one accepts a loss of the form , were already established by Bennett, Carbery, and myself using an induction on scale argument. In a later paper of Bourgain-Guth these losses were removed by “epsilon removal lemmas” to recover Theorme 2, but only in the case when all the hypersurfaces involved had curvatures bounded away from zero.

There are two main new ingredients in the proof of Theorem 2. The first is to replace the usual induction on scales scheme to establish multilinear restriction by a “ball inflation” induction on scales scheme that more closely resembles the proof of decoupling theorems. In particular, we actually prove the more general family of estimates

where denotes the local energies

(actually for technical reasons it is more convenient to use a smoother weight than the strict cutoff to the disk ). With logarithmic losses, it is not difficult to establish this estimate by an upward induction on . To avoid such losses we use the heat flow monotonicity method. Here we run into the issue that the extension operators are complex-valued rather than non-negative, and thus would not be expected to obey many good montonicity properties. However, the local energies can be expressed in terms of the magnitude squared of what is essentially the Gabor transform of , and these are non-negative; furthermore, the dispersion relation associated to the extension operators implies that these Gabor transforms propagate along tubes, so that the situation becomes quite similar (up to several additional lower order error terms) to that in the multilinear Kakeya problem. (This can be viewed as a continuous version of the usual wave packet decomposition method used to relate restriction and Kakeya problems, which when combined with the heat flow monotonicity method allows for one to use a continuous version of induction on scales methods that do not concede any logarithmic factors.)

Finally, one can combine the curved multilinear Kakeya result with the multilinear restriction result to obtain estimates for multilinear oscillatory integrals away from the endpoint. Again, this sort of implication was already established in the previous paper of Bennett, Carbery, and myself, but the arguments there had some epsilon losses in the exponents; here we were able to run the argument more carefully and avoid these losses.

Earlier this month, Hao Huang (who, incidentally, was a graduate student here at UCLA) gave a remarkably short proof of a long-standing problem in theoretical computer science known as the sensitivity conjecture. See for instance this blog post of Gil Kalai for further discussion and links to many other online discussions of this result. One formulation of the theorem proved is as follows. Define the -dimensional hypercube graph to be the graph with vertex set , and with every vertex joined to the vertices , where is the standard basis of .

Theorem 1 (Lower bound on maximum degree of induced subgraphs of hypercube)Let be a set of at least vertices in . Then there is a vertex in that is adjacent (in ) to at least other vertices in .

The bound (or more precisely, ) is completely sharp, as shown by Chung, Furedi, Graham, and Seymour; we describe this example below the fold. When combined with earlier reductions of Gotsman-Linial and Nisan-Szegedy; we give these below the fold also.

Let be the adjacency matrix of (where we index the rows and columns directly by the vertices in , rather than selecting some enumeration ), thus when for some , and otherwise. The above theorem then asserts that if is a set of at least vertices, then the minor of has a row (or column) that contains at least non-zero entries.

The key step to prove this theorem is the construction of rather curious variant of the adjacency matrix :

Proposition 2There exists a matrix which is entrywise dominated by in the sense that

Assuming this proposition, the proof of Theorem 1 can now be quickly concluded. If we view as a linear operator on the -dimensional space of functions of , then by hypothesis this space has a -dimensional subspace on which acts by multiplication by . If is a set of at least vertices in , then the space of functions on has codimension at most in , and hence intersects non-trivially. Thus the minor of also has as an eigenvalue (this can also be derived from the Cauchy interlacing inequalities), and in particular this minor has operator norm at least . By Schur’s test, this implies that one of the rows or columns of this matrix has absolute values summing to at least , giving the claim.

Remark 3The argument actually gives a strengthening of Theorem 1: there exists a vertex of with the property that for every natural number , there are at least paths of length in the restriction of to that start from . Indeed, if we let be an eigenfunction of on , and let be a vertex in that maximises the value of , then for any we have that the component of is equal to ; on the other hand, by the triangle inequality, this component is at most times the number of length paths in starting from , giving the claim.

This argument can be viewed as an instance of a more general “interlacing method” to try to control the behaviour of a graph on all large subsets by first generating a matrix on with very good spectral properties, which are then partially inherited by the minor of by interlacing inequalities. In previous literature using this method (see e.g., this survey of Haemers, or this paper of Wilson), either the original adjacency matrix , or some non-negatively weighted version of that matrix, was used as the controlling matrix ; the novelty here is the use of signed controlling matrices. It will be interesting to see what further variants and applications of this method emerge in the near future. (Thanks to Anurag Bishoi in the comments for these references.)

The “magic” step in the above argument is constructing . In Huang’s paper, is constructed recursively in the dimension in a rather simple but mysterious fashion. Very recently, Roman Karasev gave an interpretation of this matrix in terms of the exterior algebra on . In this post I would like to give an alternate interpretation in terms of the operation of *twisted convolution*, which originated in the theory of the Heisenberg group in quantum mechanics.

Firstly note that the original adjacency matrix , when viewed as a linear operator on , is a convolution operator

where

is the counting measure on the standard basis , and denotes the ordinary convolution operation

As is well known, this operation is commutative and associative. Thus for instance the square of the adjacency operator is also a convolution operator

where the convolution kernel is moderately complicated:

The factor in this expansion comes from combining the two terms and , which both evaluate to .

More generally, given any bilinear form , one can define the *twisted convolution*

of two functions . This operation is no longer commutative (unless is symmetric). However, it remains associative; indeed, one can easily compute that

In particular, if we define the twisted convolution operator

then the square is also a twisted convolution operator

and the twisted convolution kernel can be computed as

For general bilinear forms , this twisted convolution is just as messy as is. But if we take the specific bilinear form

then for and for , and the above twisted convolution simplifies to

and now is very simple:

Thus the only eigenvalues of are and . The matrix is entrywise dominated by in the sense of (1), and in particular has trace zero; thus the and eigenvalues must occur with equal multiplicity, so in particular the eigenvalue occurs with multiplicity since the matrix has dimensions . This establishes Proposition 2.

Remark 4Twisted convolution is actually just a component of ordinary convolution, but not on the original group ; instead it relates to convolution on a Heisenberg group extension of this group. More specifically, define the Heisenberg group to be the set of pairs with group lawand inverse operation

(one can dispense with the negative signs here if desired, since we are in characteristic two). Convolution on is defined in the usual manner: one has

for any . Now if is a function on the original group , we can define the lift by the formula

and then by chasing all the definitions one soon verifies that

for any , thus relating twisted convolution to Heisenberg group convolution .

Remark 5With the twisting by the specific bilinear form given by (2), convolution by and now anticommute rather than commute. This makes the twisted convolution algebra isomorphic to a Clifford algebra (the real or complex algebra generated by formal generators subject to the relations for ) rather than the commutative algebra more familiar to abelian Fourier analysis. This connection to Clifford algebra (also observed independently by Tom Mrowka and by Daniel Matthews) may be linked to the exterior algebra interpretation of the argument in the recent preprint of Karasev mentioned above.

