Vaughan Jones

Vaughan Jones, who made fundamental contributions in operator algebras and knot theory (in particular developing a surprising connection between the two), died this week, aged 67.

Vaughan and I grew up in extremely culturally similar countries, worked in adjacent areas of mathematics, shared (as of this week) a coauthor in Dima Shylakhtenko, started out our career with the same postdoc position (as UCLA Hedrick Assistant Professors, sixteen years apart) and even ended up in sister campuses of the University of California, but surprisingly we only interacted occasionally, via chance meetings at conferences or emails on some committee business. I found him extremely easy to get along with when we did meet, though, perhaps because of our similar cultural upbringing.

I have not had much occasion to directly use much of Vaughan’s mathematical contributions, but I did very much enjoy reading his influential 1999 preprint on planar algebras (which, for some odd reason has never been formally published). Traditional algebra notation is one-dimensional in nature, with algebraic expressions being described by strings of mathematical symbols; a linear operator , for instance, might appear in the middle of such a string, taking in an input on the right and returning an output on its right that might then be fed into some other operation. There are a few mathematical notations which are two-dimensional, such as the commutative diagrams in homological algebra, the tree expansions of solutions to nonlinear PDE (particularly stochastic nonlinear PDE), or the Feynman diagrams and Penrose graphical notations from physics, but these are the exception rather than the rule, and the notation is often still concentrated on a one-dimensional complex of vertices and edges (or arrows) in the plane. Planar algebras, by contrast, fully exploit the topological nature of the plane; a planar “operator” (or “operad”) inhabits some punctured region of the plane, such as an annulus, with “inputs” entering from the inner boundaries of the region and “outputs” emerging from the outer boundary. These algebras arose for Vaughan in both operator theory and knot theory, and have since been used in some other areas of mathematics such as representation theory and homology. I myself have not found a direct use for this type of algebra in my own work, but nevertheless I found the mere possibility of higher dimensional notation being the natural choice for a given mathematical problem to be conceptually liberating.

Zarankiewicz’s problem for semilinear hypergraphs

Abdul Basit, Artem Chernikov, Sergei Starchenko, Chiu-Minh Tran and I have uploaded to the arXiv our paper Zarankiewicz’s problem for semilinear hypergraphs. This paper is in the spirit of a number of results in extremal graph theory in which the bounds for various graph-theoretic problems or results can be greatly improved if one makes some additional hypotheses regarding the structure of the graph, for instance by requiring that the graph be “definable” with respect to some theory with good model-theoretic properties.

A basic motivating example is the question of counting the number of incidences between points and lines (or between points and other geometric objects). Suppose one has points and lines in a space. How many incidences can there be between these points and lines? The utterly trivial bound is , but by using the basic fact that two points determine a line (or two lines intersect in at most one point), a simple application of Cauchy-Schwarz improves this bound to . In graph theoretic terms, the point is that the bipartite incidence graph between points and lines does not contain a copy of (there does not exist two points and two lines that are all incident to each other). Without any other further hypotheses, this bound is basically sharp: consider for instance the collection of points and lines in a finite plane , that has incidences (one can make the situation more symmetric by working with a projective plane rather than an affine plane). If however one considers lines in the real plane , the famous Szemerédi-Trotter theorem improves the incidence bound further from to . Thus the incidence graph between real points and lines contains more structure than merely the absence of .

More generally, bounding on the size of bipartite graphs (or multipartite hypergraphs) not containing a copy of some complete bipartite subgraph (or in the hypergraph case) is known as Zarankiewicz’s problem. We have results for all and all orders of hypergraph, but for sake of this post I will focus on the bipartite case.

In our paper we improve the bound to a near-linear bound in the case that the incidence graph is “semilinear”. A model case occurs when one considers incidences between points and axis-parallel rectangles in the plane. Now the condition is not automatic (it is of course possible for two distinct points to both lie in two distinct rectangles), so we impose this condition by fiat:

Theorem 1 Suppose one has points and axis-parallel rectangles in the plane, whose incidence graph contains no ‘s, for some large .

• (i) The total number of incidences is .
• (ii) If all the rectangles are dyadic, the bound can be improved to .
• (iii) The bound in (ii) is best possible (up to the choice of implied constant).

We don’t know whether the bound in (i) is similarly tight for non-dyadic boxes; the usual tricks for reducing the non-dyadic case to the dyadic case strangely fail to apply here. One can generalise to higher dimensions, replacing rectangles by polytopes with faces in some fixed finite set of orientations, at the cost of adding several more logarithmic factors; also, one can replace the reals by other ordered division rings, and replace polytopes by other sets of bounded “semilinear descriptive complexity”, e.g., unions of boundedly many polytopes, or which are cut out by boundedly many functions that enjoy coordinatewise monotonicity properties. For certain specific graphs we can remove the logarithmic factors entirely. We refer to the preprint for precise details.

The proof techniques are combinatorial. The proof of (i) relies primarily on the order structure of to implement a “divide and conquer” strategy in which one can efficiently control incidences between points and rectangles by incidences between approximately points and boxes. For (ii) there is additional order-theoretic structure one can work with: first there is an easy pruning device to reduce to the case when no rectangle is completely contained inside another, and then one can impose the “tile partial order” in which one dyadic rectangle is less than another if and . The point is that this order is “locally linear” in the sense that for any two dyadic rectangles , the set is linearly ordered, and this can be exploited by elementary double counting arguments to obtain a bound which eventually becomes after optimising certain parameters in the argument. The proof also suggests how to construct the counterexample in (iii), which is achieved by an elementary iterative construction.

Fractional free convolution powers

Dimitri Shlyakhtenko and I have uploaded to the arXiv our paper Fractional free convolution powers. For me, this project (which we started during the 2018 IPAM program on quantitative linear algebra) was motivated by a desire to understand the behavior of the minor process applied to a large random Hermitian matrix , in which one takes the successive upper left minors of and computes their eigenvalues in non-decreasing order. These eigenvalues are related to each other by the Cauchy interlacing inequalities

for , and are often arranged in a triangular array known as a Gelfand-Tsetlin pattern, as discussed in these previous blog posts.

When is large and the matrix is a random matrix with empirical spectral distribution converging to some compactly supported probability measure on the real line, then under suitable hypotheses (e.g., unitary conjugation invariance of the random matrix ensemble ), a “concentration of measure” effect occurs, with the spectral distribution of the minors for for any fixed converging to a specific measure that depends only on and . The reason for this notation is that there is a surprising description of this measure when is a natural number, namely it is the free convolution of copies of , pushed forward by the dilation map . For instance, if is the Wigner semicircular measure , then . At the random matrix level, this reflects the fact that the minor of a GUE matrix is again a GUE matrix (up to a renormalizing constant).

