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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 {{\bf R}^d}. Recall that {L^p({\bf R}^d)} is the space of {p^{th}}-power integrable functions {f: {\bf R}^d \rightarrow {\bf C}}, quotiented out by almost everywhere equivalence, with the usual modifications when {p=\infty}. If {f \in L^1({\bf R}^d)} then the Fourier transform {\hat f: {\bf R}^d \rightarrow {\bf C}} will be defined in this course by the formula

\displaystyle  \hat f(\xi) := \int_{{\bf R}^d} f(x) e^{-2\pi i x \cdot \xi}\ dx. \ \ \ \ \ (1)

From the dominated convergence theorem we see that {\hat f} is a continuous function; from the Riemann-Lebesgue lemma we see that it goes to zero at infinity. Thus {\hat f} lies in the space {C_0({\bf R}^d)} of continuous functions that go to zero at infinity, which is a subspace of {L^\infty({\bf R}^d)}. Indeed, from the triangle inequality it is obvious that

\displaystyle  \|\hat f\|_{L^\infty({\bf R}^d)} \leq \|f\|_{L^1({\bf R}^d)}. \ \ \ \ \ (2)

If {f \in L^1({\bf R}^d) \cap L^2({\bf R}^d)}, then Plancherel’s theorem tells us that we have the identity

\displaystyle  \|\hat f\|_{L^2({\bf R}^d)} = \|f\|_{L^2({\bf R}^d)}. \ \ \ \ \ (3)

Because of this, there is a unique way to extend the Fourier transform {f \mapsto \hat f} from {L^1({\bf R}^d) \cap L^2({\bf R}^d)} to {L^2({\bf R}^d)}, in such a way that it becomes a unitary map from {L^2({\bf R}^d)} to itself. By abuse of notation we continue to denote this extension of the Fourier transform by {f \mapsto \hat f}. 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 {f} leaves {L^1({\bf R}^d)}; 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 {L^1({\bf R}^d) \cap L^2({\bf R}^d)}, or in the Schwartz class {{\mathcal S}({\bf R}^d)}, or test function class {C^\infty_c({\bf R}^d)}) can be extended to functions in “rough” function spaces such as {L^2({\bf R}^d)} by standard limiting arguments.

By (2), (3), and the Riesz-Thorin interpolation theorem, we also obtain the Hausdorff-Young inequality

\displaystyle  \|\hat f\|_{L^{p'}({\bf R}^d)} \leq \|f\|_{L^p({\bf R}^d)} \ \ \ \ \ (4)

for all {1 \leq p \leq 2} and {f \in L^1({\bf R}^d) \cap L^2({\bf R}^d)}, where {2 \leq p' \leq \infty} is the dual exponent to {p}, defined by the usual formula {\frac{1}{p} + \frac{1}{p'} = 1}. (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 {L^p({\bf R}^d) \rightarrow L^{p'}({\bf R}^d)}. (The situation with {p>2} is much worse; see below the fold.)

The restriction problem asks, for a given exponent {1 \leq p \leq 2} and a subset {S} of {{\bf R}^d}, whether it is possible to meaningfully restrict the Fourier transform {\hat f} of a function {f \in L^p({\bf R}^d)} to the set {S}. If the set {S} has positive Lebesgue measure, then the answer is yes, since {\hat f} lies in {L^{p'}({\bf R}^d)} and therefore has a meaningful restriction to {S} even though functions in {L^{p'}} are only defined up to sets of measure zero. But what if {S} has measure zero? If {p=1}, then {\hat f \in C_0({\bf R}^d)} is continuous and therefore can be meaningfully restricted to any set {S}. At the other extreme, if {p=2} and {f} is an arbitrary function in {L^2({\bf R}^d)}, then by Plancherel’s theorem, {\hat f} is also an arbitrary function in {L^2({\bf R}^d)}, and thus has no well-defined restriction to any set {S} of measure zero.

It was observed by Stein (as reported in the Ph.D. thesis of Charlie Fefferman) that for certain measure zero subsets {S} of {{\bf R}^d}, such as the sphere {S^{d-1} := \{ \xi \in {\bf R}^d: |\xi| = 1\}}, one can obtain meaningful restrictions of the Fourier transforms of functions {f \in L^p({\bf R}^d)} for certain {p} between {1} and {2}, thus demonstrating that the Fourier transform of such functions retains more structure than a typical element of {L^{p'}({\bf R}^d)}:

Theorem 1 (Preliminary {L^2} restriction theorem) If {d \geq 2} and {1 \leq p < \frac{4d}{3d+1}}, then one has the estimate

\displaystyle  \| \hat f \|_{L^2(S^{d-1}, d\sigma)} \lesssim_{d,p} \|f\|_{L^p({\bf R}^d)}

for all Schwartz functions {f \in {\mathcal S}({\bf R}^d)}, where {d\sigma} denotes surface measure on the sphere {S^{d-1}}. In particular, the restriction {\hat f|_S} can be meaningfully defined by continuous linear extension to an element of {L^2(S^{d-1},d\sigma)}.