Remark 6One could replace the form (2) in this argument by any other bilinear form that obeyed the relations and for . However, this additional level of generality does not add much; any such will differ from by an antisymmetric form (so that for all , which in characteristic two implied that for all ), and such forms can always be decomposed as , where . As such, the matrices and are conjugate, with the conjugation operator being the diagonal matrix with entries at each vertex .

Remark 7(Added later) This remark combines the two previous remarks. One can view any of the matrices in Remark 6 as components of a single canonical matrix that is still of dimensions , but takes values in the Clifford algebra from Remark 5; with this “universal algebra” perspective, one no longer needs to make any arbitrary choices of form . More precisely, let denote the vector space of functions from the hypercube to the Clifford algebra; as a real vector space, this is a dimensional space, isomorphic to the direct sum of copies of , as the Clifford algebra is itself dimensional. One can then define a canonical Clifford adjacency operator on this space bywhere are the generators of . This operator can either be identified with a Clifford-valued matrix or as a real-valued matrix. In either case one still has the key algebraic relations and , ensuring that when viewed as a real matrix, half of the eigenvalues are equal to and half equal to . One can then use this matrix in place of any of the to establish Theorem 1 (noting that Schur’s test continues to work for Clifford-valued matrices because of the norm structure on ).

To relate to the real matrices , first observe that each point in the hypercube can be associated with a one-dimensional real subspace (i.e., a line) in the Clifford algebra by the formula

for any (note that this definition is well-defined even if the are out of order or contain repetitions). This can be viewed as a discrete line bundle over the hypercube. Since for any , we see that the -dimensional real linear subspace of of sections of this bundle, that is to say the space of functions such that for all , is an invariant subspace of . (Indeed, using the left-action of the Clifford algebra on , which commutes with , one can naturally identify with , with the left action of acting purely on the first factor and acting purely on the second factor.) Any trivialisation of this line bundle lets us interpret the restriction of to as a real matrix. In particular, given one of the bilinear forms from Remark 6, we can identify with by identifying any real function with the lift defined by

whenever . A somewhat tedious computation using the properties of then eventually gives the intertwining identity

and so is conjugate to .

Let be some domain (such as the real numbers). For any natural number , let denote the space of symmetric real-valued functions on variables , thus

for any permutation . For instance, for any natural numbers , the elementary symmetric polynomials

will be an element of . With the pointwise product operation, becomes a commutative real algebra. We include the case , in which case consists solely of the real constants.

Given two natural numbers , one can “lift” a symmetric function of variables to a symmetric function of variables by the formula

where ranges over all injections from to (the latter formula making it clearer that is symmetric). Thus for instance

and

Also we have

With these conventions, we see that vanishes for , and is equal to if . We also have the transitivity

if .

The lifting map is a linear map from to , but it is not a ring homomorphism. For instance, when , one has

In general, one has the identity

for all natural numbers and , , where range over all injections , with . Combinatorially, the identity (2) follows from the fact that given any injections and with total image of cardinality , one has , and furthermore there exist precisely triples of injections , , such that and .

Example 1When , one haswhich is just a restatement of the identity

Note that the coefficients appearing in (2) do not depend on the final number of variables . We may therefore abstract the role of from the law (2) by introducing the real algebra of formal sums

where for each , is an element of (with only finitely many of the being non-zero), and with the formal symbol being formally linear, thus

and

for and scalars , and with multiplication given by the analogue

of (2). Thus for instance, in this algebra we have

and

Informally, is an abstraction (or “inverse limit”) of the concept of a symmetric function of an unspecified number of variables, which are formed by summing terms that each involve only a bounded number of these variables at a time. One can check (somewhat tediously) that is indeed a commutative real algebra, with a unit . (I do not know if this algebra has previously been studied in the literature; it is somewhat analogous to the abstract algebra of finite linear combinations of Schur polynomials, with multiplication given by a Littlewood-Richardson rule. )

For natural numbers , there is an obvious specialisation map from to , defined by the formula

Thus, for instance, maps to and to . From (2) and (3) we see that this map is an algebra homomorphism, even though the maps and are not homomorphisms. By inspecting the component of we see that the homomorphism is in fact surjective.

Now suppose that we have a measure on the space , which then induces a product measure on every product space . To avoid degeneracies we will assume that the integral is strictly positive. Assuming suitable measurability and integrability hypotheses, a function can then be integrated against this product measure to produce a number

In the event that arises as a lift of another function , then from Fubini’s theorem we obtain the formula

is an element of the formal algebra , then

Note that by hypothesis, only finitely many terms on the right-hand side are non-zero.

Now for a key observation: whereas the left-hand side of (6) only makes sense when is a natural number, the right-hand side is meaningful when takes a fractional value (or even when it takes negative or complex values!), interpreting the binomial coefficient as a polynomial in . As such, this suggests a way to introduce a “virtual” concept of a symmetric function on a fractional power space for such values of , and even to integrate such functions against product measures , even if the fractional power does not exist in the usual set-theoretic sense (and similarly does not exist in the usual measure-theoretic sense). More precisely, for arbitrary real or complex , we now *define* to be the space of abstract objects

with and (and now interpreted as formal symbols, with the structure of a commutative real algebra inherited from , thus

In particular, the multiplication law (2) continues to hold for such values of , thanks to (3). Given any measure on , we formally define a measure on with regards to which we can integrate elements of by the formula (6) (providing one has sufficient measurability and integrability to make sense of this formula), thus providing a sort of “fractional dimensional integral” for symmetric functions. Thus, for instance, with this formalism the identities (4), (5) now hold for fractional values of , even though the formal space no longer makes sense as a set, and the formal measure no longer makes sense as a measure. (The formalism here is somewhat reminiscent of the technique of dimensional regularisation employed in the physical literature in order to assign values to otherwise divergent integrals. See also this post for an unrelated abstraction of the integration concept involving integration over supercommutative variables (and in particular over fermionic variables).)

Example 2Suppose is a probability measure on , and is a random variable; on any power , we let be the usual independent copies of on , thus for . Then for any real or complex , the formal integralcan be evaluated by first using the identity

(cf. (1)) and then using (6) and the probability measure hypothesis to conclude that

For a natural number, this identity has the probabilistic interpretation

whenever are jointly independent copies of , which reflects the well known fact that the sum has expectation and variance . One can thus view (7) as an abstract generalisation of (8) to the case when is fractional, negative, or even complex, despite the fact that there is no sensible way in this case to talk about independent copies of in the standard framework of probability theory.