As first observed by Bercovici and Voiculescu and developed further by Nica and Speicher, among other authors, the notion of a free convolution power of can be extended to non-integer , thus giving the notion of a “fractional free convolution power”. This notion can be defined in several different ways. One of them proceeds via the Cauchy transform

of the measure , and can be defined by solving the Burgers-type equation

with initial condition (see this previous blog post for a derivation). This equation can be solved explicitly using the -transform of , defined by solving the equation

for sufficiently large , in which case one can show that

(In the case of the semicircular measure , the -transform is simply the identity: .)

Nica and Speicher also gave a free probability interpretation of the fractional free convolution power: if is a noncommutative random variable in a noncommutative probability space with distribution , and is a real projection operator free of with trace , then the “minor” of (viewed as an element of a new noncommutative probability space whose elements are minors , with trace ) has the law of (we give a self-contained proof of this in an appendix to our paper). This suggests that the minor process (or fractional free convolution) can be studied within the framework of free probability theory.

One of the known facts about integer free convolution powers is monotonicity of the free entropy

and free Fisher information

which were introduced by Voiculescu as free probability analogues of the classical probability concepts of differential entropy and classical Fisher information. (Here we correct a small typo in the normalization constant of Fisher entropy as presented in Voiculescu’s paper.) Namely, it was shown by Shylakhtenko that the quantity is monotone non-decreasing for integer , and the Fisher information is monotone non-increasing for integer . This is the free probability analogue of the corresponding monotonicities for differential entropy and classical Fisher information that was established by Artstein, Ball, Barthe, and Naor, answering a question of Shannon.

Our first main result is to extend the monotonicity results of Shylakhtenko to fractional . We give two proofs of this fact, one using free probability machinery, and a more self contained (but less motivated) proof using integration by parts and contour integration. The free probability proof relies on the concept of the free score of a noncommutative random variable, which is the analogue of the classical score. The free score, also introduced by Voiculescu, can be defined by duality as measuring the perturbation with respect to semicircular noise, or more precisely

whenever is a polynomial and is a semicircular element free of . If has an absolutely continuous law for a sufficiently regular , one can calculate explicitly as , where is the Hilbert transform of , and the Fisher information is given by the formula

One can also define a notion of relative free score relative to some subalgebra of noncommutative random variables.

The free score interacts very well with the free minor process , in particular by standard calculations one can establish the identity

whenever is a noncommutative random variable, is an algebra of noncommutative random variables, and is a real projection of trace that is free of both and . The monotonicity of free Fisher information then follows from an application of Pythagoras’s theorem (which implies in particular that conditional expectation operators are contractions on ). The monotonicity of free entropy then follows from an integral representation of free entropy as an integral of free Fisher information along the free Ornstein-Uhlenbeck process (or equivalently, free Fisher information is essentially the rate of change of free entropy with respect to perturbation by semicircular noise). The argument also shows when equality holds in the monotonicity inequalities; this occurs precisely when is a semicircular measure up to affine rescaling.

After an extensive amount of calculation of all the quantities that were implicit in the above free probability argument (in particular computing the various terms involved in the application of Pythagoras’ theorem), we were able to extract a self-contained proof of monotonicity that relied on differentiating the quantities in and using the differential equation (1). It turns out that if for sufficiently regular , then there is an identity

where is the kernel

and . It is not difficult to show that is a positive semi-definite kernel, which gives the required monotonicity. It would be interesting to obtain some more insightful interpretation of the kernel and the identity (2).

These monotonicity properties hint at the minor process being associated to some sort of “gradient flow” in the parameter. We were not able to formalize this intuition; indeed, it is not clear what a gradient flow on a varying noncommutative probability space even means. However, after substantial further calculation we were able to formally describe the minor process as the Euler-Lagrange equation for an intriguing Lagrangian functional that we conjecture to have a random matrix interpretation. We first work in “Lagrangian coordinates”, defining the quantity on the “Gelfand-Tsetlin pyramid”

by the formula

which is well defined if the density of is sufficiently well behaved. The random matrix interpretation of is that it is the asymptotic location of the eigenvalue of the upper left minor of a random matrix with asymptotic empirical spectral distribution and with unitarily invariant distribution, thus is in some sense a continuum limit of Gelfand-Tsetlin patterns. Thus for instance the Cauchy interlacing laws in this asymptotic limit regime become

After a lengthy calculation (involving extensive use of the chain rule and product rule), the equation (1) is equivalent to the Euler-Lagrange equation

where is the Lagrangian density

Thus the minor process is formally a critical point of the integral . The quantity measures the mean eigenvalue spacing at some location of the Gelfand-Tsetlin pyramid, and the ratio measures mean eigenvalue drift in the minor process. This suggests that this Lagrangian density is some sort of measure of entropy of the asymptotic microscale point process emerging from the minor process at this spacing and drift. There is work of Metcalfe demonstrating that this point process is given by the Boutillier bead model, so we conjecture that this Lagrangian density somehow measures the entropy density of this process.

The Ionescu-Wainger multiplier theorem and the adeles

I’ve just uploaded to the arXiv my paper The Ionescu-Wainger multiplier theorem and the adeles“. This paper revisits a useful multiplier theorem of Ionescu and Wainger on “major arc” Fourier multiplier operators on the integers (or lattices ), and strengthens the bounds while also interpreting it from the viewpoint of the adelic integers (which were also used in my recent paper with Krause and Mirek).

For simplicity let us just work in one dimension. Any smooth function then defines a discrete Fourier multiplier operator for any by the formula

where is the Fourier transform on ; similarly, any test function defines a continuous Fourier multiplier operator by the formula

where . In both cases we refer to as the symbol of the multiplier operator .

We will be interested in discrete Fourier multiplier operators whose symbols are supported on a finite union of arcs. One way to construct such operators is by “folding” continuous Fourier multiplier operators into various target frequencies. To make this folding operation precise, given any continuous Fourier multiplier operator , and any frequency , we define the discrete Fourier multiplier operator for any frequency shift by the formula

or equivalently

More generally, given any finite set , we can form a multifrequency projection operator on by the formula

thus

This construction gives discrete Fourier multiplier operators whose symbol can be localised to a finite union of arcs. For instance, if is supported on , then is a Fourier multiplier whose symbol is supported on the set .

