Abstract
We establish Fourier extension estimates for compact subsets of the hyperbolic hyperboloid in three dimensions via polynomial partitioning.
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1 Introduction
In this article, we establish Fourier extension estimates for compact subsets of the hyperbolic, or one-sheeted, hyperboloid in three dimensions. This surface may be defined as the set of points \((\tau ,\xi ) \in \mathbb {R}\times \mathbb {R}^2\) satisfying the relation \(\tau ^2 = 1 + \xi _1^2-\xi _2^2\). Setting \(\phi (\xi ) := \sqrt{1+\xi _1^2-\xi _2^2}\) and \(\Omega := \{\xi \in \mathbb {R}^2 : 1+\xi _1^2-\xi _2^2 \ge 0\}\), we may restrict our attention to the graph
We aim to adapt the polynomial partitioning method of Guth [5] to obtain extension estimates for a bounded subset of \(\Sigma \) near (1, 0), which we denote by \(\Sigma _1\). Use of the parabolic scalings \(P_r(\tau ,\xi ) := (r^{-2}\tau ,r^{-1}\xi )\) in Guth’s argument presents an immediate obstacle here, as hyperboloids are not invariant under such transformations. To overcome this minor issue, we will simultaneously prove extension estimates for all parabolic rescalings of \(\Sigma _1\) with constants uniform in the scaling parameter. Toward that end, let \(U := \{\xi : |\xi | \le \delta _0/10\}\), where \(\delta _0 > 0\) is a small constant to be chosen later, and for each \(r \in (0,1]\), let \(\phi _r(\xi ) := r^{-2}(\phi (r\xi )-1)\) and
Each \(\Sigma _r\) is the image of \(\Sigma _1 \cap \{(\tau ,\xi ) : \xi \in rU\}\) under the parabolic scaling \(P_r\), and the ‘\(-1\)’ in \(\phi _r\) just makes \(\Sigma _r\) converge to the hyperbolic paraboloid \(\Sigma _0 := \{(\frac{1}{2}(\xi _1^2-\xi _2^2),\xi ) : \xi \in U\}\) as \(r\rightarrow 0\). We associate to \(\Sigma _r\) the extension operator
Theorem 1.1
If \(q > 13/4\) and \(p > (q/2)'\), then \(\mathcal {E}_r : L^p(U) \rightarrow L^q(\mathbb {R}^3)\) with operator norm bounded uniformly in r.
Remark 1.2
The bilinear and bilinear-to-linear theories for \(\mathcal {E}_1\) appear in a separate article [1] of Stovall, Oliveira e Silva, and the author. Using the bilinear machinery and Theorem 1.1, boundedness of \(\mathcal {E}_r\) on the parabolic scaling line \(p = (q/2)'\) (for \(q > 13/4\)) can also be proved. See [1, Remark 5.2], as well as [8, 9], and [6] for arguments of this type.
Theorem 1.1 can be compared to several recent developments in the restriction/extension theory for hyperbolic surfaces in three dimensions. Cho and Lee [3] generalized Guth’s argument in [5] to the hyperbolic paraboloid, proving strong type (p, q) extension estimates in the range \(q > 13/4\), \(p \ge q\). Later work of Kim [6] and Stovall [8] brought those estimates to the scaling line \(p =(q/2)'\). (Letting \(r \rightarrow 0\) and applying Fatou’s lemma, Theorem 1.1 reproves the off-scaling extension estimates for the hyperbolic paraboloid.) Recently, Buschenhenke–Müller–Vargas [2] and Guo–Oh [4] independently obtained extension estimates for all smooth compact surfaces in \(\mathbb {R}^3\) with negative Gaussian curvature using polynomial partitioning. In particular, Theorem 1.1 is now (essentially) a special case of their results, which were announced after the completion of the arXiv preprint version of the present article.
The rest of the article is organized as follows: In Sect. 2, we adapt the notion of ‘broad points’ in [5] to the hyperbolic hyperboloid, motivating our definition through the geometry of the surface. In Sect. 3, we use Kim’s argument in [6] to reduce Theorem 1.1 to Theorem 2.1, an estimate on the contribution to \(\mathcal {E}_r\) from broad points. Finally, in Sect. 4, the heart of the article, we prove Theorem 2.1 using polynomial partitioning as in [5].