In this particular case, the quantity (7) is non-negative for every nonnegative , which looks plausible given the form of the left-hand side. Unfortunately, this sort of non-negativity does not always hold; for instance, if has mean zero, one can check that

and the right-hand side can become negative for . This is a shame, because otherwise one could hope to start endowing with some sort of commutative von Neumann algebra type structure (or the abstract probability structure discussed in this previous post) and then interpret it as a genuine measure space rather than as a virtual one. (This failure of positivity is related to the fact that the characteristic function of a random variable, when raised to the power, need not be a characteristic function of any random variable once is no longer a natural number: “fractional convolution” does not preserve positivity!) However, one vestige of positivity remains: if is non-negative, then so is

One can wonder what the point is to all of this abstract formalism and how it relates to the rest of mathematics. For me, this formalism originated implicitly in an old paper I wrote with Jon Bennett and Tony Carbery on the multilinear restriction and Kakeya conjectures, though we did not have a good language for working with it at the time, instead working first with the case of natural number exponents and appealing to a general extrapolation theorem to then obtain various identities in the fractional case. The connection between these fractional dimensional integrals and more traditional integrals ultimately arises from the simple identity

(where the right-hand side should be viewed as the fractional dimensional integral of the unit against ). As such, one can manipulate powers of ordinary integrals using the machinery of fractional dimensional integrals. A key lemma in this regard is

Lemma 3 (Differentiation formula)Suppose that a positive measure on depends on some parameter and varies by the formula

for some function . Let be any real or complex number. Then, assuming sufficient smoothness and integrability of all quantities involved, we have

for all that are independent of . If we allow to now depend on also, then we have the more general total derivative formula

again assuming sufficient amounts of smoothness and regularity.

*Proof:* We just prove (10), as (11) then follows by same argument used to prove the usual product rule. By linearity it suffices to verify this identity in the case for some symmetric function for a natural number . By (6), the left-hand side of (10) is then

Differentiating under the integral sign using (9) we have

and similarly

where are the standard copies of on :

By the product rule, we can thus expand (12) as

where we have suppressed the dependence on for brevity. Since , we can write this expression using (6) as

where is the symmetric function

But from (2) one has

and the claim follows.

Remark 4It is also instructive to prove this lemma in the special case when is a natural number, in which case the fractional dimensional integral can be interpreted as a classical integral. In this case, the identity (10) is immediate from applying the product rule to (9) to conclude thatOne could in fact derive (10) for arbitrary real or complex from the case when is a natural number by an extrapolation argument; see the appendix of my paper with Bennett and Carbery for details.

Let us give a simple PDE application of this lemma as illustration:

Proposition 5 (Heat flow monotonicity)Let be a solution to the heat equation with initial data a rapidly decreasing finite non-negative Radon measure, or more explicitlyfor al . Then for any , the quantity

is monotone non-decreasing in for , constant for , and monotone non-increasing for .

*Proof:* By a limiting argument we may assume that is absolutely continuous, with Radon-Nikodym derivative a test function; this is more than enough regularity to justify the arguments below.

For any , let denote the Radon measure

Then the quantity can be written as a fractional dimensional integral

Observe that

and thus by Lemma 3 and the product rule

where we use for the variable of integration in the factor space of .

To simplify this expression we will take advantage of integration by parts in the variable. Specifically, in any direction , we have

and hence by Lemma 3

Multiplying by and integrating by parts, we see that

where we use the Einstein summation convention in . Similarly, if is any reasonable function depending only on , we have

and hence on integration by parts

We conclude that

and thus by (13)

The choice of that then achieves the most cancellation turns out to be (this cancels the terms that are linear or quadratic in the ), so that . Repeating the calculations establishing (7), one has

and

where is the random variable drawn from with the normalised probability measure . Since , one thus has

This expression is clearly non-negative for , equal to zero for , and positive for , giving the claim. (One could simplify here as if desired, though it is not strictly necessary to do so for the proof.)

Remark 6As with Remark 4, one can also establish the identity (14) first for natural numbers by direct computation avoiding the theory of fractional dimensional integrals, and then extrapolate to the case of more general values of . This particular identity is also simple enough that it can be directly established by integration by parts without much difficulty, even for fractional values of .

A more complicated version of this argument establishes the non-endpoint multilinear Kakeya inequality (without any logarithmic loss in a scale parameter ); this was established in my previous paper with Jon Bennett and Tony Carbery, but using the “natural number first” approach rather than using the current formalism of fractional dimensional integration. However, the arguments can be translated into this formalism without much difficulty; we do so below the fold. (To simplify the exposition slightly we will not address issues of establishing enough regularity and integrability to justify all the manipulations, though in practice this can be done by standard limiting arguments.)

The following situation is very common in modern harmonic analysis: one has a large scale parameter (sometimes written as in the literature for some small scale parameter , or as for some large radius ), which ranges over some unbounded subset of (e.g. all sufficiently large real numbers , or all powers of two), and one has some positive quantity depending on that is known to be of *polynomial size* in the sense that

for all in the range and some constant , and one wishes to obtain a *subpolynomial upper bound* for , by which we mean an upper bound of the form

for all and all in the range, where can depend on but is independent of . In many applications, this bound is nearly tight in the sense that one can easily establish a matching lower bound

in which case the property of having a subpolynomial upper bound is equivalent to that of being *subpolynomial size* in the sense that

for all and all in the range. It would naturally be of interest to tighten these bounds further, for instance to show that is polylogarithmic or even bounded in size, but a subpolynomial bound is already sufficient for many applications.

Let us give some illustrative examples of this type of problem:

Example 1 (Kakeya conjecture)Here ranges over all of . Let be a fixed dimension. For each , we pick a maximal -separated set of directions . We let be the smallest constant for which one has the Kakeya inequalitywhere is a -tube oriented in the direction . The Kakeya maximal function conjecture is then equivalent to the assertion that has a subpolynomial upper bound (or equivalently, is of subpolynomial size). Currently this is only known in dimension .

Example 2 (Restriction conjecture for the sphere)Here ranges over all of . Let be a fixed dimension. We let be the smallest constant for which one has the restriction inequalityfor all bounded measurable functions on the unit sphere equipped with surface measure , where is the ball of radius centred at the origin. The restriction conjecture of Stein for the sphere is then equivalent to the assertion that has a subpolynomial upper bound (or equivalently, is of subpolynomial size). Currently this is only known in dimension .

Example 3 (Multilinear Kakeya inequality)Again ranges over all of . Let be a fixed dimension, and let be compact subsets of the sphere which aretransversein the sense that there is a uniform lower bound for the wedge product of directions for (equivalently, there is no hyperplane through the origin that intersects all of the ). For each , we let be the smallest constant for which one has the multilinear Kakeya inequalitywhere for each , is a collection of infinite tubes in of radius oriented in a direction in , which are separated in the sense that for any two tubes in , either the directions of differ by an angle of at least , or are disjoint; and is our notation for the geometric mean

The multilinear Kakeya inequality of Bennett, Carbery, and myself establishes that is of subpolynomial size; a later argument of Guth improves this further by showing that is bounded (and in fact comparable to ).

