Abstract
In a recent paper, Du and Zhang (Ann Math 189:837–861, 2019) proved a fractal Fourier restriction estimate and used it to establish the sharp \(L^2\) estimate on the Schrödinger maximal function in \(\mathbb R^n\), \(n \ge 2\). In this paper, we show that the Du–Zhang estimate is the endpoint of a family of fractal restriction estimates such that each member of the family (other than the original) implies a sharp Kakeya result in \(\mathbb R^n\) that is closely related to the polynomial Wolff axioms. We also prove that all the estimates of our family are true in \(\mathbb R^2\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \(Ef=E_{\mathcal P} f\) be the extension operator associated with the unit paraboloid \({\mathcal P} = \{ \xi \in \mathbb R^n : \xi _n = \xi _1^2 + \ldots + \xi _{n-1}^2 \le 1 \}\) in \(\mathbb R^n\):
where \(\mathbb B^{n-1}\) is the unit ball in \(\mathbb R^{n-1}\).
Our starting point is the following fractal restriction theorem of Du and Zhang [4]. (Throughout this paper, we denote a cube in \(\mathbb R^n\) of center x and side-length r by \(\widetilde{B}(x,r)\).)
Theorem1-A
(Du and Zhang [4, Corollary 1.6]) Suppose \(n \ge 2\), \(1 \le \alpha \le n\), \(R \ge 1\), \(X= \cup _k \widetilde{B}_k\) is a union of lattice unit cubes in \(\widetilde{B}(0,R) \subset \mathbb R^n\), and
where the sup is taken over all pairs \((x',r) \in \mathbb R^n \times [1,\infty )\) satisfying \(\widetilde{B}(x',r) \subset \widetilde{B}(0,R)\). Then to every \(\epsilon > 0\) there is a constant \(C_\epsilon \) such that
for all \(f \in L^2(\mathbb B^{n-1})\).
In [4], Theorem 1 was used to derive the sharp \(L^2\) estimate on the Schrödinger maximal function (see [4, Theorem 1.3] and the paragraph following the statement of [4, Corollary 1.6]). The authors of [4] also used Theorem 1 to obtain new results on the Hausdorff dimension of the sets where Schrödinger solutions diverge (see [11]), achieve progress on Falconer’s distance set conjecture in geometric measure theory (see [6]), and improve on the decay estimates of spherical means of Fourier transforms of measures (see [16]).
The purpose of this paper is threefold:
-
Show that Theorem 1 is a borderline sharp Kakeya result in the sense that (1) is the endpoint of a family of estimates (see (2) in the statement of Conjecture 1.1) such that each member of the family (other than (1)) implies a certain sharp Kakeya result that we will formulate in §3 below.
-
Show that the sharp Kakeya result is true in certain cases in \(\mathbb R^3\); see Theorem 4.1.
-
Prove Conjecture 1.1 in \(\mathbb R^2\) (see Theorem 5.1) in the hope that this will shed some light on whether it would be possible to modify the Du-Zhang argument to also prove it in higher dimensions and consequently obtain the Kakeya result without having to pass through the restriction conjecture.
Conjectute 1.1
(when \(\beta =2/n\) or \(n=2\), this is a theorem) Suppose n, \(\alpha \), R, X, and \(\gamma \) are as in the statement of Theorem 1.
Let \(\beta \) be a parameter satisfying \(1/n \le \beta \le 2/n\), and define the exponent p by
Then to every \(\epsilon > 0\) there is a constant \(C_\epsilon \) such that
for all \(f \in L^p(\mathbb B^{n-1})\).
We note that when \(\beta = 2/n\), (2) becomes (1), so, to prove Conjecture 1.1 we need to perform the following trade: lower the power of \(\gamma \) in (1) from 2/n to \(\beta \) in return for raising the Lebesgue space exponent from 2 to p.
We will show below that if (2) holds for any \(\beta < 2/n\), then we obtain the sharp Kakeya result of §3.
As noted above, in dimension \(n=2\), (2) is true for all \(1/2 \le \beta \le 1\) (and hence Conjecture 1.1 is a theorem in the plane). We will prove this in the last three sections of the paper by using weighted bilinear restriction estimates and the broad-narrow strategy of [1].
Before we discuss the implications of Conjecture 1.1 to the Kakeya problem, it will be convenient to write (2) in an equivalent form, which is, perhaps, more user-friendly. This is the purpose of the next section.
2 Writing (2) in an Equivalent Form
Suppose \(n \ge 1\) and \(0 < \alpha \le n\). Following [12] (see also [3] and [13]), for Lebesgue measurable functions \(H: \mathbb R^n \rightarrow [0,1]\), we define
where \(B(x_0,R)\) denotes the ball in \(\mathbb R^n\) of center \(x_0\) and radius R. We say H is a weight of fractal dimension \(\alpha \) if \(A_\alpha (H) < \infty \). We note that \(A_\beta (H) \le A_\alpha (H)\) if \(\beta \ge \alpha \), so we are not really assigning a dimension to the function H; the phrase “H is a weight of dimension \(\alpha \)” is merely another way for us to say that \(A_\alpha (H) < \infty \).
Proposition 2.1
Suppose n, \(\alpha \), R, X, \(\gamma \), \(\beta \), and p are as in the statement of Conjecture 1.1. Then the estimate (2) holds if and only if to every \(\epsilon > 0\) there is a constant \(C_\epsilon \) such that
for all functions \(f \in L^p(\mathbb B^{n-1})\) and weights H of fractal dimension \(\alpha \).
Proof
Let H be the characteristic function of X. By the definition of \(\gamma \), we have
for all \(x_0 \in \mathbb R^n\) and \(r \ge 1\). Thus H is a weight on \(\mathbb R^n\) of fractal dimension \(\alpha \), and \(A_\alpha (H) \le 3^\alpha \gamma \). This immediately shows that (3) implies (2).