There are a body of results relating the theory of discrete Fourier multiplier operators such as or with the theory of their continuous counterparts. For instance we have the basic result of Magyar, Stein, and Wainger:

Proposition 1 (Magyar-Stein-Wainger sampling principle) Let and .

• (i) If is a smooth function supported in , then , where denotes the operator norm of an operator .
• (ii) More generally, if is a smooth function supported in for some natural number , then .

When the implied constant in these bounds can be set to equal . In the paper of Magyar, Stein, and Wainger it was posed as an open problem as to whether this is the case for other ; in an appendix to this paper I show that the answer is negative if is sufficiently close to or , but I do not know the full answer to this question.

This proposition allows one to get a good multiplier theory for symbols supported near cyclic groups ; for instance it shows that a discrete Fourier multiplier with symbol for a fixed test function is bounded on , uniformly in and . For many applications in discrete harmonic analysis, one would similarly like a good multiplier theory for symbols supported in “major arc” sets such as

and in particular to get a good Littlewood-Paley theory adapted to major arcs. (This is particularly the case when trying to control “true complexity zero” expressions for which the minor arc contributions can be shown to be negligible; my recent paper with Krause and Mirek is focused on expressions of this type.) At present we do not have a good multiplier theory that is directly adapted to the classical major arc set (1) (though I do not know of rigorous negative results that show that such a theory is not possible); however, Ionescu and Wainger were able to obtain a useful substitute theory in which (1) was replaced by a somewhat larger set that had better multiplier behaviour. Starting with a finite collection of pairwise coprime natural numbers, and a natural number , one can form the major arc type set

where consists of all rational points in the unit circle of the form where is the product of at most elements from and is an integer. For suitable choices of and not too large, one can make this set (2) contain the set (1) while still having a somewhat controlled size (very roughly speaking, one chooses to consist of (small powers of) large primes between and for some small constant , together with something like the product of all the primes up to (raised to suitable powers)).

In the regime where is fixed and is small, there is a good theory:

Theorem 2 (Ionescu-Wainger theorem, rough version) If is an even integer or the dual of an even integer, and is supported on for a sufficiently small , then

There is a more explicit description of how small needs to be for this theorem to work (roughly speaking, it is not much more than what is needed for all the arcs in (2) to be disjoint), but we will not give it here. The logarithmic loss of was reduced to by Mirek. In this paper we refine the bound further to

when or for some integer . In particular there is no longer any logarithmic loss in the cardinality of the set .

The proof of (3) follows a similar strategy as to previous proofs of Ionescu-Wainger type. By duality we may assume . We use the following standard sequence of steps:

• (i) (Denominator orthogonality) First one splits into various pieces depending on the denominator appearing in the element of , and exploits “superorthogonality” in to estimate the norm by the norm of an appropriate square function.
• (ii) (Nonconcentration) One expands out the power of the square function and estimates it by a “nonconcentrated” version in which various factors that arise in the expansion are “disjoint”.
• (iii) (Numerator orthogonality) We now decompose based on the numerators appearing in the relevant elements of , and exploit some residual orthogonality in this parameter to reduce to estimating a square-function type expression involving sums over various cosets .
• (iv) (Marcinkiewicz-Zygmund) One uses the Marcinkiewicz-Zygmund theorem relating scalar and vector valued operator norms to eliminate the role of the multiplier .
• (v) (Rubio de Francia) Use a reverse square function estimate of Rubio de Francia type to conclude.

The main innovations are that of using the probabilistic decoupling method to remove some logarithmic losses in (i), and recent progress on the Erdos-Rado sunflower conjecture (as discussed in this recent post) to improve the bounds in (ii). For (i), the key point is that one can express a sum such as

where is the set of -element subsets of an index set , and are various complex numbers, as an average

where is a random partition of into subclasses (chosen uniformly over all such partitions), basically because every -element subset of has a probability exactly of being completely shattered by such a random partition. This “decouples” the index set into a Cartesian product which is more convenient for application of the superorthogonality theory. For (ii), the point is to efficiently obtain estimates of the form

where are various non-negative quantities, and a sunflower is a collection of sets that consist of a common “core” and disjoint “petals” . The other parts of the argument are relatively routine; see for instance this survey of Pierce for a discussion of them in the simple case .

In this paper we interpret the Ionescu-Wainger multiplier theorem as being essentially a consequence of various quantitative versions of the Shannon sampling theorem. Recall that this theorem asserts that if a (Schwartz) function has its Fourier transform supported on , then can be recovered uniquely from its restriction . In fact, as can be shown from a little bit of routine Fourier analysis, if we narrow the support of the Fourier transform slightly to for some , then the restriction has the same behaviour as the original function, in the sense that

for all ; see Theorem 4.18 of this paper of myself with Krause and Mirek. This is consistent with the uncertainty principle, which suggests that such functions should behave like a constant at scales .

The quantitative sampling theorem (4) can be used to give an alternate proof of Proposition 1(i), basically thanks to the identity

whenever is Schwartz and has Fourier transform supported in , and is also supported on ; this identity can be easily verified from the Poisson summation formula. A variant of this argument also yields an alternate proof of Proposition 1(ii), where the role of is now played by , and the standard embedding of into is now replaced by the embedding of into ; the analogue of (4) is now

whenever is Schwartz and has Fourier transform supported in , and is endowed with probability Haar measure.

The locally compact abelian groups and can all be viewed as projections of the adelic integers (the product of the reals and the profinite integers ). By using the Ionescu-Wainger multiplier theorem, we are able to obtain an adelic version of the quantitative sampling estimate (5), namely

whenever , is Schwartz-Bruhat and has Fourier transform supported on for some sufficiently small (the precise bound on depends on in a fashion not detailed here). This allows one obtain an “adelic” extension of the Ionescu-Wainger multiplier theorem, in which the operator norm of any discrete multiplier operator whose symbol is supported on major arcs can be shown to be comparable to the operator norm of an adelic counterpart to that multiplier operator; in principle this reduces “major arc” harmonic analysis on the integers to “low frequency” harmonic analysis on the adelic integers , which is a simpler setting in many ways (mostly because the set of major arcs (2) is now replaced with a product set ).