Example 4 (Multilinear restriction theorem)Once again ranges over all of . Let be a fixed dimension, and let be compact subsets of the sphere which aretransverseas in the previous example. For each , we let be the smallest constant for which one has the multilinear restriction inequalityfor all bounded measurable functions on for . Then the multilinear restriction theorem of Bennett, Carbery, and myself establishes that is of subpolynomial size; it is known to be bounded for (as can be easily verified from Plancherel’s theorem), but it remains open whether it is bounded for any .

Example 5 (Decoupling for the paraboloid)now ranges over the square numbers. Let , and subdivide the unit cube into cubes of sidelength . For any , define the extension operatorsand

for and . We also introduce the weight function

For any , let be the smallest constant for which one has the decoupling inequality

The decoupling theorem of Bourgain and Demeter asserts that is of subpolynomial size for all in the optimal range .

Example 6 (Decoupling for the moment curve)now ranges over the natural numbers. Let , and subdivide into intervals of length . For any , define the extension operatorsand more generally

for . For any , let be the smallest constant for which one has the decoupling inequality

It was shown by Bourgain, Demeter, and Guth that is of subpolynomial size for all in the optimal range , which among other things implies the Vinogradov main conjecture (as discussed in this previous post).

It is convenient to use asymptotic notation to express these estimates. We write , , or to denote the inequality for some constant independent of the scale parameter , and write for . We write to denote a bound of the form where as along the given range of . We then write for , and for . Then the statement that is of polynomial size can be written as

while the statement that has a subpolynomial upper bound can be written as

and similarly the statement that is of subpolynomial size is simply

Many modern approaches to bounding quantities like in harmonic analysis rely on some sort of *induction on scales* approach in which is bounded using quantities such as for some exponents . For instance, suppose one is somehow able to establish the inequality

for all , and suppose that is also known to be of polynomial size. Then this implies that has a subpolynomial upper bound. Indeed, one can iterate this inequality to show that

for any fixed ; using the polynomial size hypothesis one thus has

for some constant independent of . As can be arbitrarily large, we conclude that for any , and hence is of subpolynomial size. (This sort of iteration is used for instance in my paper with Bennett and Carbery to derive the multilinear restriction theorem from the multilinear Kakeya theorem.)

Exercise 7If is of polynomial size, and obeys the inequalityfor any fixed , where the implied constant in the notation is independent of , show that has a subpolynomial upper bound. This type of inequality is used to equate various linear estimates in harmonic analysis with their multilinear counterparts; see for instance this paper of myself, Vargas, and Vega for an early example of this method.

In more recent years, more sophisticated induction on scales arguments have emerged in which one or more auxiliary quantities besides also come into play. Here is one example, this time being an abstraction of a short proof of the multilinear Kakeya inequality due to Guth. Let be the quantity in Example 3. We define similarly to for any , except that we now also require that the diameter of each set is at most . One can then observe the following estimates:

- (Triangle inequality) For any , we have
- (Multiplicativity) For any , one has
- (Loomis-Whitney inequality) We have

These inequalities now imply that has a subpolynomial upper bound, as we now demonstrate. Let be a large natural number (independent of ) to be chosen later. From many iterations of (6) we have

and hence by (7) (with replaced by ) and (5)

where the implied constant in the exponent does not depend on . As can be arbitrarily large, the claim follows. We remark that a nearly identical scheme lets one deduce decoupling estimates for the three-dimensional cone from that of the two-dimensional paraboloid; see the final section of this paper of Bourgain and Demeter.

Now we give a slightly more sophisticated example, abstracted from the proof of decoupling of the paraboloid by Bourgain and Demeter, as described in this study guide after specialising the dimension to and the exponent to the endpoint (the argument is also more or less summarised in this previous post). (In the cited papers, the argument was phrased only for the non-endpoint case , but it has been observed independently by many experts that the argument extends with only minor modifications to the endpoint .) Here we have a quantity that we wish to show is of subpolynomial size. For any and , one can define an auxiliary quantity . The precise definitions of and are given in the study guide (where they are called and respectively, setting and ) but will not be of importance to us for this discussion. Suffice to say that the following estimates are known:

- (Crude upper bound for ) is of polynomial size: .
- (Bilinear reduction, using parabolic rescaling) For any , one has
- (Crude upper bound for ) For any one has
- (Application of multilinear Kakeya and decoupling) If are sufficiently small (e.g. both less than ), then

In all of these bounds the implied constant exponents such as or are independent of and , although the implied constants in the notation can depend on both and . Here we gloss over an annoying technicality in that quantities such as , , or might not be an integer (and might not divide evenly into ), which is needed for the application to decoupling theorems; this can be resolved by restricting the scales involved to powers of two and restricting the values of to certain rational values, which introduces some complications to the later arguments below which we shall simply ignore as they do not significantly affect the numerology.

It turns out that these estimates imply that is of subpolynomial size. We give the argument as follows. As is known to be of polynomial size, we have some for which we have the bound

for all . We can pick to be the minimal exponent for which this bound is attained: thus

We will call this the *upper exponent* of . We need to show that . We assume for contradiction that . Let be a sufficiently small quantity depending on to be chosen later. From (10) we then have

for any sufficiently small . A routine iteration then gives

for any that is independent of , if is sufficiently small depending on . A key point here is that the implied constant in the exponent is uniform in (the constant comes from summing a convergent geometric series). We now use the crude bound (9) followed by (11) and conclude that

Applying (8) we then have

If we choose sufficiently large depending on (which was assumed to be positive), then the negative term will dominate the term. If we then pick sufficiently small depending on , then finally sufficiently small depending on all previous quantities, we will obtain for some strictly less than , contradicting the definition of . Thus cannot be positive, and hence has a subpolynomial upper bound as required.

Exercise 8Show that one still obtains a subpolynomial upper bound if the estimate (10) is replaced withfor some constant , so long as we also improve (9) to

(This variant of the argument lets one handle the non-endpoint cases of the decoupling theorem for the paraboloid.)

To establish decoupling estimates for the moment curve, restricting to the endpoint case for sake of discussion, an even more sophisticated induction on scales argument was deployed by Bourgain, Demeter, and Guth. The proof is discussed in this previous blog post, but let us just describe an abstract version of the induction on scales argument. To bound the quantity , some auxiliary quantities are introduced for various exponents and and , with the following bounds:

- (Crude upper bound for ) is of polynomial size: .
- (Multilinear reduction, using non-isotropic rescaling) For any and , one has
- (Crude upper bound for ) For any and one has
- (Hölder) For and one has
- (Rescaled decoupling hypothesis) For , one has
- (Lower dimensional decoupling) If and , then
- (Multilinear Kakeya) If and , then

It is now substantially less obvious that these estimates can be combined to demonstrate that is of subpolynomial size; nevertheless this can be done. A somewhat complicated arrangement of the argument (involving some rather unmotivated choices of expressions to induct over) appears in my previous blog post; I give an alternate proof later in this post.