Proof: Fix {d,p,f}. We expand out

\displaystyle  \| \hat f \|_{L^2(S^{d-1}, d\sigma)}^2 = \int_{S^{d-1}} |\hat f(\xi)|^2\ d\sigma(\xi).

From (1) and Fubini’s theorem, the right-hand side may be expanded as

\displaystyle  \int_{{\bf R}^d} \int_{{\bf R}^d} f(x) \overline{f}(y) (d\sigma)^\vee(y-x)\ dx dy

where the inverse Fourier transform {(d\sigma)^\vee} of the measure {d\sigma} is defined by the formula

\displaystyle  (d\sigma)^\vee(x) := \int_{S^{d-1}} e^{2\pi i x \cdot \xi}\ d\sigma(\xi).

In other words, we have the identity

\displaystyle  \| \hat f \|_{L^2(S^{d-1}, d\sigma)}^2 = \langle f, f * (d\sigma)^\vee \rangle_{L^2({\bf R}^d)}, \ \ \ \ \ (5)

using the Hermitian inner product {\langle f, g\rangle_{L^2({\bf R}^d)} := \int_{{\bf R}^d} \overline{f(x)} g(x)\ dx}. Since the sphere {S^{d-1}} have bounded measure, we have from the triangle inequality that

\displaystyle  (d\sigma)^\vee(x) \lesssim_d 1. \ \ \ \ \ (6)

Also, from the method of stationary phase (as covered in the previous class 247A), or Bessel function asymptotics, we have the decay

\displaystyle  (d\sigma)^\vee(x) \lesssim_d |x|^{-(d-1)/2} \ \ \ \ \ (7)

for any {x \in {\bf R}^d} (note that the bound already follows from (6) unless {|x| \geq 1}). We remark that the exponent {-\frac{d-1}{2}} here can be seen geometrically from the following considerations. For {|x|>1}, the phase {e^{2\pi i x \cdot \xi}} on the sphere is stationary at the two antipodal points {x/|x|, -x/|x|} of the sphere, and constant on the tangent hyperplanes to the sphere at these points. The wavelength of this phase is proportional to {1/|x|}, so the phase would be approximately stationary on a cap formed by intersecting the sphere with a {\sim 1/|x|} 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 {\sim 1/|x|^{1/2}} in the directions of the {d-1}-dimensional tangent space, so the cap will have surface measure {\sim |x|^{-(d-1)/2}}, which leads to the prediction (7). We combine (6), (7) into the unified estimate

\displaystyle  (d\sigma)^\vee(x) \lesssim_d \langle x\rangle^{-(d-1)/2}, \ \ \ \ \ (8)

where the “Japanese bracket” {\langle x\rangle} is defined as {\langle x \rangle := (1+|x|^2)^{1/2}}. Since {\langle x \rangle^{-\alpha}} lies in {L^p({\bf R}^d)} precisely when {p > \frac{d}{\alpha}}, we conclude that

\displaystyle  (d\sigma)^\vee \in L^q({\bf R}^d) \hbox{ iff } q > \frac{d}{(d-1)/2}.

Applying Young’s convolution inequality, we conclude (after some arithmetic) that

\displaystyle  \| f * (d\sigma)^\vee \|_{L^{p'}({\bf R}^d)} \lesssim_{p,d} \|f\|_{L^p({\bf R}^d)}

whenever {1 \leq p < \frac{4d}{3d+1}}, and the claim now follows from (5) and Hölder’s inequality. \Box

Remark 2 By using the Hardy-Littlewood-Sobolev inequality in place of Young’s convolution inequality, one can also establish this result for {p = \frac{4d}{3d+1}}.

Motivated by this result, given any Radon measure {\mu} on {{\bf R}^d} and any exponents {1 \leq p,q \leq \infty}, we use {R_\mu(p \rightarrow q)} to denote the claim that the restriction estimate

\displaystyle  \| \hat f \|_{L^q({\bf R}^d, \mu)} \lesssim_{d,p,q,\mu} \|f\|_{L^p({\bf R}^d)} \ \ \ \ \ (9)

for all Schwartz functions {f}; if {S} is a {k}-dimensional submanifold of {{\bf R}^d} (possibly with boundary), we write {R_S(p \rightarrow q)} for {R_\mu(p \rightarrow q)} where {\mu} is the {k}-dimensional surface measure on {S}. Thus, for instance, we trivially always have {R_S(1 \rightarrow \infty)}, while Theorem 1 asserts that {R_{S^{d-1}}(p \rightarrow 2)} holds whenever {1 \leq p < \frac{4d}{3d+1}}. 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.)