Pointwise ergodic theorems for non-conventional bilinear polynomial averages

Ben Krause, Mariusz Mirek, and I have uploaded to the arXiv our paper Pointwise ergodic theorems for non-conventional bilinear polynomial averages. This paper is a contribution to the decades-long program of extending the classical ergodic theorems to “non-conventional” ergodic averages. Here, the focus is on pointwise convergence theorems, and in particular looking for extensions of the pointwise ergodic theorem of Birkhoff:

Theorem 1 (Birkhoff ergodic theorem) Let be a measure-preserving system (by which we mean is a -finite measure space, and is invertible and measure-preserving), and let for any . Then the averages converge pointwise for -almost every .

Pointwise ergodic theorems have an inherently harmonic analysis content to them, as they are closely tied to maximal inequalities. For instance, the Birkhoff ergodic theorem is closely tied to the Hardy-Littlewood maximal inequality.

The above theorem was generalized by Bourgain (conceding the endpoint , where pointwise almost everywhere convergence is now known to fail) to polynomial averages:

Theorem 2 (Pointwise ergodic theorem for polynomial averages) Let be a measure-preserving system, and let for any . Let be a polynomial with integer coefficients. Then the averages converge pointwise for -almost every .

For bilinear averages, we have a separate 1990 result of Bourgain (for functions), extended to other spaces by Lacey, and with an alternate proof given, by Demeter:

Theorem 3 (Pointwise ergodic theorem for two linear polynomials) Let be a measure-preserving system with finite measure, and let , for some with . Then for any integers , the averages converge pointwise almost everywhere.

It has been an open question for some time (see e.g., Problem 11 of this survey of Frantzikinakis) to extend this result to other bilinear ergodic averages. In our paper we are able to achieve this in the partially linear case:

Theorem 4 (Pointwise ergodic theorem for one linear and one nonlinear polynomial) Let be a measure-preserving system, and let , for some with . Then for any polynomial of degree , the averages converge pointwise almost everywhere.

We actually prove a bit more than this, namely a maximal function estimate and a variational estimate, together with some additional estimates that “break duality” by applying in certain ranges with , but we will not discuss these extensions here. A good model case to keep in mind is when and (which is the case we started with). We note that norm convergence for these averages was established much earlier by Furstenberg and Weiss (in the case at least), and in fact norm convergence for arbitrary polynomial averages is now known thanks to the work of Host-Kra, Leibman, and Walsh.

Our proof of Theorem 4 is much closer in spirit to Theorem 2 than to Theorem 3. The property of the averages shared in common by Theorems 2, 4 is that they have “true complexity zero”, in the sense that they can only be only be large if the functions involved are “major arc” or “profinite”, in that they behave periodically over very long intervals (or like a linear combination of such periodic functions). In contrast, the average in Theorem 3 has “true complexity one”, in the sense that they can also be large if are “almost periodic” (a linear combination of eigenfunctions, or plane waves), and as such all proofs of the latter theorem have relied (either explicitly or implicitly) on some form of time-frequency analysis. In principle, the true complexity zero property reduces one to study the behaviour of averages on major arcs. However, until recently the available estimates to quantify this true complexity zero property were not strong enough to achieve a good reduction of this form, and even once one was in the major arc setting the bilinear averages in Theorem 4 were still quite complicated, exhibiting a mixture of both continuous and arithmetic aspects, both of which being genuinely bilinear in nature.

After applying standard reductions such as the Calderón transference principle, the key task is to establish a suitably “scale-invariant” maximal (or variational) inequality on the integer shift system (in which with counting measure, and ). A model problem is to establish the maximal inequality

where ranges over powers of two and is the bilinear operator

The single scale estimate

or equivalently (by duality)

is immediate from Hölder’s inequality; the difficulty is how to take the supremum over scales .

The first step is to understand when the single-scale estimate (2) can come close to equality. A key example to keep in mind is when , , where is a small modulus, are such that , is a smooth cutoff to an interval of length , and is also supported on and behaves like a constant on intervals of length . Then one can check that (barring some unusual cancellation) (2) is basically sharp for this example. A remarkable result of Peluse and Prendiville (generalised to arbitrary nonlinear polynomials by Peluse) asserts, roughly speaking, that this example basically the only way in which (2) can be saturated, at least when are supported on a common interval of length and are normalised in rather than . (Strictly speaking, the above paper of Peluse and Prendiville only says something like this regarding the factors; the corresponding statement for was established in a subsequent paper of Peluse and Prendiville.) The argument requires tools from additive combinatorics such as the Gowers uniformity norms, and hinges in particular on the “degree lowering argument” of Peluse and Prendiville, which I discussed in this previous blog post. Crucially for our application, the estimates are very quantitative, with all bounds being polynomial in the ratio between the left and right hand sides of (2) (or more precisely, the -normalized version of (2)).

For our applications we had to extend the inverse theory of Peluse and Prendiville to an theory. This turned out to require a certain amount of “sleight of hand”. Firstly, one can dualise the theorem of Peluse and Prendiville to show that the “dual function”

can be well approximated in by a function that has Fourier support on “major arcs” if enjoy control. To get the required extension to in the aspect one has to improve the control on the error from to ; this can be done by some interpolation theory combined with the useful Fourier multiplier theory of Ionescu and Wainger on major arcs. Then, by further interpolation using recent improving estimates of Han, Kovac, Lacey, Madrid, and Yang for linear averages such as , one can relax the hypothesis on to an hypothesis, and then by undoing the duality one obtains a good inverse theorem for (2) for the function ; a modification of the arguments also gives something similar for .

Using these inverse theorems (and the Ionescu-Wainger multiplier theory) one still has to understand the “major arc” portion of (1); a model case arises when are supported near rational numbers with for some moderately large . The inverse theory gives good control (with an exponential decay in ) on individual scales , and one can leverage this with a Rademacher-Menshov type argument (see e.g., this blog post) and some closer analysis of the bilinear Fourier symbol of to eventually handle all “small” scales, with ranging up to say where for some small constant and large constant . For the “large” scales, it becomes feasible to place all the major arcs simultaneously under a single common denominator , and then a quantitative version of the Shannon sampling theorem allows one to transfer the problem from the integers to the locally compact abelian group . Actually it was conceptually clearer for us to work instead with the adelic integers , which is the inverse limit of the . Once one transfers to the adelic integers, the bilinear operators involved split up as tensor products of the “continuous” bilinear operator

on , and the “arithmetic” bilinear operator

on the profinite integers , equipped with probability Haar measure . After a number of standard manipulations (interpolation, Fubini’s theorem, Hölder’s inequality, variational inequalities, etc.) the task of estimating this tensor product boils down to establishing an improving estimate

for some . Splitting the profinite integers into the product of the -adic integers , it suffices to establish this claim for each separately (so long as we keep the implied constant equal to for sufficiently large ). This turns out to be possible using an arithmetic version of the Peluse-Prendiville inverse theorem as well as an arithmetic improving estimate for linear averaging operators which ultimately arises from some estimates on the distribution of polynomials on the -adic field , which are a variant of some estimates of Kowalski and Wright.