To prove the reverse implication, we follow [4, Proof of Theorem 2.2].
We consider a covering \(\{ \widetilde{B} \}\) of B(0, R) by unit lattice cubes. Since every unit cube is contained in a ball of radius \(\sqrt{n}\), we have \(\int _{\widetilde{B}} H(x) dx \le A_\alpha (H) n^{\alpha /2}\), so, if we define \(v(\widetilde{B})=A_\alpha (H)^{-1} \int _{\widetilde{B}} H(x) dx\) and \(V_k = \{ \widetilde{B} : 2^{k-1} < n^{-\alpha /2} v(\widetilde{B}) \le 2^k \}\), then
We note that
for all \(\widetilde{B} \in V_k\), where we have used the assumptions \(\beta \le 2/n \le 1\) and \(\Vert H \Vert _{L^\infty } \le 1\).
The vast majority of the sets \(V_k\) are negligible for us. In fact, letting \(k_1\) be the sup of the set \(\{ k \in \mathbb Z :2^k \le R^{-1000n/\beta } \}\), we see that
where we used (4) on the line before the last, and the fact that \(2^{k_1} \le R^{-1000n/\beta }\) on the last line. Therefore, we only need to estimate
Letting \(k_0 \in \{ k_1+1, k_1+2, \ldots , 0 \}\) be the integer satisfying
we see that
Since \(-k_1 {\mathop {\sim }\limits ^{\textstyle <}}\log (2R)\), it follows that we only need to estimate
We start by using the uncertainty principle in the following form. Let \(d\sigma \) be the pushforward of the \((n-1)\)-dimensional Lebesgue measure under the map \(T : \mathbb B^{n-1} \rightarrow {\mathcal P}\) given by \(T(\omega )= (\omega , |\omega |^2)\). Since the measure \(d\sigma \) is compactly supported and \(Ef=\widehat{gd\sigma }\), where g is the function on \({\mathcal P}\) defined by the equation \(f= g \circ T\), it follows that there is a non-negative rapidly decaying function \(\psi \) on \(\mathbb R^n\) such that
where \(c(\widetilde{B})\) is the center of \(\widetilde{B}\). Thus
From (4) we know that \(\int _{\widetilde{B}} H(x) dx {\mathop {\sim }\limits ^{\textstyle <}}A_\alpha (H)^\beta 2^{k_0 \beta }\) for all \(\widetilde{B} \in V_{k_0}\). Also,
and
so
where \(V= \cup _{\widetilde{B} \in V_{k_0}} B(c(\widetilde{B}),R^\epsilon )\).
We now let \(\{ \widetilde{B}^* \}\) be the set of all the unit lattice cubes that intersect V, and \(X= \cup \, \widetilde{B}^*\). We plan to apply (2) on this set X, but we first need to estimate \(\gamma \).
Let \(B_r\) be a ball in \(\mathbb R^n\) of radius \(r \ge R^\epsilon \) (if \(1 \le r \le R^\epsilon \), then, clearly, \(\# \{ \widetilde{B}^* : \widetilde{B}^* \subset B_r \} {\mathop {\sim }\limits ^{\textstyle <}}R^{n \epsilon }\)), and \(V_r\) the subset of \(V_{k_0}\) that consists of all unit cubes \(\widetilde{B}\) such that \(B(c(\widetilde{B}),2R^\epsilon ) \cap B_r \not = \emptyset \). If \(B_r\) intersects any of the cubes \(\widetilde{B}^*\) that make up X, then \(B_r\) intersects \(B(c(\widetilde{B}),2R^\epsilon )\) for some \(\widetilde{B} \in V_r\). Therefore,
Our assumption \(r \ge R^\epsilon \), tells us that
so (using (6))
On the other hand,
so \(\#(V_r) {\mathop {\sim }\limits ^{\textstyle <}}R^{n \epsilon } 2^{-k_0} r^\alpha \), and so
Therefore, \(\gamma {\mathop {\sim }\limits ^{\textstyle <}}R^{2n \epsilon } 2^{-k_0}\).
Applying (2), we now obtain
which, combined with (5) and (7), implies that
which is our desired estimate (3).
3 Conjecture 1.1 Implies a Sharp Kakeya Result
Let \(\Omega \) be a subset of \(\mathbb R^n\) that obeys the following property: there is a number \(\alpha \) between 1 and n such that
for all balls \(B_R\) in \(\mathbb R^n\) of radius \(R \ge 1\). (Given \(E \subset \mathbb R^n\) a Lebesgue measurable set, we let |E| denote its Lebesgue measure.)
For large L, we divide the unit paraboloid \({\mathcal P}\) into finitely overlapping caps \(\theta _j\) each of radius \(L^{-1}\), and we associate with each \(\theta _j\) a family \(\mathbb T_j\) of parallel \(1 \times L\) tubes that tile \(\mathbb R^n\) and point in the direction normal to \(\theta _j\) at its center. We let N be the cardinality of the set
It is easy to see that the Kakeya conjecture (in its maximal operator form) implies the following bound on N: to every \(\epsilon > 0\) there is a constant \(C_\epsilon \) such that
for all \(L \ge 1\). In fact, [2, Proposition 2.2] presents a proof of the fact that the Kakeya conjecture implies (10) in the case when \(\Omega \) is a neighborhood of an algebraic variety. This proof easily extends to general sets \(\Omega \) satisfying (8). (For the connection between neighborhoods of algebraic varieties and the condition (8), we refer the reader to [14].)
We note that (10) implies that if \(\Omega \cap B(0,5L)\) contains at least one tube from each direction (i.e. at least one tube from each of the \(\sim L^{n-1}\) families \(\mathbb T_j\)), then \(\alpha = n\).