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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 {\delta > 0} be a radius. For each {j = 1,\dots,d}, let {\mathbb{T}_j} denote a finite family of infinite tubes {T_j} in {{\bf R}^d} of radius {\delta}. Assume the following axiom:

  • (i) (Transversality) whenever {T_j \in \mathbb{T}_j} is oriented in the direction of a unit vector {n_j} for {j =1,\dots,d}, we have

    \displaystyle  \left|\bigwedge_{j=1}^d n_j\right| \geq A^{-1}

    for some {A>0}, where we use the usual Euclidean norm on the wedge product {\bigwedge^d {\bf R}^d}.

Then, for any {p \geq \frac{1}{d-1}}, one has

\displaystyle  \left\| \prod_{j=1}^d \sum_{T_j \in \mathbb{T}_j} 1_{T_j} \right\|_{L^p({\bf R}^d)} \lesssim_{A,p} \delta^{\frac{d}{p}} \prod_{j \in [d]} \# \mathbb{T}_j. \ \ \ \ \ (1)

where {L^p({\bf R}^d)} are the usual Lebesgue norms with respect to Lebesgue measure, {1_{T_j}} denotes the indicator function of {T_j}, and {\# \mathbb{T}_j} denotes the cardinality of {\mathbb{T}_j}.

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 {O(1)} (and sets {\delta} to be small), but allows the tubes {T_j} to be curved in a {C^2} 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 {\delta^{-\varepsilon}} or {\log^{O(1)} \frac{1}{\delta}}), so long as one stays away from the endpoint case {p=\frac{1}{d-1}} (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 {\log^{O(1)} \frac{1}{\delta}} (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 {\frac{1}{d-1} < p \leq \infty} be an exponent, and let {A \geq 2} be a parameter. Let {M} be a sufficiently large natural number, depending only on {d}. For {j \in [d]}, let {U_j} be an open subset of {B^{d-1}(0,A)}, and let {h_j: U_j \rightarrow {\bf R}} be a smooth function obeying the following axioms:

  • (i) (Regularity) For each {j \in [d]} and {\xi \in U_j}, one has

    \displaystyle  |\nabla_\xi^{\otimes m} \otimes h_j(\xi)| \leq A \ \ \ \ \ (2)

    for all {1 \leq m \leq M}.

  • (ii) (Transversality) One has

    \displaystyle  \left| \bigwedge_{j \in [d]} (-\nabla_\xi h_j(\xi_j),1) \right| \geq A^{-1}

    whenever {\xi_j \in U_j} for {j \in [d]}.

Let {U_{j,1/A} \subset U_j} be the sets

\displaystyle  U_{j,1/A} := \{ \xi \in U_j: B^{d-1}(\xi,1/A) \subset U_j \}. \ \ \ \ \ (3)

Then one has

\displaystyle  \left\| \prod_{j \in [d]} {\mathcal E}_j f_j \right\|_{L^{2p}({\bf R}^d)} \leq A^{O(1)} \left(d-1-\frac{1}{p}\right)^{-O(1)} \prod_{j \in [d]} \|f_j \|_{L^2(U_{j,1/A})}

for any {f_j \in L^2(U_{j,1/A} \rightarrow {\bf C})}, {j \in [d]}, extended by zero outside of {U_{j,1/A}}, and {{\mathcal E}_j} denotes the extension operator

\displaystyle  {\mathcal E}_j f_j( x', x_d ) := \int_{U_j} e^{2\pi i (x' \xi^T + x_d h_j(\xi))} f_j(\xi)\ d\xi.

Local versions of such estimate, in which {L^{2p}({\bf R}^d)} is replaced with {L^{2p}(B^d(0,R))} for some {R \geq 2}, and one accepts a loss of the form {\log^{O(1)} R}, 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

\displaystyle  \left\| \prod_{j \in [d]} E_{r}[{\mathcal E}_j f_j] \right\|_{L^{p}({\bf R}^d)} \leq A^{O(1)} \left(d-1 - \frac{1}{p}\right)^{O(1)} r^{\frac{d}{p}} \prod_{j \in [d]} \| f_j \|_{L^2(U_{j,1/A})}^2

where {E_r} denotes the local energies

\displaystyle  E_{r}[f](x',x_d) := \int_{B^{d-1}(x',r)} |f(y',x_d)|^2\ dy'

(actually for technical reasons it is more convenient to use a smoother weight than the strict cutoff to the disk {B^{d-1}(x',r)}). With logarithmic losses, it is not difficult to establish this estimate by an upward induction on {r}. To avoid such losses we use the heat flow monotonicity method. Here we run into the issue that the extension operators {{\mathcal E}_j f_j} are complex-valued rather than non-negative, and thus would not be expected to obey many good montonicity properties. However, the local energies {E_r[{\mathcal E}_j f_j]} can be expressed in terms of the magnitude squared of what is essentially the Gabor transform of {{\mathcal E}_j f_j}, and these are non-negative; furthermore, the dispersion relation associated to the extension operators {{\mathcal E}_j f_j} 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.

