You are currently browsing the tag archive for the ‘multilinear Kakeya conjecture’ tag.
This set of notes focuses on the restriction problem in Fourier analysis. Introduced by Elias Stein in the 1970s, the restriction problem is a key model problem for understanding more general oscillatory integral operators, and which has turned out to be connected to many questions in geometric measure theory, harmonic analysis, combinatorics, number theory, and PDE. Only partial results on the problem are known, but these partial results have already proven to be very useful or influential in many applications.
We work in a Euclidean space . Recall that
is the space of
-power integrable functions
, quotiented out by almost everywhere equivalence, with the usual modifications when
. If
then the Fourier transform
will be defined in this course by the formula
From the dominated convergence theorem we see that is a continuous function; from the Riemann-Lebesgue lemma we see that it goes to zero at infinity. Thus
lies in the space
of continuous functions that go to zero at infinity, which is a subspace of
. Indeed, from the triangle inequality it is obvious that
If , then Plancherel’s theorem tells us that we have the identity
Because of this, there is a unique way to extend the Fourier transform from
to
, in such a way that it becomes a unitary map from
to itself. By abuse of notation we continue to denote this extension of the Fourier transform by
. Strictly speaking, this extension is no longer defined in a pointwise sense by the formula (1) (indeed, the integral on the RHS ceases to be absolutely integrable once
leaves
; we will return to the (surprisingly difficult) question of whether pointwise convergence continues to hold (at least in an almost everywhere sense) later in this course, when we discuss Carleson’s theorem. On the other hand, the formula (1) remains valid in the sense of distributions, and in practice most of the identities and inequalities one can show about the Fourier transform of “nice” functions (e.g., functions in
, or in the Schwartz class
, or test function class
) can be extended to functions in “rough” function spaces such as
by standard limiting arguments.
By (2), (3), and the Riesz-Thorin interpolation theorem, we also obtain the Hausdorff-Young inequality
for all and
, where
is the dual exponent to
, defined by the usual formula
. (One can improve this inequality by a constant factor, with the optimal constant worked out by Beckner, but the focus in these notes will not be on optimal constants.) As a consequence, the Fourier transform can also be uniquely extended as a continuous linear map from
. (The situation with
is much worse; see below the fold.)
The restriction problem asks, for a given exponent and a subset
of
, whether it is possible to meaningfully restrict the Fourier transform
of a function
to the set
. If the set
has positive Lebesgue measure, then the answer is yes, since
lies in
and therefore has a meaningful restriction to
even though functions in
are only defined up to sets of measure zero. But what if
has measure zero? If
, then
is continuous and therefore can be meaningfully restricted to any set
. At the other extreme, if
and
is an arbitrary function in
, then by Plancherel’s theorem,
is also an arbitrary function in
, and thus has no well-defined restriction to any set
of measure zero.
It was observed by Stein (as reported in the Ph.D. thesis of Charlie Fefferman) that for certain measure zero subsets of
, such as the sphere
, one can obtain meaningful restrictions of the Fourier transforms of functions
for certain
between
and
, thus demonstrating that the Fourier transform of such functions retains more structure than a typical element of
:
Theorem 1 (Preliminary
restriction theorem) If
and
, then one has the estimate
for all Schwartz functions
, where
denotes surface measure on the sphere
. In particular, the restriction
can be meaningfully defined by continuous linear extension to an element of
.
Proof: Fix . We expand out
From (1) and Fubini’s theorem, the right-hand side may be expanded as
where the inverse Fourier transform of the measure
is defined by the formula
In other words, we have the identity
using the Hermitian inner product . Since the sphere
have bounded measure, we have from the triangle inequality that
Also, from the method of stationary phase (as covered in the previous class 247A), or Bessel function asymptotics, we have the decay
for any (note that the bound already follows from (6) unless
). We remark that the exponent
here can be seen geometrically from the following considerations. For
, the phase
on the sphere is stationary at the two antipodal points
of the sphere, and constant on the tangent hyperplanes to the sphere at these points. The wavelength of this phase is proportional to
, so the phase would be approximately stationary on a cap formed by intersecting the sphere with a
neighbourhood of the tangent hyperplane to one of the stationary points. As the sphere is tangent to second order at these points, this cap will have diameter
in the directions of the
-dimensional tangent space, so the cap will have surface measure
, which leads to the prediction (7). We combine (6), (7) into the unified estimate
where the “Japanese bracket” is defined as
. Since
lies in
precisely when
, we conclude that
Applying Young’s convolution inequality, we conclude (after some arithmetic) that
whenever , and the claim now follows from (5) and Hölder’s inequality.
Remark 2 By using the Hardy-Littlewood-Sobolev inequality in place of Young’s convolution inequality, one can also establish this result for
.
Motivated by this result, given any Radon measure on
and any exponents
, we use
to denote the claim that the restriction estimate
for all Schwartz functions ; if
is a
-dimensional submanifold of
(possibly with boundary), we write
for
where
is the
-dimensional surface measure on
. Thus, for instance, we trivially always have
, while Theorem 1 asserts that
holds whenever
. We will not give a comprehensive survey of restriction theory in these notes, but instead focus on some model results that showcase some of the basic techniques in the field. (I have a more detailed survey on this topic from 2003, but it is somewhat out of date.)
Read the rest of this entry »
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 1 When
, one has
which 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 2 Suppose
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 integral
can 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 4 It 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 that
One 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 explicitly
for 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 6 As 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.)