In the special case when \(\Omega \) is a neighborhood of an algebraic variety, this bound on N was proved by Guth [7] in \(\mathbb R^3\), conjectured by Guth [8] to be true in \(\mathbb R^n\) for all \(n \ge 3\), and proved by Zahl [17] in \(\mathbb R^4\); see also [9]. The conjecture of [8] was then settled in all dimensions by Katz and Rogers in [10].
In this section we prove that Conjecture 1.1 about the extension operator implies that all sets \(\Omega \subset \mathbb R^n\) that satisfy the dimensionality condition (8) will also possess the Kakeya property (10). Here is the precise statement.
Theorem 3.1
Suppose (3) (or equivalently (2)) holds for some \(1/n \le \beta < 2/n\). Then (10) holds for all Lebesgue measurable sets \(\Omega \subset \mathbb R^n\) that obey (8).
Proof
We first write the set J as \(\{ j_1, j_2, \ldots , j_N \}\), and for each \(1 \le l \le N\), we let \(T_l\) be a tube from \(\mathbb T_{j_l}\) that lies in \(\Omega \cap B(0,5L) = \Omega \cap B_{5L}\). Then
Recall that \(\mathbb T_{j_l}\) is a family of parallel \(1 \times L\) tubes that tile \(\mathbb R^n\) and point in the direction normal to the \(L^{-1}\)-cap \(\theta _{j_l}\). The projection of \(\theta _{j_l}\) into \(\mathbb B^{n-1}\) is an \(L^{-1}\)-ball. We denote this ball by \(B_l\) and let \(\omega _l\) be its center and \(\chi _l\) its characteristic function. Then
for all \(x=(x_1,x_2) \in \mathbb R^{n-1} \times \mathbb R\). Since \(|\omega -\omega _l| \le L^{-1}\) for all \(\omega \in B_l\), it follows that \(|E \chi _l(x)| {\mathop {\sim }\limits ^{\textstyle >}}|B_l| \sim L^{-(n-1)}\) on the set \(\{ x \in \mathbb R^n : |x_1 + 2x_2 \omega _l| {\mathop {\sim }\limits ^{\textstyle<}}L \text{ and } |x_2| {\mathop {\sim }\limits ^{\textstyle <}}L^2 \}\), and hence \(|E \chi _l(Lx)| {\mathop {\sim }\limits ^{\textstyle >}}L^{-(n-1)}\) on the set \(\{ x \in \mathbb R^n : |x_1 + 2x_2 \omega _l| {\mathop {\sim }\limits ^{\textstyle<}}1 \text{ and } |x_2| {\mathop {\sim }\limits ^{\textstyle <}}L \}\). Since \(|\omega _l| \le 1\), this last set contains a \(1 \times L\) tube \(\widetilde{T}_l\) that is parallel to the normal vector of the cap \(\theta _{j_l}\) at its center \((\omega _l, |\omega _l|^2)\). Moreover,
for all \(x \in \mathbb R^n\).
The tube \(\widetilde{T}_l\) is parallel to the tube \(T_l\) that we chose at the beginning of the proof and has the same dimensions, so \(T_l = v + \widetilde{T}_l\) for some vector \(v \in \mathbb R^n\), and so
for all \(x \in \mathbb R^n\). Defining the function \(f_l\) on \(\mathbb R^{n-1}\) by
we see that \(Ef_l(x) = E\chi _l(x-Lv)\), so that
for all \(x \in \mathbb R^n\). Returning to (11) and letting \(H = \chi _\Omega \), we arrive at
Next, we let \(\epsilon _l = \pm 1\) be random signs, define the function \(f: \mathbb B^{n-1} \rightarrow \mathbb C\) by \(f = \sum _{l=1}^N \epsilon _l f_l\), and use Khintchin’s inequality to get
where \(\mathbb E\) is the expectation sign. Since \(p \ge 2\), we can apply Hölder’s inequality in the inner integral to get
Applying the change of variables \(u=Lx\) and defining the weight \(H^*\) by \(H^*(u)= H(x)= H(u/L)\), this becomes
so that
We note that
if \(R \ge L\). On the other hand, if \(R \le L\), then
Therefore,
We are now in a good shape to apply (3), which tells us that
Inserting this back in (12), we get
so that
so that
Therefore,
But
so
At this point, it might be helpful for the reader to observe how the above argument breaks down in the \(p=2\) case: recalling that
we see that \(\beta = 2/n\) and (13) becomes \(1 {\mathop {\sim }\limits ^{\textstyle <}}L^{2\epsilon }\), which tells us nothing.
4 Proof of (10) in the Regime \(1 \le \alpha \le 2\) in \(\mathbb R^3\)
The fact that the Kakeya conjecture is true in \(\mathbb R^2\) tells us that (10) is also true there. In this section, we use Wolff’s hairbrush argument from [15], as adapted by Guth in [7], to prove the following bound on N.
Theorem 4.1
In \(\mathbb R^3\), we have
Proof
Let \(\Omega \) be a subset of \(\mathbb R^3\) that obeys (8). As we did in the previous section, for large L, we consider a decomposition \(\{ \theta _j \}\) of \({\mathcal P}\) into finitely overlapping caps each of radius \(L^{-1}\), and we associate with each \(\theta _j\) a family \(\mathbb T_j\) of parallel \(1 \times L\) tubes that tile \(\mathbb R^3\) and point in the direction of the normal vector \(v_j\) of \({\mathcal P}\) at the center of \(\theta _j\). The quantity N that we need to estimate is the cardinality of the set J as defined in (9).
For each \(j \in J\), we let \(T_j\) be a member of \(\mathbb T_j\) that lies in \(\Omega \cap B(0,5L)\), and \(S= \{ T_j \}\). Of course, \(N = \#(S)\).
We tile \(\Omega \cap B(0,5L)\) by unit lattice cubes \(\widetilde{B}\). Then (8) tells us that
Also, each tube \(T_j\) intersects \(\sim L\) of the cubes \(\widetilde{B}\).