These examples indicate a general strategy to establish that some quantity is of subpolynomial size, by

- (i) Introducing some family of related auxiliary quantities, often parameterised by several further parameters;
- (ii) establishing as many bounds between these quantities and the original quantity as possible; and then
- (iii) appealing to some sort of “induction on scales” to conclude.

The first two steps (i), (ii) depend very much on the harmonic analysis nature of the quantities and the related auxiliary quantities, and the estimates in (ii) will typically be proven from various harmonic analysis inputs such as Hölder’s inequality, rescaling arguments, decoupling estimates, or Kakeya type estimates. The final step (iii) requires no knowledge of where these quantities come from in harmonic analysis, but the iterations involved can become extremely complicated.

In this post I would like to observe that one can clean up and made more systematic this final step (iii) by passing to upper exponents (12) to eliminate the role of the parameter (and also “tropicalising” all the estimates), and then taking similar limit superiors to eliminate some other less important parameters, until one is left with a simple linear programming problem (which, among other things, could be amenable to computer-assisted proving techniques). This method is analogous to that of passing to a simpler asymptotic limit object in many other areas of mathematics (for instance using the Furstenberg correspondence principle to pass from a combinatorial problem to an ergodic theory problem, as discussed in this previous post). We use the limit superior exclusively in this post, but many of the arguments here would also apply with one of the other generalised limit functionals discussed in this previous post, such as ultrafilter limits.

For instance, if is the upper exponent of a quantity of polynomial size obeying (4), then a comparison of the upper exponent of both sides of (4) one arrives at the scalar inequality

from which it is immediate that , giving the required subpolynomial upper bound. Notice how the passage to upper exponents converts the estimate to a simpler inequality .

Exercise 9Repeat Exercise 7 using this method.

Similarly, given the quantities obeying the axioms (5), (6), (7), and assuming that is of polynomial size (which is easily verified for the application at hand), we see that for any real numbers , the quantity is also of polynomial size and hence has some upper exponent ; meanwhile itself has some upper exponent . By reparameterising we have the homogeneity

for any . Also, comparing the upper exponents of both sides of the axioms (5), (6), (7) we arrive at the inequalities

For any natural number , the third inequality combined with homogeneity gives , which when combined with the second inequality gives , which on combination with the first estimate gives . Sending to infinity we obtain as required.

Now suppose that , obey the axioms (8), (9), (10). For any fixed , the quantity is of polynomial size (thanks to (9) and the polynomial size of ), and hence has some upper exponent ; similarly has some upper exponent . (Actually, strictly speaking our axioms only give an upper bound on so we have to temporarily admit the possibility that , though this will soon be eliminated anyway.) Taking upper exponents of all the axioms we then conclude that

for all and .

Assume for contradiction that , then , and so the statement (20) simplifies to

At this point we can eliminate the role of and simplify the system by taking a second limit superior. If we write

then on taking limit superiors of the previous inequalities we conclude that

for all ; in particular . We take advantage of this by taking a further limit superior (or “upper derivative”) in the limit to eliminate the role of and simplify the system further. If we define

so that is the best constant for which as , then is finite, and by inserting this “Taylor expansion” into the right-hand side of (21) and conclude that

This leads to a contradiction when , and hence as desired.

Exercise 10Redo Exercise 8 using this method.

The same strategy now clarifies how to proceed with the more complicated system of quantities obeying the axioms (13)–(19) with of polynomial size. Let be the exponent of . From (14) we see that for fixed , each is also of polynomial size (at least in upper bound) and so has some exponent (which for now we can permit to be ). Taking upper exponents of all the various axioms we can now eliminate and arrive at the simpler axioms

for all , , and , with the lower dimensional decoupling inequality

for and , and the multilinear Kakeya inequality

for and .

As before, if we assume for sake of contradiction that then the first inequality simplifies to

We can then again eliminate the role of by taking a second limit superior as , introducing

and thus getting the simplified axiom system

for and , and

for and .

In view of the latter two estimates it is natural to restrict attention to the quantities for . By the axioms (22), these quantities are of the form . We can then eliminate the role of by taking another limit superior

The axioms now simplify to

It turns out that the inequality (27) is strongest when , thus

From the last two inequalities (28), (29) we see that a special role is likely to be played by the exponents

for and

for . From the convexity (25) and a brief calculation we have

for , hence from (28) we have

Similarly, from (25) and a brief calculation we have

for ; the same bound holds for if we drop the term with the factor, thanks to (24). Thus from (29) we have

for , again with the understanding that we omit the first term on the right-hand side when . Finally, (26) gives

Let us write out the system of equations we have obtained in full:

We can then eliminate the variables one by one. Inserting (33) into (32) we obtain

which simplifies to

Inserting this into (34) gives

which when combined with (35) gives

which simplifies to

Iterating this we get

for all and

for all . In particular

which on insertion into (36), (37) gives

which is absurd if . Thus and so must be of subpolynomial growth.

Remark 11(This observation is essentially due to Heath-Brown.) If we let denote the column vector with entries (arranged in whatever order one pleases), then the above system of inequalities (32)–(36) (using (37) to handle the appearance of in (36)) readsfor some explicit square matrix with non-negative coefficients, where the inequality denotes pointwise domination, and is an explicit vector with non-positive coefficients that reflects the effect of (37). It is possible to show (using (24), (26)) that all the coefficients of are negative (assuming the counterfactual situation of course). Then we can iterate this to obtain

for any natural number . This would lead to an immediate contradiction if the Perron-Frobenius eigenvalue of exceeds because would now grow exponentially; this is typically the situation for “non-endpoint” applications such as proving decoupling inequalities away from the endpoint. In the endpoint situation discussed above, the Perron-Frobenius eigenvalue is , with having a non-trivial projection to this eigenspace, so the sum now grows at least linearly, which still gives the required contradiction for any . So it is important to gather “enough” inequalities so that the relevant matrix has a Perron-Frobenius eigenvalue greater than or equal to (and in the latter case one needs non-trivial injection of an induction hypothesis into an eigenspace corresponding to an eigenvalue ). More specifically, if is the spectral radius of and is a left Perron-Frobenius eigenvector, that is to say a non-negative vector, not identically zero, such that , then by taking inner products of (38) with we obtain

If this leads to a contradiction since is negative and is non-positive. When one still gets a contradiction as long as is strictly negative.

Remark 12(This calculation is essentially due to Guo and Zorin-Kranich.) Here is a concrete application of the Perron-Frobenius strategy outlined above to the system of inequalities (32)–(37). Consider the weighted sumI had secretly calculated the weights , as coming from the left Perron-Frobenius eigenvector of the matrix described in the previous remark, but for this calculation the precise provenance of the weights is not relevant. Applying the inequalities (31), (30) we see that is bounded by

(with the convention that the term is absent); this simplifies after some calculation to the bound

and this and (37) then leads to the required contradiction.