Higher uniformity of bounded multiplicative functions in short intervals on average

Kaisa Matomäki, Maksym Radziwill, Joni Teräväinen, Tamar Ziegler and I have uploaded to the arXiv our paper Higher uniformity of bounded multiplicative functions in short intervals on average. This paper (which originated from a working group at an AIM workshop on Sarnak’s conjecture) focuses on the local Fourier uniformity conjecture for bounded multiplicative functions such as the Liouville function . One form of this conjecture is the assertion that

as for any fixed and any that goes to infinity as , where is the (normalized) Gowers uniformity norm. Among other things this conjecture implies (logarithmically averaged version of) the Chowla and Sarnak conjectures for the Liouville function (or the Möbius function), see this previous blog post.

The conjecture gets more difficult as increases, and also becomes more difficult the more slowly grows with . The conjecture is equivalent to the assertion

which was proven (for arbitrarily slowly growing ) in a landmark paper of Matomäki and Radziwill, discussed for instance in this blog post.

For , the conjecture is equivalent to the assertion

This remains open for sufficiently slowly growing (and it would be a major breakthrough in particular if one could obtain this bound for as small as for any fixed , particularly if applicable to more general bounded multiplicative functions than , as this would have new implications for a generalization of the Chowla conjecture known as the Elliott conjecture). Recently, Kaisa, Maks and myself were able to establish this conjecture in the range (in fact we have since worked out in the current paper that we can get as small as ). In our current paper we establish Fourier uniformity conjecture for higher for the same range of . This in particular implies local orthogonality to polynomial phases,

where denotes the polynomials of degree at most , but the full conjecture is a bit stronger than this, establishing the more general statement

for any degree filtered nilmanifold and Lipschitz function , where now ranges over polynomial maps from to . The method of proof follows the same general strategy as in the previous paper with Kaisa and Maks. (The equivalence of (4) and (1) follows from the inverse conjecture for the Gowers norms, proven in this paper.) We quickly sketch first the proof of (3), using very informal language to avoid many technicalities regarding the precise quantitative form of various estimates. If the estimate (3) fails, then we have the correlation estimate

for many and some polynomial depending on . The difficulty here is to understand how can depend on . We write the above correlation estimate more suggestively as

Because of the multiplicativity at small primes , one expects to have a relation of the form

for many for which for some small primes . (This can be formalised using an inequality of Elliott related to the Turan-Kubilius theorem.) This gives a relationship between and for “edges” in a rather sparse “graph” connecting the elements of say . Using some graph theory one can locate some non-trivial “cycles” in this graph that eventually lead (in conjunction to a certain technical but important “Chinese remainder theorem” step to modify the to eliminate a rather serious “aliasing” issue that was already discussed in this previous post) to obtain functional equations of the form

for some large and close (but not identical) integers , where should be viewed as a first approximation (ignoring a certain “profinite” or “major arc” term for simplicity) as “differing by a slowly varying polynomial” and the polynomials should now be viewed as taking values on the reals rather than the integers. This functional equation can be solved to obtain a relation of the form

for some real number of polynomial size, and with further analysis of the relation (5) one can make basically independent of . This simplifies (3) to something like

and this is now of a form that can be treated by the theorem of Matomäki and Radziwill (because is a bounded multiplicative function). (Actually because of the profinite term mentioned previously, one also has to insert a Dirichlet character of bounded conductor into this latter conclusion, but we will ignore this technicality.)

Now we apply the same strategy to (4). For abelian the claim follows easily from (3), so we focus on the non-abelian case. One now has a polynomial sequence attached to many , and after a somewhat complicated adaptation of the above arguments one again ends up with an approximate functional equation

where the relation is rather technical and will not be detailed here. A new difficulty arises in that there are some unwanted solutions to this equation, such as

for some , which do not necessarily lead to multiplicative characters like as in the polynomial case, but instead to some unfriendly looking “generalized multiplicative characters” (think of as a rough caricature). To avoid this problem, we rework the graph theory portion of the argument to produce not just one functional equation of the form (6)for each , but many, leading to dilation invariances

for a “dense” set of . From a certain amount of Lie algebra theory (ultimately arising from an understanding of the behaviour of the exponential map on nilpotent matrices, and exploiting the hypothesis that is non-abelian) one can conclude that (after some initial preparations to avoid degenerate cases) must behave like for some central element of . This eventually brings one back to the multiplicative characters that arose in the polynomial case, and the arguments now proceed as before.

We give two applications of this higher order Fourier uniformity. One regards the growth of the number

of length sign patterns in the Liouville function. The Chowla conjecture implies that , but even the weaker conjecture of Sarnak that for some remains open. Until recently, the best asymptotic lower bound on was , due to McNamara; with our result, we can now show for any (in fact we can get for any ). The idea is to repeat the now-standard argument to exploit multiplicativity at small primes to deduce Chowla-type conjectures from Fourier uniformity conjectures, noting that the Chowla conjecture would give all the sign patterns one could hope for. The usual argument here uses the “entropy decrement argument” to eliminate a certain error term (involving the large but mean zero factor ). However the observation is that if there are extremely few sign patterns of length , then the entropy decrement argument is unnecessary (there isn’t much entropy to begin with), and a more low-tech moment method argument (similar to the derivation of Chowla’s conjecture from Sarnak’s conjecture, as discussed for instance in this post) gives enough of Chowla’s conjecture to produce plenty of length sign patterns. If there are not extremely few sign patterns of length then we are done anyway. One quirk of this argument is that the sign patterns it produces may only appear exactly once; in contrast with preceding arguments, we were not able to produce a large number of sign patterns that each occur infinitely often.

The second application is to obtain cancellation for various polynomial averages involving the Liouville function or von Mangoldt function , such as

or

where are polynomials of degree at most , no two of which differ by a constant (the latter is essential to avoid having to establish the Chowla or Hardy-Littlewood conjectures, which of course remain open). Results of this type were previously obtained by Tamar Ziegler and myself in the “true complexity zero” case when the polynomials had distinct degrees, in which one could use the theory of Matomäki and Radziwill; now that higher is available at the scale we can now remove this restriction.