We now define the function \(f : \{ \widetilde{B} \} \rightarrow \mathbb Z\) by
Then
So, by Cauchy–Schwarz and (14),
and so
which means that the set
has cardinality \({\mathop {\sim }\limits ^{\textstyle >}}N^2 L^{2 - \alpha }\). Therefore, the set
has cardinality
If \(C_1 N^2 L^{2 - \alpha } \le 5 C_2 N L\), then \(N \le (5C_2/C_1) L^{\alpha -1}\) and the theorem will be proved. So, we may assume that \(N \ge C_3 L^{\alpha -1}\) for some large constant \(C_3\). Therefore, \(\#(X) {\mathop {\sim }\limits ^{\textstyle >}}N^2 L^{2 - \alpha }\).
For \(l \in \mathbb N\), we define \(X_l\) to be the subset of X for which
Since the angle between any two tubes in our set S ranges between \(L^{-1}\) and 1, it follows by the pigeonhole principle that \(\#(X) {\mathop {\sim }\limits ^{\textstyle <}}(\log L) \#(X_{l_0})\) for some \(l_0 \in \mathbb N\). Denoting \(2^{l_0} L^{-1}\) by \(\theta \), and \(X_{l_0}\) by \(X'\), we have \(L^{-1} \le \theta \le 1\) and \(\#(X') {\mathop {\sim }\limits ^{\textstyle >}}N^2 L^{2 - \alpha } (\log L)^{-1}\).
There are N tubes in S. By the pigeonhole principle, one of the tubes must appear in \({\mathop {\sim }\limits ^{\textstyle >}}N^2 L^{2 - \alpha } (\log L)^{-1}/N = N L^{2 - \alpha } (\log L)^{-1}\) of the elements of \(X'\). We call this tube T, and we define
Let v be the direction of the tube T. Since the angle between v and \(v_j\) is \(\sim \theta \), it follows that \(|T \cap T_j| {\mathop {\sim }\limits ^{\textstyle <}}\theta ^{-1}\). So, the set \(\{ \widetilde{B} : (\widetilde{B}, T, T_j) \in X' \}\) has cardinality \({\mathop {\sim }\limits ^{\textstyle <}}\theta ^{-1}\), and so
To finish the proof, we need to also have an upper bound on \(\#(\mathbb H)\). We first observe that
where B is a box in \(\mathbb R^3\) of dimensions \(L \times \theta L \times \theta L\). Since B can be covered by \(\sim L/(\theta L)\) balls of radius \(\theta L\), and since \(\theta L \ge 1\), the dimensionality property (8) tells us that
Next, we use the (by now) standard fact that the tubes \(T_j\) in \(\mathbb H\) are morally disjoint (see [7, Lemma 4.9] for a very nice explanation of this idea) to see that
Therefore,
Comparing the lower and upper bounds we now have on the cardinality of \(\mathbb H\), we conclude that
Therefore,
If \(\alpha \ge 2\), then the fact that \(\theta \le 1\) tells us that
If \(1 \le \alpha < 2\), then the fact that \(\theta \ge 1/L\) tells us that
It might be interesting for the reader to observe that the sharp result that we get in the case \(1 \le \alpha < 2\) is due to the fact that we are using ‘substantial’ information about \(\theta \) (namely, \(\theta \ge 1/L\)), whereas in the case \(2 \le \alpha \le 3\) we only can use the relatively ‘unsubstantial’ information that \(\theta \le 1\).
We note that if \(\Omega \subset \mathbb R^3\) obeys (8) and \(\Omega \cap B(0,5L)\) contains at least one tube from each direction (i.e. at least one tube from each of the \(\sim L^2\) families \(\mathbb T_j\)), then Theorem 4.1 implies that \(\alpha \ge 5/2\) (cf. [15]).
5 Proof of Conjecture 1.1 in the Plane
The rest of the paper is concerned in proving that Conjecture 2.1 is true in \(\mathbb R^2\). In view of Proposition 2.1, this task will be accomplished as soon as we prove Theorem 5.1 below.
We alert the reader that the extension operator in Theorem 5.1 is the one associated with the unit circle \(\mathbb S^1 \subset \mathbb R^2\) and is given by
for \(f \in L^1(\sigma )\), where \(\sigma \) is induced Lebesgue measure on \(\mathbb S^1\). The proof for the extension operator associated with the unit parabola is similar (and a little easier).
Theorem 5.1
Suppose \(1 \le \alpha \le 2\) and \(R \ge 1\). Let \(\beta \) be a parameter satisfying \(1/2 \le \beta \le 1\), and define the exponent p by
Then to every \(\epsilon > 0\) there is a constant \(C_\epsilon \) such that
for all functions \(f \in L^p(\sigma )\) and weights H of fractal dimension \(\alpha \).
The proof of Theorem 5.1 will use ideas from [5, 12, 16], and [4]. The overarching idea, however, is the broad-narrow strategy of [1]. Implementing this strategy involves
-
proving a bilinear estimate (see (22) in Subsection 7.1 below) that will be used to control Ef on the broad set
-
proving a linear estimate (see (28) in Subsection 7.2 below) that will be used to establish (15) when the function f is supported on an arc of small size (i.e. \(\sigma \)-measure), which will provide the base of a recursive process
-
carrying out a recursive process on the size of the function’s support that will establish (15) for general f.
The main new idea in the proof of Theorem 5.1 is a localization of the weight argument that will help us in deriving the bilinear estimate (22). We use this argument to take advantage of the locally constant property of the Fourier transform, and we will end this section by formulating the intuition that lies behind it in a lemma.
Given a function \(f : \mathbb R^n \rightarrow \mathbb C\) and a number \(K > 0\), we say that f is essentially constant at scale K if there is a constant C such that
for all cubes \(Q_K \subset \mathbb R^n\) of side-length K.