Exercise 13

- (i) Extend the above analysis to also cover the non-endpoint case . (One will need to establish the claim for .)
- (ii) Modify the argument to deal with the remaining cases by dropping some of the steps.

While talking mathematics with a postdoc here at UCLA (March Boedihardjo) we came across the following matrix problem which we managed to solve, but the proof was cute and the process of discovering it was fun, so I thought I would present the problem here as a puzzle without revealing the solution for now.

The problem involves word maps on a matrix group, which for sake of discussion we will take to be the special orthogonal group of real matrices (one of the smallest matrix groups that contains a copy of the free group, which incidentally is the key observation powering the Banach-Tarski paradox). Given any abstract word of two generators and their inverses (i.e., an element of the free group ), one can define the word map simply by substituting a pair of matrices in into these generators. For instance, if one has the word , then the corresponding word map is given by

for . Because contains a copy of the free group, we see the word map is non-trivial (not equal to the identity) if and only if the word itself is nontrivial.

Anyway, here is the problem:

Problem.Does there exist a sequence of non-trivial word maps that converge uniformly to the identity map?

To put it another way, given any , does there exist a non-trivial word such that for all , where denotes (say) the operator norm, and denotes the identity matrix in ?

As I said, I don’t want to spoil the fun of working out this problem, so I will leave it as a challenge. Readers are welcome to share their thoughts, partial solutions, or full solutions in the comments below.

Note: this post is not required reading for this course, or for the sequel course in the winter quarter.

In a Notes 2, we reviewed the classical construction of Leray of global weak solutions to the Navier-Stokes equations. We did not quite follow Leray’s original proof, in that the notes relied more heavily on the machinery of Littlewood-Paley projections, which have become increasingly common tools in modern PDE. On the other hand, we did use the same “exploiting compactness to pass to weakly convergent subsequence” strategy that is the standard one in the PDE literature used to construct weak solutions.

As I discussed in a previous post, the manipulation of sequences and their limits is analogous to a “cheap” version of nonstandard analysis in which one uses the Fréchet filter rather than an ultrafilter to construct the nonstandard universe. (The manipulation of generalised functions of Columbeau-type can also be comfortably interpreted within this sort of cheap nonstandard analysis.) Augmenting the manipulation of sequences with the right to pass to subsequences whenever convenient is then analogous to a sort of “lazy” nonstandard analysis, in which the implied ultrafilter is never actually constructed as a “completed object“, but is instead lazily evaluated, in the sense that whenever membership of a given subsequence of the natural numbers in the ultrafilter needs to be determined, one either passes to that subsequence (thus placing it in the ultrafilter) or the complement of the sequence (placing it out of the ultrafilter). This process can be viewed as the initial portion of the transfinite induction that one usually uses to construct ultrafilters (as discussed using a voting metaphor in this post), except that there is generally no need in any given application to perform the induction for any uncountable ordinal (or indeed for most of the countable ordinals also).

On the other hand, it is also possible to work directly in the orthodox framework of nonstandard analysis when constructing weak solutions. This leads to an approach to the subject which is largely equivalent to the usual subsequence-based approach, though there are some minor technical differences (for instance, the subsequence approach occasionally requires one to work with separable function spaces, whereas in the ultrafilter approach the reliance on separability is largely eliminated, particularly if one imposes a strong notion of saturation on the nonstandard universe). The subject acquires a more “algebraic” flavour, as the quintessential analysis operation of taking a limit is replaced with the “standard part” operation, which is an algebra homomorphism. The notion of a sequence is replaced by the distinction between standard and nonstandard objects, and the need to pass to subsequences disappears entirely. Also, the distinction between “bounded sequences” and “convergent sequences” is largely eradicated, particularly when the space that the sequences ranged in enjoys some compactness properties on bounded sets. Also, in this framework, the notorious non-uniqueness features of weak solutions can be “blamed” on the non-uniqueness of the nonstandard extension of the standard universe (as well as on the multiple possible ways to construct nonstandard mollifications of the original standard PDE). However, many of these changes are largely cosmetic; switching from a subsequence-based theory to a nonstandard analysis-based theory does *not* seem to bring one significantly closer for instance to the global regularity problem for Navier-Stokes, but it could have been an alternate path for the historical development and presentation of the subject.

In any case, I would like to present below the fold this nonstandard analysis perspective, quickly translating the relevant components of real analysis, functional analysis, and distributional theory that we need to this perspective, and then use it to re-prove Leray’s theorem on existence of global weak solutions to Navier-Stokes.

The celebrated decomposition theorem of Fefferman and Stein shows that every function of bounded mean oscillation can be decomposed in the form

modulo constants, for some , where are the Riesz transforms. A technical note here a function in BMO is defined only up to constants (as well as up to the usual almost everywhere equivalence); related to this, if is an function, then the Riesz transform is well defined as an element of , but is also only defined up to constants and almost everywhere equivalence.

The original proof of Fefferman and Stein was indirect (relying for instance on the Hahn-Banach theorem). A constructive proof was later given by Uchiyama, and was in fact the topic of the second post on this blog. A notable feature of Uchiyama’s argument is that the construction is quite nonlinear; the vector-valued function is defined to take values on a sphere, and the iterative construction to build these functions from involves repeatedly projecting a potential approximant to this function to the sphere (also, the high-frequency components of this approximant are constructed in a manner that depends nonlinearly on the low-frequency components, which is a type of technique that has become increasingly common in analysis and PDE in recent years).

It is natural to ask whether the Fefferman-Stein decomposition (1) can be made linear in , in the sense that each of the depend linearly on . Strictly speaking this is easily accomplished using the axiom of choice: take a Hamel basis of , choose a decomposition (1) for each element of this basis, and then extend linearly to all finite linear combinations of these basis functions, which then cover by definition of Hamel basis. But these linear operations have no reason to be continuous as a map from to . So the correct question is whether the decomposition can be made *continuously linear* (or equivalently, boundedly linear) in , that is to say whether there exist continuous linear transformations such that

modulo constants for all . Note from the open mapping theorem that one can choose the functions to depend in a bounded fashion on (thus for some constant , however the open mapping theorem does not guarantee linearity. Using a result of Bartle and Graves one can also make the depend continuously on , but again the dependence is not guaranteed to be linear.