The sunflower lemma via Shannon entropy

A family of sets for some is a sunflower if there is a core set contained in each of the such that the petal sets are disjoint. If , let denote the smallest natural number with the property that any family of distinct sets of cardinality at most contains distinct elements that form a sunflower. The celebrated Erdös-Rado theorem asserts that is finite; in fact Erdös and Rado gave the bounds

The sunflower conjecture asserts in fact that the upper bound can be improved to . This remains open at present despite much effort (including a Polymath project); after a long series of improvements to the upper bound, the best general bound known currently is

for all , established in 2019 by Rao (building upon a recent breakthrough a month previously of Alweiss, Lovett, Wu, and Zhang). Here we remove the easy cases or in order to make the logarithmic factor a little cleaner.

Rao’s argument used the Shannon noiseless coding theorem. It turns out that the argument can be arranged in the very slightly different language of Shannon entropy, and I would like to present it here. The argument proceeds by locating the core and petals of the sunflower separately (this strategy is also followed in Alweiss-Lovett-Wu-Zhang). In both cases the following definition will be key. In this post all random variables, such as random sets, will be understood to be discrete random variables taking values in a finite range. We always use boldface symbols to denote random variables, and non-boldface for deterministic quantities.

Definition 1 (Spread set) Let . A random set is said to be -spread if one has

for all sets . A family of sets is said to be -spread if is non-empty and the random variable is -spread, where is drawn uniformly from .

The core can then be selected greedily in such a way that the remainder of a family becomes spread:

Lemma 2 (Locating the core) Let be a family of subsets of a finite set , each of cardinality at most , and let . Then there exists a “core” set of cardinality at most such that the set

has cardinality at least , and such that the family is -spread. Furthermore, if and the are distinct, then .

Proof: We may assume is non-empty, as the claim is trivial otherwise. For any , define the quantity

and let be a subset of that maximizes . Since and when , we see that . If the are distinct and , then we also have when , thus in this case we have .

Let be the set (3). Since , is non-empty. It remains to check that the family is -spread. But for any and drawn uniformly at random from one has

Since and , we obtain the claim

In view of the above lemma, the bound (2) will then follow from

Proposition 3 (Locating the petals) Let be natural numbers, and suppose that for a sufficiently large constant . Let be a finite family of subsets of a finite set , each of cardinality at most which is -spread. Then there exist such that is disjoint.

Indeed, to prove (2), we assume that is a family of sets of cardinality greater than for some ; by discarding redundant elements and sets we may assume that is finite and that all the are contained in a common finite set . Apply Lemma 2 to find a set of cardinality such that the family is -spread. By Proposition 3 we can find such that are disjoint; since these sets have cardinality , this implies that the are distinct. Hence form a sunflower as required.

Remark 4 Proposition 3 is easy to prove if we strengthen the condition on to . In this case, we have for every , hence by the union bound we see that for any with there exists such that is disjoint from the set , which has cardinality at most . Iterating this, we obtain the conclusion of Proposition 3 in this case. This recovers a bound of the form , and by pursuing this idea a little further one can recover the original upper bound (1) of Erdös and Rado.

It remains to prove Proposition 3. In fact we can locate the petals one at a time, placing each petal inside a random set.

Proposition 5 (Locating a single petal) Let the notation and hypotheses be as in Proposition 3. Let be a random subset of , such that each lies in with an independent probability of . Then with probability greater than , contains one of the .

To see that Proposition 5 implies Proposition 3, we randomly partition into by placing each into one of the , chosen uniformly and independently at random. By Proposition 5 and the union bound, we see that with positive probability, it is simultaneously true for all that each contains one of the . Selecting one such for each , we obtain the required disjoint petals.

We will prove Proposition 5 by gradually increasing the density of the random set and arranging the sets to get quickly absorbed by this random set. The key iteration step is

Proposition 6 (Refinement inequality) Let and . Let be a random subset of a finite set which is -spread, and let be a random subset of independent of , such that each lies in with an independent probability of . Then there exists another random subset of with the same distribution as , such that and

Note that a direct application of the first moment method gives only the bound

but the point is that by switching from to an equivalent we can replace the factor by a quantity significantly smaller than .

One can iterate the above proposition, repeatedly replacing with (noting that this preserves the -spread nature ) to conclude

Corollary 7 (Iterated refinement inequality) Let , , and . Let be a random subset of a finite set which is -spread, and let be a random subset of independent of , such that each lies in with an independent probability of . Then there exists another random subset of with the same distribution as , such that

Now we can prove Proposition 5. Let be chosen shortly. Applying Corollary 7 with drawn uniformly at random from the , and setting , or equivalently , we have

In particular, if we set , so that , then by choice of we have , hence

In particular with probability at least , there must exist such that , giving the proposition.

It remains to establish Proposition 6. This is the difficult step, and requires a clever way to find the variant of that has better containment properties in than does. The main trick is to make a conditional copy of that is conditionally independent of subject to the constraint . The point here is that this constrant implies the inclusions

and

Because of the -spread hypothesis, it is hard for to contain any fixed large set. If we could apply this observation in the contrapositive to we could hope to get a good upper bound on the size of and hence on thanks to (4). One can also hope to improve such an upper bound by also employing (5), since it is also hard for the random set to contain a fixed large set. There are however difficulties with implementing this approach due to the fact that the random sets are coupled with in a moderately complicated fashion. In Rao’s argument a somewhat complicated encoding scheme was created to give information-theoretic control on these random variables; below thefold we accomplish a similar effect by using Shannon entropy inequalities in place of explicit encoding. A certain amount of information-theoretic sleight of hand is required to decouple certain random variables to the extent that the Shannon inequalities can be effectively applied. The argument bears some resemblance to the “entropy compression method” discussed in this previous blog post; there may be a way to more explicitly express the argument below in terms of that method. (There is also some kinship with the method of dependent random choice, which is used for instance to establish the Balog-Szemerédi-Gowers lemma, and was also translated into information theoretic language in these unpublished notes of Van Vu and myself.)