Lemma 5.1
Suppose \(1 \le \alpha \le 2\), \(1/2 \le \beta \le 1\), \(R > K^2 \ge 1\), and Q is a box in \(\mathbb R^2\) of dimensionsFootnote 1\(R/K \times R\). Also, suppose that f is a non-negative function on \(\mathbb R^2\) that is essentially constant at scale K, and H is a weight on \(\mathbb R^2\) of fractal dimension \(\alpha \). Then
for some \(m \ge 0\) (in fact, \(m=\beta -(1/2)+(1-\beta )(\alpha -1)\)), where \(\widetilde{Q}\) is a box of dimensions \(2R/K \times 2R\) that has the same center as Q, and the implicit constant depends only on \(\alpha \) and \(\beta \) and the constant C from (16).
Proof
We tile \(\mathbb R^2\) by cubes \(\widetilde{B}_l\) of side-length K. If \(\widetilde{B}_l \cap Q \not = \emptyset \), we let \(c_l\) be the center of \(\widetilde{B}_l\) and write
where \(H': \mathbb R^2 \rightarrow [0,\infty )\) is given by
For \(y \in \widetilde{B}_l\), we have
where \(0 \le \theta \le 1\) is a parameter that will be determined later in the argument.
Next, we define the function \({\mathcal H} : \mathbb R^2 \rightarrow [0,1]\) by
and observe that
for all \(x_0 \in \mathbb R^2\) and \(r \ge K\). On the other hand, when \(1 \le r \le K\) we use the fact that
for all \(y \in \mathbb R^2\) to see that
(because \(K^{\alpha -2} \le r^{\alpha -2}\)). Therefore, \({\mathcal H}\) is a weight on \(\mathbb R^2\) of fractal dimension \(\alpha \) with
Going back to our integral, we now have
Bounding the integral on the right-hand side by Cauchy–Schwarz, this becomes
But \(\widetilde{Q}\) can be covered by \(\sim K\) balls of radius R/K, so
and so
We now determine \(\theta \) by solving the equation \((1+\theta )/2=\beta \), which gives \(\theta =2\beta -1\), and we arrive at
with \(m=\beta -(1/2)+(1-\beta )(\alpha -1)\).
6 Preliminaries for the Proof of Theorem 5.1
This section contains basic facts that we need to prove Theorem 5.1 that we include to make the paper as self-contained as possible.
6.1 The \(L^1\) Norm of a Rapidly Decaying Function over a Box
In the rigorous version of the localization argument that we described in the previous section, instead of integrating over a proper \(R/K \times R\) box, we will be integrating against a Schwartz function that is essentially supported on such a box. It is easy to see that (17) continues to be true in this case. Here are the details.
Lemma 6.1
Suppose \(0 < \alpha \le n\), \(R \ge K^2 \ge 1\), and \(\Psi \) is a non-negative Schwartz function on \(\mathbb R^n\). Then
for all weights H on \(\mathbb R^n\) of fractal dimension \(\alpha \).
Proof
Suppose \(R_1, \ldots , R_n > 0\) and \(\Psi \) is a non-negative Schwartz function. For \(l= 0, 1, 2, \ldots \), we let \(\chi _l\) be the characteristic function of the box in \(\mathbb R^n\) of center 0 and dimensions \(2^{l+1} R_1 \times \ldots \times 2^{l+1} R_n\), and \(B_l=B(0,2^l)\). Then
for all \(x, \nu \in \mathbb R^n\) and \(N \in \mathbb N\), so that
where \(P_l\) is the box in \(\mathbb R^n\) of center \(\nu \) and dimensions \(2^{l+1} R_1 \times \ldots \times 2^{l+1} R_n\).
In the special case \(R_1= \ldots = R_{n-1} = R/K\) and \(R_n=R\) with \(R \ge K^2 \ge 1\) (as in (17)), this gives
for all weights H on \(\mathbb R^n\) of fractal dimension \(\alpha \).
6.2 A Property of \(R/K \times \cdots \times R/K \times R\) Boxes
Suppose \(R \ge K^2 \ge 1\), Q is an \(R/K \times \cdots \times R/K \times R\) box in \(\mathbb R^n\). A box \(Q^* \subset \mathbb R^n\) of dimensions \((R/K)^{-1} \times \cdots \times (R/K)^{-1} \times R^{-1}\) and with the same axes as Q is called a dual box of Q. This subsection is about the following observation.
Lemma 6.2
Suppose \(Q^*\) is a dual box of Q whose \((R/K)^{-1} \times \cdots \times (R/K)^{-1}\)-face is tangent to the unit sphere \(\mathbb S^{n-1} \subset \mathbb R^n\) at some point e. Then \(Q^*\) lies in the \(R^{-1}\)-neighborhood of \(\mathbb S^{n-1}\).
Proof
Let \(\delta = K^{-1}\). Then \(Q^*\) has dimensions \((R \delta )^{-1} \times \ldots \times (R \delta )^{-1} \times R^{-1}\) and its \((R \delta )^{-1} \times \ldots \times (R \delta )^{-1}\)-face is tangent to \(\mathbb S^{n-1}\) at e.
Without any loss of generality, we may assume that \(e=(0, \ldots , 0, 1)\).
Suppose \(y \in Q^*\). Then
so that
so that
so that
where we have used the fact that
6.3 The Kakeya Information Underlying the Bilinear Estimate
Suppose \(\delta > 0\), \(R \ge \delta ^{-1}\), and \(J_1\) and \(J_2\) are subsets of the circular arc \(\{ e^{i\theta } : \pi /4 \le \theta \le 3\pi /4 \}\) such that \(\text{ Dist }(J_1,J_2) \ge 3\delta \).
Let \(N_1\) and \(N_2\) be the \(R^{-1}\)-neighborhoods of \(J_1\) and \(J_2\), respectively. In this subsection, we derive the following well-known bound on the Lebesgue measure of the set \((x+N_1) \cap N_2\) for \(x \in \mathbb R^2\).
Lemma 6.3
We have
for a.e. \(x \in \mathbb R^2\).