It is generally accepted folklore that continuous linear dependence is known to be impossible, but I had difficulty recently tracking down an explicit proof of this assertion in the literature (if anyone knows of a reference, I would be glad to know of it). The closest I found was a proof of a similar statement in this paper of Bourgain and Brezis, which I was able to adapt to establish the current claim. The basic idea is to average over the symmetries of the decomposition, which in the case of (1) are translation invariance, rotation invariance, and dilation invariance. This effectively makes the operators invariant under all these symmetries, which forces them to themselves be linear combinations of the identity and Riesz transform operators; however, no such non-trivial linear combination maps to , and the claim follows. Formal details of this argument (which we phrase in a dual form in order to avoid some technicalities) appear below the fold.

In the previous set of notes we developed a theory of “strong” solutions to the Navier-Stokes equations. This theory, based around viewing the Navier-Stokes equations as a perturbation of the linear heat equation, has many attractive features: solutions exist locally, are unique, depend continuously on the initial data, have a high degree of regularity, can be continued in time as long as a sufficiently high regularity norm is under control, and tend to enjoy the same sort of conservation laws that classical solutions do. However, it is a major open problem as to whether these solutions can be extended to be (forward) global in time, because the norms that we know how to control globally in time do not have high enough regularity to be useful for continuing the solution. Also, the theory becomes degenerate in the inviscid limit .

However, it is possible to construct “weak” solutions which lack many of the desirable features of strong solutions (notably, uniqueness, propagation of regularity, and conservation laws) but can often be constructed globally in time even when one us unable to do so for strong solutions. Broadly speaking, one usually constructs weak solutions by some sort of “compactness method”, which can generally be described as follows.

- Construct a sequence of “approximate solutions” to the desired equation, for instance by developing a well-posedness theory for some “regularised” approximation to the original equation. (This theory often follows similar lines to those in the previous set of notes, for instance using such tools as the contraction mapping theorem to construct the approximate solutions.)
- Establish some
*uniform*bounds (over appropriate time intervals) on these approximate solutions, even in the limit as an approximation parameter is sent to zero. (Uniformity is key;*non-uniform*bounds are often easy to obtain if one puts enough “mollification”, “hyper-dissipation”, or “discretisation” in the approximating equation.) - Use some sort of “weak compactness” (e.g., the Banach-Alaoglu theorem, the Arzela-Ascoli theorem, or the Rellich compactness theorem) to extract a subsequence of approximate solutions that converge (in a topology weaker than that associated to the available uniform bounds) to a limit. (Note that there is no reason
*a priori*to expect such limit points to be unique, or to have any regularity properties beyond that implied by the available uniform bounds..) - Show that this limit solves the original equation in a suitable weak sense.

The quality of these weak solutions is very much determined by the type of uniform bounds one can obtain on the approximate solution; the stronger these bounds are, the more properties one can obtain on these weak solutions. For instance, if the approximate solutions enjoy an energy identity leading to uniform energy bounds, then (by using tools such as Fatou’s lemma) one tends to obtain energy *inequalities* for the resulting weak solution; but if one somehow is able to obtain uniform bounds in a higher regularity norm than the energy then one can often recover the full energy *identity*. If the uniform bounds are at the regularity level needed to obtain well-posedness, then one generally expects to upgrade the weak solution to a strong solution. (This phenomenon is often formalised through *weak-strong uniqueness* theorems, which we will discuss later in these notes.) Thus we see that as far as attacking global regularity is concerned, both the theory of strong solutions and the theory of weak solutions encounter essentially the same obstacle, namely the inability to obtain uniform bounds on (exact or approximate) solutions at high regularities (and at arbitrary times).

For simplicity, we will focus our discussion in this notes on finite energy weak solutions on . There is a completely analogous theory for periodic weak solutions on (or equivalently, weak solutions on the torus which we will leave to the interested reader.

In recent years, a completely different way to construct weak solutions to the Navier-Stokes or Euler equations has been developed that are not based on the above compactness methods, but instead based on techniques of convex integration. These will be discussed in a later set of notes.

This is a sequel to this previous blog post, in which we discussed the effect of the heat flow evolution

on the zeroes of a time-dependent family of polynomials , with a particular focus on the case when the polynomials had real zeroes. Here (inspired by some discussions I had during a recent conference on the Riemann hypothesis in Bristol) we record the analogous theory in which the polynomials instead have zeroes on a circle , with the heat flow slightly adjusted to compensate for this. As we shall discuss shortly, a key example of this situation arises when is the numerator of the zeta function of a curve.

More precisely, let be a natural number. We will say that a polynomial

of degree (so that ) obeys the *functional equation* if the are all real and

for all , thus

and

for all non-zero . This means that the zeroes of (counting multiplicity) lie in and are symmetric with respect to complex conjugation and inversion across the circle . We say that this polynomial *obeys the Riemann hypothesis* if all of its zeroes actually lie on the circle . For instance, in the case, the polynomial obeys the Riemann hypothesis if and only if .

Such polynomials arise in number theory as follows: if is a projective curve of genus over a finite field , then, as famously proven by Weil, the associated local zeta function (as defined for instance in this previous blog post) is known to take the form

where is a degree polynomial obeying both the functional equation and the Riemann hypothesis. In the case that is an elliptic curve, then and takes the form , where is the number of -points of minus . The Riemann hypothesis in this case is a famous result of Hasse.

Another key example of such polynomials arise from rescaled characteristic polynomials

of matrices in the compact symplectic group . These polynomials obey both the functional equation and the Riemann hypothesis. The Sato-Tate conjecture (in higher genus) asserts, roughly speaking, that “typical” polyomials arising from the number theoretic situation above are distributed like the rescaled characteristic polynomials (1), where is drawn uniformly from with Haar measure.

Given a polynomial of degree with coefficients

we can evolve it in time by the formula

thus for . Informally, as one increases , this evolution accentuates the effect of the extreme monomials, particularly, and at the expense of the intermediate monomials such as , and conversely as one decreases . This family of polynomials obeys the heat-type equation

In view of the results of Marcus, Spielman, and Srivastava, it is also very likely that one can interpret this flow in terms of expected characteristic polynomials involving conjugation over the compact symplectic group , and should also be tied to some sort of “” version of Brownian motion on this group, but we have not attempted to work this connection out in detail.

It is clear that if obeys the functional equation, then so does for any other time . Now we investigate the evolution of the zeroes. Suppose at some time that the zeroes of are distinct, then

From the inverse function theorem we see that for times sufficiently close to , the zeroes of continue to be distinct (and vary smoothly in ), with

Differentiating this at any not equal to any of the , we obtain

and

and

Inserting these formulae into (2) (expanding as ) and canceling some terms, we conclude that

for sufficiently close to , and not equal to . Extracting the residue at , we conclude that

which we can rearrange as

If we make the change of variables (noting that one can make depend smoothly on for sufficiently close to ), this becomes

Intuitively, this equation asserts that the phases repel each other if they are real (and attract each other if their difference is imaginary). If obeys the Riemann hypothesis, then the are all real at time , then the Picard uniqueness theorem (applied to and its complex conjugate) then shows that the are also real for sufficiently close to . If we then define the entropy functional

then the above equation becomes a gradient flow

which implies in particular that is non-increasing in time. This shows that as one evolves time forward from , there is a uniform lower bound on the separation between the phases , and hence the equation can be solved indefinitely; in particular, obeys the Riemann hypothesis for all if it does so at time . Our argument here assumed that the zeroes of were simple, but this assumption can be removed by the usual limiting argument.