Homogenization of iterated singular integrals with applications to random quasiconformal maps

Kari Astala, Steffen Rohde, Eero Saksman and I have (finally!) uploaded to the arXiv our preprint “Homogenization of iterated singular integrals with applications to random quasiconformal maps“. This project started (and was largely completed) over a decade ago, but for various reasons it was not finalised until very recently. The motivation for this project was to study the behaviour of “random” quasiconformal maps. Recall that a (smooth) quasiconformal map is a homeomorphism that obeys the Beltrami equation

for some Beltrami coefficient ; this can be viewed as a deformation of the Cauchy-Riemann equation . Assuming that is asymptotic to at infinity, one can (formally, at least) solve for in terms of using the Beurling transform

by the Neumann series

We looked at the question of the asymptotic behaviour of if is a random field that oscillates at some fine spatial scale . A simple model to keep in mind is

where are independent random signs and is a bump function. For models such as these, we show that a homogenisation occurs in the limit ; each multilinear expression

converges weakly in probability (and almost surely, if we restrict to a lacunary sequence) to a deterministic limit, and the associated quasiconformal map similarly converges weakly in probability (or almost surely). (Results of this latter type were also recently obtained by Ivrii and Markovic by a more geometric method which is simpler, but is applied to a narrower class of Beltrami coefficients.) In the specific case (1), the limiting quasiconformal map is just the identity map , but if for instance replaces the by non-symmetric random variables then one can have significantly more complicated limits. The convergence theorem for multilinear expressions such as is not specific to the Beurling transform ; any other translation and dilation invariant singular integral can be used here.

The random expression (2) is somewhat reminiscent of a moment of a random matrix, and one can start computing it analogously. For instance, if one has a decomposition such as (1), then (2) expands out as a sum

The random fluctuations of this sum can be treated by a routine second moment estimate, and the main task is to show that the expected value

becomes asymptotically independent of .

If all the were distinct then one could use independence to factor the expectation to get

which is a relatively straightforward expression to calculate (particularly in the model (1), where all the expectations here in fact vanish). The main difficulty is that there are a number of configurations in (3) in which various of the collide with each other, preventing one from easily factoring the expression. A typical problematic contribution for instance would be a sum of the form

This is an example of what we call a non-split sum. This can be compared with the split sum

If we ignore the constraint in the latter sum, then it splits into

where

and

and one can hope to treat this sum by an induction hypothesis. (To actually deal with constraints such as requires an inclusion-exclusion argument that creates some notational headaches but is ultimately manageable.) As the name suggests, the non-split configurations such as (4) cannot be factored in this fashion, and are the most difficult to handle. A direct computation using the triangle inequality (and a certain amount of combinatorics and induction) reveals that these sums are somewhat localised, in that dyadic portions such as

exhibit power decay in (when measured in suitable function space norms), basically because of the large number of times one has to transition back and forth between and . Thus, morally at least, the dominant contribution to a non-split sum such as (4) comes from the local portion when . From the translation and dilation invariance of this type of expression then simplifies to something like

(plus negligible errors) for some reasonably decaying function , and this can be shown to converge to a weak limit as .

In principle all of these limits are computable, but the combinatorics is remarkably complicated, and while there is certainly some algebraic structure to the calculations, it does not seem to be easily describable in terms of an existing framework (e.g., that of free probability).

247B, Notes 4: almost everywhere convergence of Fourier series

This set of notes discusses aspects of one of the oldest questions in Fourier analysis, namely the nature of convergence of Fourier series.

If is an absolutely integrable function, its Fourier coefficients are defined by the formula

If is smooth, then the Fourier coefficients are absolutely summable, and we have the Fourier inversion formula

where the series here is uniformly convergent. In particular, if we define the partial summation operators

then converges uniformly to when is smooth.

What if is not smooth, but merely lies in an class for some ? The Fourier coefficients remain well-defined, as do the partial summation operators . The question of convergence in norm is relatively easy to settle:

Exercise 1

• (i) If and , show that converges in norm to . (Hint: first use the boundedness of the Hilbert transform to show that is bounded in uniformly in .)
• (ii) If or , show that there exists such that the sequence is unbounded in (so in particular it certainly does not converge in norm to . (Hint: first show that is not bounded in uniformly in , then apply the uniform boundedness principle in the contrapositive.)

The question of pointwise almost everywhere convergence turned out to be a significantly harder problem:

Theorem 2 (Pointwise almost everywhere convergence)

• (i) (Kolmogorov, 1923) There exists such that is unbounded in for almost every .
• (ii) (Carleson, 1966; conjectured by Lusin, 1913) For every , converges to as for almost every .
• (iii) (Hunt, 1967) For every and , converges to as for almost every .

Note from Hölder’s inequality that contains for all , so Carleson’s theorem covers the case of Hunt’s theorem. We remark that the precise threshold near between Kolmogorov-type divergence results and Carleson-Hunt pointwise convergence results, in the category of Orlicz spaces, is still an active area of research; see this paper of Lie for further discussion.

Carleson’s theorem in particular was a surprisingly difficult result, lying just out of reach of classical methods (as we shall see later, the result is much easier if we smooth either the function or the summation method by a tiny bit). Nowadays we realise that the reason for this is that Carleson’s theorem essentially contains a frequency modulation symmetry in addition to the more familiar translation symmetry and dilation symmetry. This basically rules out the possibility of attacking Carleson’s theorem with tools such as Calderón-Zygmund theory or Littlewood-Paley theory, which respect the latter two symmetries but not the former. Instead, tools from “time-frequency analysis” that essentially respect all three symmetries should be employed. We will illustrate this by giving a relatively short proof of Carleson’s theorem due to Lacey and Thiele. (There are other proofs of Carleson’s theorem, including Carleson’s original proof, its modification by Hunt, and a later time-frequency proof by Fefferman; see Remark 18 below.)

247B, Notes 3: pseudodifferential operators

In contrast to previous notes, in this set of notes we shall focus exclusively on Fourier analysis in the one-dimensional setting for simplicity of notation, although all of the results here have natural extensions to higher dimensions. Depending on the physical context, one can view the physical domain as representing either space or time; we will mostly think in terms of the former interpretation, even though the standard terminology of “time-frequency analysis”, which we will make more prominent use of in later notes, clearly originates from the latter.