This is a blog version of a talk I recently gave at the IPAM workshop on “The Kakeya Problem, Restriction Problem, and Sum-product Theory”.

Note: the discussion here will be highly non-rigorous in nature, being extremely loose in particular with asymptotic notation and with the notion of dimension. Caveat emptor.

One of the most infamous unsolved problems at the intersection of geometric measure theory, incidence combinatorics, and real-variable harmonic analysis is the Kakeya set conjecture. We will focus on the following three-dimensional case of the conjecture, stated informally as follows:

Conjecture 1 (Kakeya conjecture) Let {E} be a subset of {{\bf R}^3} that contains a unit line segment in every direction. Then {\hbox{dim}(E) = 3}.

This conjecture is not precisely formulated here, because we have not specified exactly what type of set {E} is (e.g. measurable, Borel, compact, etc.) and what notion of dimension we are using. We will deliberately ignore these technical details in this post. It is slightly more convenient for us here to work with lines instead of unit line segments, so we work with the following slight variant of the conjecture (which is essentially equivalent):

Conjecture 2 (Kakeya conjecture, again) Let {{\cal L}} be a family of lines in {{\bf R}^3} that meet {B(0,1)} and contain a line in each direction. Let {E} be the union of the restriction {\ell \cap B(0,2)} to {B(0,2)} of every line {\ell} in {{\cal L}}. Then {\hbox{dim}(E) = 3}.

As the space of all directions in {{\bf R}^3} is two-dimensional, we thus see that {{\cal L}} is an (at least) two-dimensional subset of the four-dimensional space of lines in {{\bf R}^3} (actually, it lies in a compact subset of this space, since we have constrained the lines to meet {B(0,1)}). One could then ask if this is the only property of {{\cal L}} that is needed to establish the Kakeya conjecture, that is to say if any subset of {B(0,2)} which contains a two-dimensional family of lines (restricted to {B(0,2)}, and meeting {B(0,1)}) is necessarily three-dimensional. Here we have an easy counterexample, namely a plane in {B(0,2)} (passing through the origin), which contains a two-dimensional collection of lines. However, we can exclude this case by adding an additional axiom, leading to what one might call a “strong” Kakeya conjecture:

Conjecture 3 (Strong Kakeya conjecture) Let {{\cal L}} be a two-dimensional family of lines in {{\bf R}^3} that meet {B(0,1)}, and assume the Wolff axiom that no (affine) plane contains more than a one-dimensional family of lines in {{\cal L}}. Let {E} be the union of the restriction {\ell \cap B(0,2)} of every line {\ell} in {{\cal L}}. Then {\hbox{dim}(E) = 3}.

Actually, to make things work out we need a more quantitative version of the Wolff axiom in which we constrain the metric entropy (and not just dimension) of lines that lie close to a plane, rather than exactly on the plane. However, for the informal discussion here we will ignore these technical details. Families of lines that lie in different directions will obey the Wolff axiom, but the converse is not true in general.

In 1995, Wolff established the important lower bound {\hbox{dim}(E) \geq 5/2} (for various notions of dimension, e.g. Hausdorff dimension) for sets {E} in Conjecture 3 (and hence also for the other forms of the Kakeya problem). However, there is a key obstruction to going beyond the {5/2} barrier, coming from the possible existence of half-dimensional (approximate) subfields of the reals {{\bf R}}. To explain this problem, it easiest to first discuss the complex version of the strong Kakeya conjecture, in which all relevant (real) dimensions are doubled:

Conjecture 4 (Strong Kakeya conjecture over {{\bf C}}) Let {{\cal L}} be a four (real) dimensional family of complex lines in {{\bf C}^3} that meet the unit ball {B(0,1)} in {{\bf C}^3}, and assume the Wolff axiom that no four (real) dimensional (affine) subspace contains more than a two (real) dimensional family of complex lines in {{\cal L}}. Let {E} be the union of the restriction {\ell \cap B(0,2)} of every complex line {\ell} in {{\cal L}}. Then {E} has real dimension {6}.