For any polynomial obeying the functional equation, the rescaled polynomials converge locally uniformly to as . By Rouche’s theorem, we conclude that the zeroes of converge to the equally spaced points on the circle . Together with the symmetry properties of the zeroes, this implies in particular that obeys the Riemann hypothesis for all sufficiently large positive . In the opposite direction, when , the polynomials converge locally uniformly to , so if , of the zeroes converge to the origin and the other converge to infinity. In particular, fails the Riemann hypothesis for sufficiently large negative . Thus (if ), there must exist a real number , which we call the *de Bruijn-Newman constant* of the original polynomial , such that obeys the Riemann hypothesis for and fails the Riemann hypothesis for . The situation is a bit more complicated if vanishes; if is the first natural number such that (or equivalently, ) does not vanish, then by the above arguments one finds in the limit that of the zeroes go to the origin, go to infinity, and the remaining zeroes converge to the equally spaced points . In this case the de Bruijn-Newman constant remains finite except in the degenerate case , in which case .

For instance, consider the case when and for some real with . Then the quadratic polynomial

has zeroes

and one easily checks that these zeroes lie on the circle when , and are on the real axis otherwise. Thus in this case we have (with if ). Note how as increases to , the zeroes repel each other and eventually converge to , while as decreases to , the zeroes collide and then separate on the real axis, with one zero going to the origin and the other to infinity.

The arguments in my paper with Brad Rodgers (discussed in this previous post) indicate that for a “typical” polynomial of degree that obeys the Riemann hypothesis, the expected time to relaxation to equilibrium (in which the zeroes are equally spaced) should be comparable to , basically because the average spacing is and hence by (3) the typical velocity of the zeroes should be comparable to , and the diameter of the unit circle is comparable to , thus requiring time comparable to to reach equilibrium. Taking contrapositives, this suggests that the de Bruijn-Newman constant should typically take on values comparable to (since typically one would not expect the initial configuration of zeroes to be close to evenly spaced). I have not attempted to formalise or prove this claim, but presumably one could do some numerics (perhaps using some of the examples of given previously) to explore this further.

We now approach conformal maps from yet another perspective. Given an open subset of the complex numbers , define a univalent function on to be a holomorphic function that is also injective. We will primarily be studying this concept in the case when is the unit disk .

Clearly, a univalent function on the unit disk is a conformal map from to the image ; in particular, is simply connected, and not all of (since otherwise the inverse map would violate Liouville’s theorem). In the converse direction, the Riemann mapping theorem tells us that every open simply connected proper subset of the complex numbers is the image of a univalent function on . Furthermore, if contains the origin, then the univalent function with this image becomes unique once we normalise and . Thus the Riemann mapping theorem provides a one-to-one correspondence between open simply connected proper subsets of the complex plane containing the origin, and univalent functions with and . We will focus particular attention on the univalent functions with the normalisation and ; such functions will be called schlicht functions.

One basic example of a univalent function on is the Cayley transform , which is a Möbius transformation from to the right half-plane . (The slight variant is also referred to as the Cayley transform, as is the closely related map , which maps to the upper half-plane.) One can square this map to obtain a further univalent function , which now maps to the complex numbers with the negative real axis removed. One can normalise this function to be schlicht to obtain the Koebe function

which now maps to the complex numbers with the half-line removed. A little more generally, for any we have the *rotated Koebe function*

that is a schlicht function that maps to the complex numbers with the half-line removed.

Every schlicht function has a convergent Taylor expansion

for some complex coefficients with . For instance, the Koebe function has the expansion

and similarly the rotated Koebe function has the expansion

Intuitively, the Koebe function and its rotations should be the “largest” schlicht functions available. This is formalised by the famous Bieberbach conjecture, which asserts that for any schlicht function, the coefficients should obey the bound for all . After a large number of partial results, this conjecture was eventually solved by de Branges; see for instance this survey of Korevaar or this survey of Koepf for a history.

It turns out that to resolve these sorts of questions, it is convenient to restrict attention to schlicht functions that are *odd*, thus for all , and the Taylor expansion now reads

for some complex coefficients with . One can transform a general schlicht function to an odd schlicht function by observing that the function , after removing the singularity at zero, is a non-zero function that equals at the origin, and thus (as is simply connected) has a unique holomorphic square root that also equals at the origin. If one then sets

it is not difficult to verify that is an odd schlicht function which additionally obeys the equation

Conversely, given an odd schlicht function , the formula (4) uniquely determines a schlicht function .

For instance, if is the Koebe function (1), becomes

which maps to the complex numbers with two slits removed, and if is the rotated Koebe function (2), becomes

De Branges established the Bieberbach conjecture by first proving an analogous conjecture for odd schlicht functions known as Robertson’s conjecture. More precisely, we have

Theorem 1 (de Branges’ theorem)Let be a natural number.

- (i) (Robertson conjecture) If is an odd schlicht function, then
- (ii) (Bieberbach conjecture) If is a schlicht function, then

It is easy to see that the Robertson conjecture for a given value of implies the Bieberbach conjecture for the same value of . Indeed, if is schlicht, and is the odd schlicht function given by (3), then from extracting the coefficient of (4) we obtain a formula

for the coefficients of in terms of the coefficients of . Applying the Cauchy-Schwarz inequality, we derive the Bieberbach conjecture for this value of from the Robertson conjecture for the same value of . We remark that Littlewood and Paley had conjectured a stronger form of Robertson’s conjecture, but this was disproved for by Fekete and Szegö.

To prove the Robertson and Bieberbach conjectures, one first takes a logarithm and deduces both conjectures from a similar conjecture about the Taylor coefficients of , known as the *Milin conjecture*. Next, one continuously enlarges the image of the schlicht function to cover all of ; done properly, this places the schlicht function as the initial function in a sequence of univalent maps known as a Loewner chain. The functions obey a useful differential equation known as the Loewner equation, that involves an unspecified forcing term (or , in the case that the image is a slit domain) coming from the boundary; this in turn gives useful differential equations for the Taylor coefficients of , , or . After some elementary calculus manipulations to “integrate” this equations, the Bieberbach, Robertson, and Milin conjectures are then reduced to establishing the non-negativity of a certain explicit hypergeometric function, which is non-trivial to prove (and will not be done here, except for small values of ) but for which several proofs exist in the literature.

The theory of Loewner chains subsequently became fundamental to a more recent topic in complex analysis, that of the Schramm-Loewner equation (SLE), which is the focus of the next and final set of notes.

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