In previous notes we have often performed various localisations in either physical space or Fourier space , for instance in order to take advantage of the uncertainty principle. One can formalise these operations in terms of the functional calculus of two basic operations on Schwartz functions , the position operator defined by

and the momentum operator , defined by

(The terminology comes from quantum mechanics, where it is customary to also insert a small constant on the right-hand side of (1) in accordance with de Broglie’s law. Such a normalisation is also used in several branches of mathematics, most notably semiclassical analysis and microlocal analysis, where it becomes profitable to consider the semiclassical limit , but we will not emphasise this perspective here.) The momentum operator can be viewed as the counterpart to the position operator, but in frequency space instead of physical space, since we have the standard identity

for any and . We observe that both operators are formally self-adjoint in the sense that

for all , where we use the Hermitian inner product

Clearly, for any polynomial of one real variable (with complex coefficients), the operator is given by the spatial multiplier operator

and similarly the operator is given by the Fourier multiplier operator

Inspired by this, if is any smooth function that obeys the derivative bounds

for all and (that is to say, all derivatives of grow at most polynomially), then we can define the spatial multiplier operator by the formula

one can easily verify from several applications of the Leibniz rule that maps Schwartz functions to Schwartz functions. We refer to as the symbol of this spatial multiplier operator. In a similar fashion, we define the Fourier multiplier operator associated to the symbol by the formula

For instance, any constant coefficient linear differential operators can be written in this notation as

however there are many Fourier multiplier operators that are not of this form, such as fractional derivative operators for non-integer values of , which is a Fourier multiplier operator with symbol . It is also very common to use spatial cutoffs and Fourier cutoffs for various bump functions to localise functions in either space or frequency; we have seen several examples of such cutoffs in action in previous notes (often in the higher dimensional setting ).

We observe that the maps and are ring homomorphisms, thus for instance

and

for any obeying the derivative bounds (2); also is formally adjoint to in the sense that

for , and similarly for and . One can interpret these facts as part of the functional calculus of the operators , which can be interpreted as densely defined self-adjoint operators on . However, in this set of notes we will not develop the spectral theory necessary in order to fully set out this functional calculus rigorously.

In the field of PDE and ODE, it is also very common to study variable coefficient linear differential operators

where the are now functions of the spatial variable obeying the derivative bounds (2). A simple example is the quantum harmonic oscillator Hamiltonian . One can rewrite this operator in our notation as

and so it is natural to interpret this operator as a combination of both the position operator and the momentum operator , where the symbol this operator is the function

Indeed, from the Fourier inversion formula

for any we have

and hence on multiplying by and summing we have

Inspired by this, we can introduce the Kohn-Nirenberg quantisation by defining the operator by the formula

whenever and is any smooth function obeying the derivative bounds

for all and (note carefully that the exponent in on the right-hand side is required to be uniform in ). This quantisation clearly generalises both the spatial multiplier operators and the Fourier multiplier operators defined earlier, which correspond to the cases when the symbol is a function of only or only respectively. Thus we have combined the physical space and the frequency space into a single domain, known as phase space . The term “time-frequency analysis” encompasses analysis based on decompositions and other manipulations of phase space, in much the same way that “Fourier analysis” encompasses analysis based on decompositions and other manipulations of frequency space. We remark that the Kohn-Nirenberg quantization is not the only choice of quantization one could use; see Remark 19 below.

Exercise 1

• (i) Show that for obeying (6), that does indeed map to .
• (ii) Show that the symbol is uniquely determined by the operator . That is to say, if are two functions obeying (6) with for all , then . (Hint: apply to a suitable truncation of a plane wave and then take limits.)

In principle, the quantisations are potentially very useful for such tasks as inverting variable coefficient linear operators, or to localize a function simultaneously in physical and Fourier space. However, a fundamental difficulty arises: map from symbols to operators is now no longer a ring homomorphism, in particular

in general. Fundamentally, this is due to the fact that pointwise multiplication of symbols is a commutative operation, whereas the composition of operators such as and does not necessarily commute. This lack of commutativity can be measured by introducing the commutator

of two operators , and noting from the product rule that

(In the language of Lie groups and Lie algebras, this tells us that are (up to complex constants) the standard Lie algebra generators of the Heisenberg group.) From a quantum mechanical perspective, this lack of commutativity is the root cause of the uncertainty principle that prevents one from simultaneously localizing in both position and momentum past a certain point. Here is one basic way of formalising this principle:

Exercise 2 (Heisenberg uncertainty principle) For any and , show that

(Hint: evaluate the expression in two different ways and apply the Cauchy-Schwarz inequality.) Informally, this exercise asserts that the spatial uncertainty and the frequency uncertainty of a function obey the Heisenberg uncertainty relation .

Nevertheless, one still has the correspondence principle, which asserts that in certain regimes (which, with our choice of normalisations, corresponds to the high-frequency regime), quantum mechanics continues to behave like a commutative theory, and one can sometimes proceed as if the operators (and the various operators constructed from them) commute up to “lower order” errors. This can be formalised using the pseudodifferential calculus, which we give below the fold, in which we restrict the symbol to certain “symbol classes” of various orders (which then restricts to be pseudodifferential operators of various orders), and obtains approximate identities such as

where the error between the left and right-hand sides is of “lower order” and can in fact enjoys a useful asymptotic expansion. As a first approximation to this calculus, one can think of functions as having some sort of “phase space portrait” which somehow combines the physical space representation with its Fourier representation , and pseudodifferential operators behave approximately like “phase space multiplier operators” in this representation in the sense that

Unfortunately the uncertainty principle (or the non-commutativity of and ) prevents us from making these approximations perfectly precise, and it is not always clear how to even define a phase space portrait of a function precisely (although there are certain popular candidates for such a portrait, such as the FBI transform (also known as the Gabor transform in signal processing literature), or the Wigner quasiprobability distribution, each of which have some advantages and disadvantages). Nevertheless even if the concept of a phase space portrait is somewhat fuzzy, it is of great conceptual benefit both within mathematics and outside of it. For instance, the musical score one assigns a piece of music can be viewed as a phase space portrait of the sound waves generated by that music.

To complement the pseudodifferential calculus we have the basic Calderón-Vaillancourt theorem, which asserts that pseudodifferential operators of order zero are Calderón-Zygmund operators and thus bounded on for . The standard proof of this theorem is a classic application of one of the basic techniques in harmonic analysis, namely the exploitation of almost orthogonality; the proof we will give here will achieve this through the elegant device of the Cotlar-Stein lemma.

Pseudodifferential operators (especially when generalised to higher dimensions ) are a fundamental tool in the theory of linear PDE, as well as related fields such as semiclassical analysis, microlocal analysis, and geometric quantisation. There is an even wider class of operators that is also of interest, namely the Fourier integral operators, which roughly speaking not only approximately multiply the phase space portrait of a function by some multiplier , but also move the portrait around by a canonical transformation. However, the development of theory of these operators is beyond the scope of these notes; see for instance the texts of Hormander or Eskin.

This set of notes is only the briefest introduction to the theory of pseudodifferential operators. Many texts are available that cover the theory in more detail, for instance this text of Taylor.