Proof
Since we are interested in the \(L^\infty \)-norm of the function
we let \(h \in L^1(\mathbb R^2)\) be a non-negative function and consider the integral
Writing
and applying the change of variables \(u=y-x\) in the inner integral, we see that
Changing into polar coordinates, this becomes
where \(\tilde{J_1}=N_1 \cap \mathbb S^1\) and \(\tilde{J_2}=N_2 \cap \mathbb S^1\).
We define
The Jacobian of this transformation is
So
But \(|\theta -\varphi | \le \pi /2\), so
and so
Thus
and (20) follows by duality.
7 Proof of Theorem 5.1
As the paragraph following the statement of Theorem 5.1 says, our proof of this theorem relies on ideas from [1, 5, 16, 12], and [4].
7.1 The Bilinear Estimate
Following [1, pp. 1281–1283], we write the ball B(0, R) as a disjoint union of two sets, one broad, the other narrow (see Subsection 7.3 below for the definition of these two sets). To estimate the \(L^p(H dx)\)-norm of Ef on the broad set, we consider a bilinear estimate.
For the rest of the paper, we will use the following notation. If \(\phi \) is a function on \(\mathbb R^2\) and \(\rho > 0\), then \(\phi _\rho \) is the function given by \(\phi _\rho (\cdot ) = \rho ^{-2} \phi (\rho ^{-1} \cdot )\).
Lemma 7.1
Suppose f is supported in an arc I and g is supported in an arc J with \(\sigma (I) \sim \sigma (J) \sim \delta \) and \(\delta \le \text{ Dist }(I,J) \le R^\epsilon \delta \). Also, suppose that
Then
Proof
Let \(\eta \) be a \(C_0^\infty \) function on \(\mathbb R^2\) satisfying \(|\widehat{\eta }| \ge 1\) on B(0, 1). Then
where \(F=\eta _{R^{-1}} *fd\sigma \) and \(G=\eta _{R^{-1}} *gd\sigma \).
Applying the Cauchy-Schwarz inequality in the convolution integral with respect to the measure \(|\eta _{R^{-1}}(\xi -\cdot )|d\sigma \), we see that
where in the second inequality we used the fact that
Therefore,
Since F is supported in the \(R^{-1}\)-neighborhood of I and G is supported in the \(R^{-1}\)-neighborhood of J, we see (via (21)) that F is supported in a ball of radius \((\delta /2) + (\delta /10) = (3\delta /5)\) and similarly for G. So \(F *G\) is supported in a ball of radius \((6\delta /5)\), say \(B(\xi _0,(6\delta /5))\). Via the locally constant property of the Fourier transform, this fact tells us that the Fourier transform of \(F *G\) is essentially constant at scale \(K=\delta ^{-1}\), and hence allows us to implement the localization of the weight argument that we described in Section 5 at the intuitive level, and which we now carry out rigorously.
Let \(\phi \) be a Schwartz function which is equal to 1 on B(0, 6/5). Then \(\phi _\delta (\xi -\xi _0)=\delta ^{-2}\) on \(B(\xi _0,\frac{6\delta }{5})\), so that
and
Since \(\big ( \phi _\delta (\cdot - \xi _0) \widehat{\big )}(x)= e^{-2\pi i x \cdot \xi _0} \widehat{\phi }(\delta x)\), it follows that
so that
Therefore,
For \(l = 0, 1, 2, \ldots \), we let \(B_l=B(y,2^l \delta ^{-1})\) and write
We now let \(0 \le \theta \le 1\) be a parameter that will be determined later and write
where we have used the fact that \(1/\delta \ge 1\), and we obtain
Also,
Applying the change of variables \(z=\delta (x-y)\) in the inner integral, we get
But
for all \(x_0 \in \mathbb R^n\) and \(r \ge 1\), so
for all \(x_0 \in \mathbb R^2\) and \(r \ge 1\).
For \(y \in \mathbb R^2\), define
In view of the above discussion, we have
for all \(x_0 \in \mathbb R^2\) and \(r \ge 1\). Thus \({\mathcal H}\) is a weight on \(\mathbb R^2\) of fractal dimension \(\alpha \) with
Going back to (24), we now have
Next, we let \(Q^*\) be the box in frequency space (where the circle is located) of dimensions \((R\delta )^{-1} \times R^{-1}\), centered at the origin, and with the \((R\delta )^{-1}\)-side (i.e. the long side) parallel to the line segment that connects the midpoint of I to that of J. We also let \(\{ Q_l \}\) be a tiling of \(\mathbb R^2\) by boxes dual to \(Q^*\) (i.e. each \(Q_l\) is an \(R\delta \times R\) box whose \(R\delta \)-side is parallel to the \((R\delta )^{-1}\)-side of \(Q^*\)) with centers \(\{ \nu _l \}\), \(\psi \) be a \(C_0^\infty \) function on \(\mathbb R^2\), and we define
In the definition of \(\psi _l\), we are assuming that the line joining the midpoint of I to that of J is horizontal (i.e. parallel to the \(\xi _1\)-axis). This assumption makes the presentation a little smoother and, of course, does not cost us any loss of generality.
We assume further that the Fourier transform of \(\psi \) is non-negative and satisfies \(\widehat{\psi } \ge 1/2\) on \([-1/2,1/2] \times [-1/2,1/2]\). Then
By the Schwartz decay of \(\widehat{\psi }\), we have \(\sum _{m \in \mathbb Z^2} \widehat{\psi }(\cdot - m)^k {\mathop {\sim }\limits ^{\textstyle <}}1\) for any \(k \in \mathbb N\). Also, \(\{ \nu _l \}\) is basically \(R\delta \mathbb Z \times R \mathbb Z\), so
and so
for all \(x \in \mathbb R^2\).