The argument of Wolff can be adapted to the complex case to show that all sets {E} occuring in Conjecture 4 have real dimension at least {5}. Unfortunately, this is sharp, due to the following fundamental counterexample:

Proposition 5 (Heisenberg group counterexample) Let {H \subset {\bf C}^3} be the Heisenberg group

\displaystyle  H = \{ (z_1,z_2,z_3) \in {\bf C}^3: \hbox{Im}(z_1) = \hbox{Im}(z_2 \overline{z_3}) \}

and let {{\cal L}} be the family of complex lines

\displaystyle  \ell_{s,t,\alpha} := \{ (\overline{\alpha} z + t, z, sz + \alpha): z \in {\bf C} \}

with {s,t \in {\bf R}} and {\alpha \in {\bf C}}. Then {H} is a five (real) dimensional subset of {{\bf C}^3} that contains every line in the four (real) dimensional set {{\cal L}}; however each four real dimensional (affine) subspace contains at most a two (real) dimensional set of lines in {{\cal L}}. In particular, the strong Kakeya conjecture over the complex numbers is false.

This proposition is proven by a routine computation, which we omit here. The group structure on {H} is given by the group law

\displaystyle  (z_1,z_2,z_3) \cdot (w_1,w_2,w_3) = (z_1 + w_1 + z_2 \overline{w_3} - z_3 \overline{w_2}, z_2 +w_2, z_3+w_3),

giving {E} the structure of a {2}-step simply-connected nilpotent Lie group, isomorphic to the usual Heisenberg group over {{\bf R}^2}. Note that while the Heisenberg group is a counterexample to the complex strong Kakeya conjecture, it is not a counterexample to the complex form of the original Kakeya conjecture, because the complex lines {{\cal L}} in the Heisenberg counterexample do not point in distinct directions, but instead only point in a three (real) dimensional subset of the four (real) dimensional space of available directions for complex lines. For instance, one has the one real-dimensional family of parallel lines

\displaystyle  \ell_{0,t,0} = \{ (t, z, 0): z \in {\bf C}\}

with {t \in {\bf R}}; multiplying this family of lines on the right by a group element in {H} gives other families of parallel lines, which in fact sweep out all of {{\cal L}}.

The Heisenberg counterexample ultimately arises from the “half-dimensional” (and hence degree two) subfield {{\bf R}} of {{\bf C}}, which induces an involution {z \mapsto \overline{z}} which can then be used to define the Heisenberg group {H} through the formula

\displaystyle  H = \{ (z_1,z_2,z_3) \in {\bf C}^3: z_1 - \overline{z_1} = z_2 \overline{z_3} - z_3 \overline{z_2} \}.

Analogous Heisenberg counterexamples can also be constructed if one works over finite fields {{\bf F}_{q^2}} that contain a “half-dimensional” subfield {{\bf F}_q}; we leave the details to the interested reader. Morally speaking, if {{\bf R}} in turn contained a subfield of dimension {1/2} (or even a subring or “approximate subring”), then one ought to be able to use this field to generate a counterexample to the strong Kakeya conjecture over the reals. Fortunately, such subfields do not exist; this was a conjecture of Erdos and Volkmann that was proven by Edgar and Miller, and more quantitatively by Bourgain (answering a question of Nets Katz and myself). However, this fact is not entirely trivial to prove, being a key example of the sum-product phenomenon.

We thus see that to go beyond the {5/2} dimension bound of Wolff for the 3D Kakeya problem over the reals, one must do at least one of two things:

  • (a) Exploit the distinct directions of the lines in {{\mathcal L}} in a way that goes beyond the Wolff axiom; or
  • (b) Exploit the fact that {{\bf R}} does not contain half-dimensional subfields (or more generally, intermediate-dimensional approximate subrings).

(The situation is more complicated in higher dimensions, as there are more obstructions than the Heisenberg group; for instance, in four dimensions quadric surfaces are an important obstruction, as discussed in this paper of mine.)

Various partial or complete results on the Kakeya problem over various fields have been obtained through route (a) or route (b). For instance, in 2000, Nets Katz, Izabella Laba and myself used route (a) to improve Wolff’s lower bound of {5/2} for Kakeya sets very slightly to {5/2+10^{-10}} (for a weak notion of dimension, namely upper Minkowski dimension). In 2004, Bourgain, Katz, and myself established a sum-product estimate which (among other things) ruled out approximate intermediate-dimensional subrings of {{\bf F}_p}, and then pursued route (b) to obtain a corresponding improvement {5/2+\epsilon} to the Kakeya conjecture over finite fields of prime order. The analogous (discretised) sum-product estimate over the reals was established by Bourgain in 2003, which in principle would allow one to extend the result of Katz, Laba and myself to the strong Kakeya setting, but this has not been carried out in the literature. Finally, in 2009, Dvir used route (a) and introduced the polynomial method (as discussed previously here) to completely settle the Kakeya conjecture in finite fields.