Given any finite collection of elements in some Banach space
, the triangle inequality tells us that
However, when the all “oscillate in different ways”, one expects to improve substantially upon the triangle inequality. For instance, if
is a Hilbert space and the
are mutually orthogonal, we have the Pythagorean theorem
For sake of comparison, from the triangle inequality and Cauchy-Schwarz one has the general inequality
for any finite collection in any Banach space
, where
denotes the cardinality of
. Thus orthogonality in a Hilbert space yields “square root cancellation”, saving a factor of
or so over the trivial bound coming from the triangle inequality.
More generally, let us somewhat informally say that a collection exhibits decoupling in
if one has the Pythagorean-like inequality
for any , thus one obtains almost the full square root cancellation in the
norm. The theory of almost orthogonality can then be viewed as the theory of decoupling in Hilbert spaces such as
. In
spaces for
one usually does not expect this sort of decoupling; for instance, if the
are disjointly supported one has
and the right-hand side can be much larger than when
. At the opposite extreme, one usually does not expect to get decoupling in
, since one could conceivably align the
to all attain a maximum magnitude at the same location with the same phase, at which point the triangle inequality in
becomes sharp.
However, in some cases one can get decoupling for certain . For instance, suppose we are in
, and that
are bi-orthogonal in the sense that the products
for
are pairwise orthogonal in
. Then we have
giving decoupling in . (Similarly if each of the
is orthogonal to all but
of the other
.) A similar argument also gives
decoupling when one has tri-orthogonality (with the
mostly orthogonal to each other), and so forth. As a slight variant, Khintchine’s inequality also indicates that decoupling should occur for any fixed
if one multiplies each of the
by an independent random sign
.
In recent years, Bourgain and Demeter have been establishing decoupling theorems in spaces for various key exponents of
, in the “restriction theory” setting in which the
are Fourier transforms of measures supported on different portions of a given surface or curve; this builds upon the earlier decoupling theorems of Wolff. In a recent paper with Guth, they established the following decoupling theorem for the curve
parameterised by the polynomial curve
For any ball in
, let
denote the weight
which should be viewed as a smoothed out version of the indicator function of
. In particular, the space
can be viewed as a smoothed out version of the space
. For future reference we observe a fundamental self-similarity of the curve
: any arc
in this curve, with
a compact interval, is affinely equivalent to the standard arc
.
Theorem 1 (Decoupling theorem) Let
. Subdivide the unit interval
into
equal subintervals
of length
, and for each such
, let
be the Fourier transform
of a finite Borel measure
on the arc
, where
. Then the
exhibit decoupling in
for any ball
of radius
.
Orthogonality gives the case of this theorem. The bi-orthogonality type arguments sketched earlier only give decoupling in
up to the range
; the point here is that we can now get a much larger value of
. The
case of this theorem was previously established by Bourgain and Demeter (who obtained in fact an analogous theorem for any curved hypersurface). The exponent
(and the radius
) is best possible, as can be seen by the following basic example. If
where is a bump function adapted to
, then standard Fourier-analytic computations show that
will be comparable to
on a rectangular box of dimensions
(and thus volume
) centred at the origin, and exhibit decay away from this box, with
comparable to
On the other hand, is comparable to
on a ball of radius comparable to
centred at the origin, so
is
, which is just barely consistent with decoupling. This calculation shows that decoupling will fail if
is replaced by any larger exponent, and also if the radius of the ball
is reduced to be significantly smaller than
.
This theorem has the following consequence of importance in analytic number theory:
Corollary 2 (Vinogradov main conjecture) Let
be integers, and let
. Then
Proof: By the Hölder inequality (and the trivial bound of for the exponential sum), it suffices to treat the critical case
, that is to say to show that
We can rescale this as
As the integrand is periodic along the lattice , this is equivalent to
The left-hand side may be bounded by , where
and
. Since
the claim now follows from the decoupling theorem and a brief calculation.
Using the Plancherel formula, one may equivalently (when is an integer) write the Vinogradov main conjecture in terms of solutions
to the system of equations
but we will not use this formulation here.
A history of the Vinogradov main conjecture may be found in this survey of Wooley; prior to the Bourgain-Demeter-Guth theorem, the conjecture was solved completely for , or for
and
either below
or above
, with the bulk of recent progress coming from the efficient congruencing technique of Wooley. It has numerous applications to exponential sums, Waring’s problem, and the zeta function; to give just one application, the main conjecture implies the predicted asymptotic for the number of ways to express a large number as the sum of
fifth powers (the previous best result required
fifth powers). The Bourgain-Demeter-Guth approach to the Vinogradov main conjecture, based on decoupling, is ostensibly very different from the efficient congruencing technique, which relies heavily on the arithmetic structure of the program, but it appears (as I have been told from second-hand sources) that the two methods are actually closely related, with the former being a sort of “Archimedean” version of the latter (with the intervals
in the decoupling theorem being analogous to congruence classes in the efficient congruencing method); hopefully there will be some future work making this connection more precise. One advantage of the decoupling approach is that it generalises to non-arithmetic settings in which the set
that
is drawn from is replaced by some other similarly separated set of real numbers. (A random thought – could this allow the Vinogradov-Korobov bounds on the zeta function to extend to Beurling zeta functions?)
Below the fold we sketch the Bourgain-Demeter-Guth argument proving Theorem 1.
I thank Jean Bourgain and Andrew Granville for helpful discussions.
Recent Comments