Going back to (25), we can now write
Letting \(F_l=\psi _l *F\) and \(G_l=\psi _l *G\), this becomes
By Cauchy–Schwarz,
Applying (19) from Subsection 6.1 with \(n=2\) and \(K = \delta ^{-1}\), we have
so that
Therefore,
Letting \(\beta = (1+\theta )/2\) (since \(0 \le \theta \le 1\), we have \(1/2 \le \beta \le 1\)), this becomes
We now let \(A_l\) be the support of \(F_l\), \(B_l\) be the support of \(G_l\), and define the function \(\lambda _l : \mathbb R^2 \rightarrow [0,\infty )\) by \(\lambda _l(\xi )= |(\xi -A_l) \cap B_l|\). Applying Plancherel’s theorem followed by Cauchy–Schwarz, we see that
By Young’s inequality,
so the only problem is to estimate \(\Vert \lambda _l \Vert _{L^\infty }\). We will do this by using the Kakeya bound (20) of Subsection 6.3.
Our assumptions on the arcs I and J imply that the angle between any two points in \(I \cup J\) is \({\mathop {\sim }\limits ^{\textstyle <}}R^\epsilon \delta \). Also, for each l, the function \(\psi _l\) is supported in the \((R\delta )^{-1} \times R^{-1}\) box \(Q^*\) of center (0, 0) and with the long side parallel to the line joining the midpoints of I and J. So, if \(e \in I \cup J\), then the translate \(Q^*+e\) of \(Q^*\) is contained in an \((R \delta )^{-1} \times R^{\epsilon -1}\) box with the \((R \delta )^{-1}\)-side tangent to \(\mathbb S^1\) at e. Therefore, the property of boxes of this form that was presented in Subsection 6.2 tells us that \(Q^*+e\) is contained in the \(R^{\epsilon -1}\)-neighborhood of \(\mathbb S^1\). Therefore, the sets \(A_l\) and \(B_l\) satisfy the requirements needed for us to apply (20) and conclude
Putting together what we have proved in the previous two paragraphs, we obtain
and hence
By Cauchy–Schwarz and Plancherel,
Also, by (26),
and similarly for \(\sum _{l=1}^\infty \Vert \widehat{G_l} \Vert _{L^2}^2\), so
Recalling (23), our bilinear estimate becomes
Writing
and applying (33) (see the appendix), we arrive at our desired bilinear estimate
7.2 The Linear Estimate
In this subsection, we work in \(\mathbb R^n\) with \(n \ge 2\).
Lemma 7.2
Suppose f is supported in a cap of radius \(\delta /2\). Also, suppose that
Then
Proof
Let \(\eta \) be a \(C_0^\infty \) function on \(\mathbb R^n\) satisfying \(|\widehat{\eta }| \ge 1\) on B(0, 1), and \(F=\eta _{R^{-1}} *f d\sigma \). Then
Also, let \(\psi \) be a \(C_0^\infty \) function on \(\mathbb R^n\), and \(\{ B_l \}\) be a finitely overlapping cover of \(\mathbb R^n\) by balls dual to \(B(0,\delta )\) (i.e. \(\delta ^{-1}\)-balls) with centers \(\{ \nu _l \}\), and set
We assume further that \(\widehat{\psi }\) is non-negative and \(\ge 1/2\) on the unit ball. Then
if \(|\delta (x-\tau _l)| \le 1\), i.e. if \(x \in B_l\). Thus
Since \(1/n \le \beta \le 2/n\), we can apply Hölder’s inequality with the dual exponents \(1/(1-\beta )\) and \(1/\beta \) to get
Since \(\Vert H \Vert _{L^\infty } \le 1\), we have
and hence (by the proof of (19))
Also, by Hausdorff–Young,
Therefore,
Since (27) tells us \(1/R \le \delta /10\), it follows that F is supported in a ball of radius \((\delta /2)+(\delta /10)= (3/5) \delta \), say \(B(\xi _0,3\delta /5)\). Moreover, since \(\psi _l\) is supported in \(B(0,\delta )\), it follows by Hölder’s inequality and Plancherel’s theorem that
Thus
But we know from (23) (whose proof shows that it is true in \(\mathbb R^n\) for all \(n \ge 2\)) that \(\Vert F \Vert _{L^2} {\mathop {\sim }\limits ^{\textstyle <}}\sqrt{R} \, \Vert f \Vert _{L^2(\sigma )}\), so
Writing
and using (32) (see the appendix), we now see that
which proves (28).
7.3 The Recursive Process
We let \(0< \epsilon < 10^{-2}\) and \(R \ge 1\) be two numbers satisfying \(R \ge (1000)^{1/(1-4\epsilon )}\). We also let \(\delta \) be as in Lemma 7.1 (so that \(\delta \) obeys (21)). We’re going to prove our estimate by implementing a recursive process over \(\delta \).
Base of the recursion: Here \(\delta = R^{-1/2}\). Plugging this value of \(\delta \) into (28) in dimension \(n=2\), we get
The recursive step: We state this in the following lemma.
Lemma 7.3
Suppose that the estimate
holds for every function \(f \in L^1(\sigma )\) that is supported in an arc of \(\sigma \)-measure \(\le \delta \), and \(\delta \) obeys (21). Then the estimate
holds for every function \(g \in L^1(\sigma )\) that is supported in an arc of \(\sigma \)-measure \(\le R^\epsilon \delta \), where
Proof
Suppose \(\delta \) satisfies the condition (21):
and (29) is true whenever \(f \in L^1(\sigma )\), f is supported on an arc \(I_\delta \subset \mathbb S^1\), and \(\sigma (I_\delta ) \le \delta \). We need to show that (30) is true whenever \(g \in L^1(\sigma )\), g is supported on an arc \(I_{R^\epsilon \delta } \subset \mathbb S^1\), and \(\sigma (I_{R^\epsilon \delta }) \le R^\epsilon \delta \), where
We let \(K = R^\epsilon \) and cover the support of g by K arcs \(\tau \) each of measure \(\delta \). We then write \(g= \sum _{\tau } f_\tau \) with each function \(f_\tau \) supported in the arc \(\tau \).