Below the fold, I present a heuristic argument of Nets Katz and myself, which in principle would use route (b) to establish the full (strong) Kakeya conjecture. In broad terms, the strategy is as follows:

  1. Assume that the (strong) Kakeya conjecture fails, so that there are sets {E} of the form in Conjecture 3 of dimension {3-\sigma} for some {\sigma>0}. Assume that {E} is “optimal”, in the sense that {\sigma} is as large as possible.
  2. Use the optimality of {E} (and suitable non-isotropic rescalings) to establish strong forms of standard structural properties expected of such sets {E}, namely “stickiness”, “planiness”, “local graininess” and “global graininess” (we will roughly describe these properties below). Heuristically, these properties are constraining {E} to “behave like” a putative Heisenberg group counterexample.
  3. By playing all these structural properties off of each other, show that {E} can be parameterised locally by a one-dimensional set which generates a counterexample to Bourgain’s sum-product theorem. This contradiction establishes the Kakeya conjecture.

Nets and I have had an informal version of argument for many years, but were never able to make a satisfactory theorem (or even a partial Kakeya result) out of it, because we could not rigorously establish anywhere near enough of the necessary structural properties (stickiness, planiness, etc.) on the optimal set {E} for a large number of reasons (one of which being that we did not have a good notion of dimension that did everything that we wished to demand of it). However, there is beginning to be movement in these directions (e.g. in this recent result of Guth using the polynomial method obtaining a weak version of local graininess on certain Kakeya sets). In view of this (and given that neither Nets or I have been actively working in this direction for some time now, due to many other projects), we’ve decided to distribute these ideas more widely than before, and in particular on this blog.

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In this post I would like to make some technical notes on a standard reduction used in the (Euclidean, maximal) Kakeya problem, known as the two ends reduction. This reduction (which takes advantage of the approximate scale-invariance of the Kakeya problem) was introduced by Wolff, and has since been used many times, both for the Kakeya problem and in other similar problems (e.g. by Jim Wright and myself to study curved Radon-like transforms). I was asked about it recently, so I thought I would describe the trick here. As an application I give a proof of the {d=\frac{n+1}{2}} case of the Kakeya maximal conjecture.

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Below the fold is a version of my talk “Recent progress on the Kakeya conjecture” that I gave at the Fefferman conference.

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Jordan Ellenberg, Richard Oberlin, and I have just uploaded to the arXiv the paper “The Kakeya set and maximal conjectures for algebraic varieties over finite fields“, submitted to Mathematika.  This paper builds upon some work of Dvir and later authors on the Kakeya problem in finite fields, which I have discussed in this earlier blog post.  Dvir established the following:

Kakeya set conjecture for finite fields. Let F be a finite field, and let E be a subset of F^n that contains a line in every direction.  Then E has cardinality at least c_n |F|^n for some c_n > 0.

The initial argument of Dvir gave c_n = 1/n!.  This was improved to c_n = c^n for some explicit 0 < c < 1 by Saraf and Sudan, and recently to c_n =1/2^n by Dvir, Kopparty, Saraf, and Sudan, which is within a factor 2 of the optimal result.

In our work we investigate a somewhat different set of improvements to Dvir’s result.  The first concerns the Kakeya maximal function f^*: {\Bbb P}^{n-1}(F) \to {\Bbb R} of a function f: F^n \to {\Bbb R}, defined for all directions \xi \in {\Bbb P}^{n-1}(F) in the projective hyperplane at infinity by the formula

f^*(\xi) = \sup_{\ell // \xi} \sum_{x \in \ell} |f(x)|

where the supremum ranges over all lines \ell in F^n oriented in the direction \xi.  Our first result is the endpoint L^p estimate for this operator, namely

Kakeya maximal function conjecture in finite fields. We have \| f^* \|_{\ell^n({\Bbb P}^{n-1}(F))} \leq C_n |F|^{(n-1)/n} \|f\|_{\ell^n(F^n)} for some constant C_n > 0.

This result implies Dvir’s result, since if f is the indicator function of the set E in Dvir’s result, then f^*(\xi) = |F| for every \xi \in {\Bbb P}^{n-1}(F).  However, it also gives information on more general sets E which do not necessarily contain a line in every direction, but instead contain a certain fraction of a line in a subset of directions.  The exponents here are best possible in the sense that all other \ell^p \to \ell^q mapping properties of the operator can be deduced (with bounds that are optimal up to constants) by interpolating the above estimate with more trivial estimates.  This result is the finite field analogue of a long-standing (and still open) conjecture for the Kakeya maximal function in Euclidean spaces; we rely on the polynomial method of Dvir, which thus far has not extended to the Euclidean setting (but note the very interesting variant of this method by Guth that has established the endpoint multilinear Kakeya maximal function estimate in this setting, see this blog post for further discussion).

It turns out that a direct application of the polynomial method is not sufficient to recover the full strength of the maximal function estimate; but by combining the polynomial method with the Nikishin-Maurey-Pisier-Stein “method of random rotations” (as interpreted nowadays by Stein and later by Bourgain, and originally inspired by the factorisation theorems of Nikishin, Maurey, and Pisier), one can already recover a “restricted weak type” version of the above estimate.  If one then enhances the polynomial method with the “method of multiplicities” (as introduced by Saraf and Sudan) we can then recover the full “strong type” estimate; a few more details below the fold.

It turns out that one can generalise the above results to more general affine or projective algebraic varieties over finite fields.  In particular, we showed

Kakeya maximal function conjecture in algebraic varieties. Suppose that W \subset {\Bbb P}^N is an (n-1)-dimensional algebraic variety.  Let d \geq 1 be an integer. Then we have

\| \sup_{\gamma \ni x; \gamma \not \subset W} \sum_{y \in \gamma} f(y) \|_{\ell^n_x(W(F))} \leq C_{n,d,N,W} |F|^{(n-1)/n} \|f\|_{\ell^n({\Bbb P}^N(F))}

for some constant C_{n,d,N,W} > 0, where the supremum is over all irreducible algebraic curves \gamma of degree at most d that pass through x but do not lie in W, and W(F) denotes the F-points of W.

The ordinary Kakeya maximal function conjecture corresponds to the case when N=n, W is the hyperplane at infinity, and the degree d is equal to 1.  One corollary of this estimate is a Dvir-type result: a subset of {\Bbb P}^N(F) which contains, for each x in W, an irreducible algebraic curve of degree d passing through x but not lying in W, has cardinality \gg |F|^n if |W| \gg |F|^{n-1}.  (In particular this implies a lower bound for Nikodym sets worked out by Li.)  The dependence of the implied constant on W is only via the degree of W.

The techniques used in the flat case can easily handle curves \gamma of higher degree (provided that we allow the implied constants to depend on d), but the method of random rotations does not seem to work directly on the algebraic variety W as there are usually no symmetries of this variety to exploit.  Fortunately, we can get around this by using a “random projection trick” to “flatten” W into a hyperplane (after first expressing W as the zero locus of some polynomials, and then composing with the graphing map for such polynomials), reducing the non-flat case to the flat case.

Below the fold, I wish to sketch two of the key ingredients in our arguments, the random rotations method and the random projections trick.  (We of course also use some algebraic geometry, but mostly low-tech stuff, on the level of Bezout’s theorem, though we do need one non-trivial result of Kleiman (from SGA6), that asserts that bounded degree varieties can be cut out by a bounded number of polynomials of bounded degree.)

[Update, March 14: See also Jordan’s own blog post on our paper.]

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In 1917, Soichi Kakeya posed the following problem:

Kakeya needle problem. What is the least amount of area required to continuously rotate a unit line segment in the plane by a full rotation (i.e. by 360^\circ)?

In 1928, Besicovitch showed that given any \varepsilon > 0, there exists a planar set of area at most \varepsilon within which a unit needle can be continuously rotated; the proof relies on the construction of what is now known as a Besicovitch set – a set of measure zero in the plane which contains a unit line segment in every direction.  So the answer to the Kakeya needle problem is “zero”.

I was recently asked (by Claus Dollinger) whether one can take \varepsilon = 0; in other words,

Question. Does there exist a set of measure zero within which a unit line segment can be continuously rotated by a full rotation?

This question does not seem to be explicitly answered in the literature.  In the papers of von Alphen and of Cunningham, it is shown that it is possible to continuously rotate a unit line segment inside a set of arbitrarily small measure and of uniformly bounded diameter; this result is of course implied by a positive answer to the above question (since continuous functions on compact sets are bounded), but the converse is not true.

Below the fold, I give the answer to the problem… but perhaps readers may wish to make a guess as to what the answer is first before proceeding, to see how good their real analysis intuition is.  (To partially prevent spoilers for those reading this post via RSS, I will be whitening the text; you will have to highlight the text in order to see it.  Unfortunately, I do not know how to white out the LaTeX in such a way that it is visible upon highlighting, so RSS readers may wish to stop reading right now; but I suppose one can view the LaTeX as supplying hints to the problem, without giving away the full solution.)

[Update, March 13: a non-whitened version of this article can be found as part of this book.]

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One of my favourite family of conjectures (and one that has preoccupied a significant fraction of my own research) is the family of Kakeya conjectures in geometric measure theory and harmonic analysis.  There are many (not quite equivalent) conjectures in this family.  The cleanest one to state is the set conjecture:

Kakeya set conjecture: Let n \geq 1, and let E \subset {\Bbb R}^n contain a unit line segment in every direction (such sets are known as Kakeya sets or Besicovitch sets).  Then E has Hausdorff dimension and Minkowski dimension equal to n.