Following [1] and [7], for \(x \in \mathbb R^2\), we define the significant set of x by
Then
so that
The narrow set \({\mathcal N}\) and the broad set \({\mathcal B}\) are now defined as
We will estimate \(\int _{\mathcal N} |Eg(x)|^p H(x) dx\) by induction and \(\int _{\mathcal B} |Eg(x)|^p H(x) dx\) by using the bilinear estimate.
To every \(x \in {\mathcal B}\) there are two caps \(\tau _x, \tau _x' \in S(x)\) so that \(\text{ Dist }(\tau _x,\tau _x') \ge \delta \). Writing
we see that
Using the bilinear estimate (22), it follows that
Therefore,
Combining the narrow and broad estimates, we arrive at (30).
The recursion: Starting with the base of the induction, where \(\delta = R^{-1/2}\) and \(C=C_L\), and applying Lemma 7.3k times, we arrive at an estimate that holds for every function \(f \in L^1(\sigma )\) that is supported on an arc of \(\sigma \)-measure \(\le \delta _k = R^{k \epsilon }\delta = R^{k \epsilon }/\sqrt{R}\), with constant
At the step before the last, \(k=(1/(2\epsilon )) -2\) and \(\delta _k= R^{[(1/(2\epsilon )) -2] \epsilon }/\sqrt{R}=R^{-2\epsilon }\), which is a valid value of \(\delta \) (i.e. \(\delta _k=R^{-2\epsilon }\) obeys (28), because \(10 R^\epsilon \le 1/R^{-2\epsilon } \le R^{1-2\epsilon }/10\)). Applying Lemma 7.3 one last time, we get the estimate
for every function \(f \in L^1(\sigma )\) that is supported on an arc of \(\sigma \)-measure \(\le R^{-\epsilon }\), where the constant C satisfies
Since the circle \(\mathbb S^1\) can be covered by \(\sim R^\epsilon \) such arcs, (15) follows and Theorem 5.1 is proved.
Notes
Boxes of such dimensions are a common feature in this context; see [4, Subsection 3.2] and Subsection 6.2 below.
References
Bourgain, J., Guth, L.: Bounds on oscillatory integral operators based on multilinear estimates. Geom. Funct. Anal. 21, 1239–1295 (2011)
Demeter, C.: On the restriction theorem for paraboloid in \(\mathbb{R} ^4\). Colloq. Math. 156, 301–311 (2019)
Du, X., Guth, L., Ou, Y., Wang, H., Wilson, B., Zhang, R.: Weighted restriction estimates and application to Falconer distance set problem. Am. J. Math. 143, 175–211 (2021)
Du, X., Zhang, R.: Sharp \(L^2\) estimate of Schrödinger maximal function in higher dimensions. Ann. Math. 189, 837–861 (2019)
Erdoǧan, M.B.: A note on the Fourier transform of fractal measures. Math. Res. Lett. 11, 299–313 (2004)
Falconer, K.J.: On the Hausdorff dimensions of distance sets. Mathematika 32, 206–212 (1985)
Guth, L.: A restriction estimate using polynomial partitioning. J. Am. Math. Soc. 29, 371–413 (2016)
Guth, L.: Restriction estimates using polynomial partitioning \(II\). Acta Math. 221, 81–142 (2018)
Guth, L., Zahl, J.: Polynomial Wolff axioms and Kakeya-type estimates in \(\mathbb{R} ^4\). Proc. Lond. Math. Soc. 117, 192–220 (2018)
Katz, N., Rogers, K.: On the polynomial Wolff axioms. Geom. Funct. Anal. 28(2018), 1706–1716 (2019). (465–506)
Sjögren, P., Sjölin, P.: Convergence properties for the time-dependent Schrödinger equation. Ann. Acad. Sci. Fenn. 14, 13–25 (1989)
Shayya, B.: Weighted restriction estimates using polynomial partitioning. Proc. Lond. Math. Soc. (3) 115, 545–598 (2017)
Shayya, B.: Fourier restriction in low fractal dimensions. Proc. Edinb. Math. Soc. 64, 373–407 (2021)
Wongkew, R.: Volumes of tubular neighbourhoods of real algebraic varieties. Pac. J. Math. 159, 177–184 (1993)
Wolff, T.: An improved bound for Kakeya type maximal functions. Rev. Mat. Iberoamericana 11, 651–674 (1995)
Wolff, T.: Decay of circular means of Fourier transforms of measures. Int. Math. Res. Not. 10, 547–567 (1999)
Zahl, J.: A discretized Severi-type theorem with applications to harmonic analysis. Geom. Funct. Anal. 28, 1131–1181 (2018)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Stephane Jaffard.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix: Calculation Giving the Right Exponent for the Restriction estimate
Appendix: Calculation Giving the Right Exponent for the Restriction estimate
Suppose \(0 < \delta \le 1\), \(1 \le \alpha \le n\), \(1/n \le \beta \le 2/n\), \(\sigma \) is induced Lebesgue measure on the unit sphere \(\mathbb S^{n-1} \subset \mathbb R^n\), and \(f, g \in L^1(\sigma )\) are functions satisfying \(\sigma (\text{ supp } \, f), \sigma (\text{ supp } \, g) \le \delta ^{n-1}\). We are looking for an exponent \(p \ge 2\) so that
and
We have
and
so
so \((n-1)(p-2) = (n-\alpha )((2/n)-\beta )\), and so
Therefore, (32) holds with the above value of p. Using (32), we now have
which is the inequality in (33).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Shayya, B. A Family of Fractal Fourier Restriction Estimates with Implications on the Kakeya Problem. J Fourier Anal Appl 30, 8 (2024). https://doi.org/10.1007/s00041-023-10065-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-023-10065-9