One reason why I find these conjectures fascinating is the sheer variety of mathematical fields that arise both in the partial results towards this conjecture, and in the applications of those results to other problems.  See for instance this survey of Wolff, my Notices article and this article of Łaba on the connections between this problem and other problems in Fourier analysis, PDE, and additive combinatorics; there have even been some connections to number theory and to cryptography.  At the other end of the pipeline, the mathematical tools that have gone into the proofs of various partial results have included:

[This list is not exhaustive.]

Very recently, I was pleasantly surprised to see yet another mathematical tool used to obtain new progress on the Kakeya conjecture, namely (a generalisation of) the famous Ham Sandwich theorem from algebraic topology.  This was recently used by Guth to establish a certain endpoint multilinear Kakeya estimate left open by the work of Bennett, Carbery, and myself.  With regards to the Kakeya set conjecture, Guth’s arguments assert, roughly speaking, that the only Kakeya sets that can fail to have full dimension are those which obey a certain “planiness” property, which informally means that the line segments that pass through a typical point in the set must be essentially coplanar. (This property first surfaced in my paper with Katz and Łaba.)  Guth’s arguments can be viewed as a partial analogue of Dvir’s arguments in the finite field setting (which I discussed in this blog post) to the Euclidean setting; in particular, both arguments rely crucially on the ability to create a polynomial of controlled degree that vanishes at or near a large number of points.  Unfortunately, while these arguments fully settle the Kakeya conjecture in the finite field setting, it appears that some new ideas are still needed to finish off the problem in the Euclidean setting.  Nevertheless this is an interesting new development in the long history of this conjecture, in particular demonstrating that the polynomial method can be successfully applied to continuous Euclidean problems (i.e. it is not confined to the finite field setting).

In this post I would like to sketch some of the key ideas in Guth’s paper, in particular the role of the Ham Sandwich theorem (or more precisely, a polynomial generalisation of this theorem first observed by Gromov).

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One of my favourite unsolved problems in mathematics is the Kakeya conjecture in geometric measure theory. This conjecture is descended from the

Kakeya needle problem. (1917) What is the least area in the plane required to continuously rotate a needle of unit length and zero thickness around completely (i.e. by 360^\circ)?

For instance, one can rotate a unit needle inside a unit disk, which has area \pi/4. By using a deltoid one requires only \pi/8 area.

In 1928, Besicovitch showed that that in fact one could rotate a unit needle using an arbitrarily small amount of positive area. This unintuitive fact was a corollary of two observations. The first, which is easy, is that one can translate a needle using arbitrarily small area, by sliding the needle along the direction it points in for a long distance (which costs zero area), turning it slightly (costing a small amount of area), sliding back, and then undoing the turn. The second fact, which is less obvious, can be phrased as follows. Define a Kakeya set in {\Bbb R}^2 to be any set which contains a unit line segment in each direction. (See this Java applet of mine, or the wikipedia page, for some pictures of such sets.)

Theorem. (Besicovitch, 1919) There exists Kakeya sets {\Bbb R}^2 of arbitrarily small area (or more precisely, Lebesgue measure).

In fact, one can construct such sets with zero Lebesgue measure. On the other hand, it was shown by Davies that even though these sets had zero area, they were still necessarily two-dimensional (in the sense of either Hausdorff dimension or Minkowski dimension). This led to an analogous conjecture in higher dimensions:

Kakeya conjecture. A Besicovitch set in {\Bbb R}^n (i.e. a subset of {\Bbb R}^n that contains a unit line segment in every direction) has Minkowski and Hausdorff dimension equal to n.

This conjecture remains open in dimensions three and higher (and gets more difficult as the dimension increases), although many partial results are known. For instance, when n=3, it is known that Besicovitch sets have Hausdorff dimension at least 5/2 and (upper) Minkowski dimension at least 5/2 + 10^{-10}. See my Notices article for a general survey of this problem (and its connections with Fourier analysis, additive combinatorics, and PDE), my paper with Katz for a more technical survey, and Wolff’s survey for a systematic treatment of the field (up to about 1998 or so).

In 1999, Wolff proposed a simpler finite field analogue of the Kakeya conjecture as a model problem that avoided all the technical issues involving Minkowski and Hausdorff dimension. If F^n is a vector space over a finite field F, define a Kakeya set to be a subset of F^n which contains a line in every direction.

Finite field Kakeya conjecture. Let E \subset F^n be a Kakeya set. Then E has cardinality at least c_n |F|^n, where c_n > 0 depends only on n.

This conjecture has had a significant influence in the subject, in particular inspiring work on the sum-product phenomenon in finite fields, which has since proven to have many applications in number theory and computer science. Modulo minor technicalities, the progress on the finite field Kakeya conjecture was, until very recently, essentially the same as that of the original “Euclidean” Kakeya conjecture.

Last week, the finite field Kakeya conjecture was proven using a beautifully simple argument by Zeev Dvir, using the polynomial method in algebraic extremal combinatorics. The proof is so short that I can present it in full here.

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