Notation and Terminology As is standard, we write \(A \lesssim B\) or \(A = O(B)\) if there exists a constant \(C > 0\) such that \(A \le CB\). Generally, an implicit constant is not allowed to depend on any parameters present in the article. In particular, constants never depend on the parabolic scaling parameter r. There are exceptions: In Sect. 4, constants may depend on the exponent \(\varepsilon \) from Theorems 2.1 and 4.1. To highlight dependence on a parameter s, we will sometimes write \(\lesssim _s\) in place of \(\lesssim \). Likewise, we write \( c \ll 1\) to mean that c is sufficiently small, and we use subscripts to indicate dependence on parameters. A number \(\delta \) is ‘dyadic’ if \(\delta = 2^j\) for some \(j \in \mathbb {Z}\), and an interval I is ‘dyadic’ if \(I = [k2^j, (k+1)2^j)\) for some \(j,k \in \mathbb {Z}\). If u, v are geometric objects that form an angle, such as two lines or a vector and a plane, then \(\angle (u,v)\) denotes the measure of their (smallest) angle. Finally, ‘hyperboloid’ always means the hyperbolic (one-sheeted) hyperboloid.
2 Broad Points and the Geometry of the Hyperboloid
In this section, we adapt the notion of ‘broad points’ to the hyperboloid. Informally, given a function \(f \in L^1(U)\), a point \((t,x) \in \mathbb {R}\times \mathbb {R}^2\) is ‘broad’ for \(\mathcal {E}_rf\) if there exist small, well-separated squares \(\tau _1,\tau _2 \subseteq U\) such that \(f\chi _{\tau _1}\) and \(f\chi _{\tau _2}\) contribute significantly to \(\mathcal {E}_rf(t,x)\); otherwise (t, x) is ‘narrow’. To estimate \(\mathcal {E}_rf\), it suffices to bound the contributions from broad and narrow points separately. The narrow contribution will be handled by a parabolic rescaling argument, since (morally) its Fourier transform is supported in a small rectangular cap in \(\Sigma _r\). The broad contribution will be handled by polynomial partitioning, using, in particular, some techniques from bilinear restriction theory. In the latter argument, the precise separation condition imposed on the squares \(\tau _1,\tau _2\) will be crucial for ensuring that their lifts to \(\Sigma _r\) are appropriately transverse. Our choice of this condition will be motivated by the geometry of the hyperboloid, which we now describe.
First, the basic symmetries of the hyperboloid are the Lorentz transformations, linear maps on \(\mathbb {R}\times \mathbb {R}^2\) that preserve the quadratic form \((\tau ,\xi ) \mapsto \tau ^2 - \xi _1^2 +\xi _2^2\). Concretely, the spatial rotations
boosts
and dilations
will be of particular use to us. We define a measure \(d\mu \) on \(\Sigma \) by setting
for g continuous and compactly supported. This measure is Lorentz invariant in the following sense: If L is a Lorentz transformation and \({\text {supp}}g \subseteq \Sigma \) and \(L^{-1}({\text {supp}}g)\subseteq \Sigma \), then
We also record the following notation for later use. Given a Lorentz transformation L and \(\xi \in \Omega \), let
where \(\pi (\tau ,\xi ) := \xi \) is the projection to the spatial coordinates. If \({L}(\phi (\xi ),\xi ) \in \Sigma \) (equivalently, if \(e_1\cdot L(\phi (\xi ),\xi ) \ge 0\)), then \(\overline{ML}(\xi ) = \overline{M}(\overline{L}(\xi ))\) for any other Lorentz transformation M. In particular, if \(V \subseteq \Omega \) and \(L(\phi (\xi ),\xi ) \in \Sigma \) for \(\xi \in V\), then \(\overline{L}\) is invertible on V with \(\overline{L}^{-1}(\zeta ) = \overline{L^{-1}}(\zeta )\) for \(\zeta \in \overline{L}(V)\).
Second, the (hyperbolic) hyperboloid is doubly ruled. The aforementioned separation condition will be adapted to this structure: Informally, two small squares \(\tau _1,\tau _2 \subseteq U\) will be ‘separated’ if their lifts to the hyperboloid do not intersect a common line contained in the surface. While the precise version of this condition will be stated in Sect. 4, we record a few preparatory details here. The Lorentz norm of \((\tau ,\xi ) \in \mathbb {R}\times \mathbb {R}^2\) is defined as
It is clearly Lorentz invariant, and if \((\tau ,\xi ),(\tau ',\xi ') \in \Sigma \), then \(\llbracket (\tau ,\xi )-(\tau ',\xi ')\rrbracket = 0\) if and only if \((\tau ,\xi )\) and \((\tau ',\xi ')\) belong to a common line contained \(\Sigma \). The latter property can be checked by using the formulae
which parametrize the lines \(\ell _{(\tau ,\xi )}^\pm \subset \Sigma \) that intersect at \((\tau ,\xi ) \in \Sigma \). We also define the Lorentz separation of \(\xi ,\zeta \in \Omega \) as the quantity
which can be viewed as the ‘distance’ between \((\phi (\xi ),\xi )\) and \((\phi (\zeta ),\zeta )\) modulo the rulings of \(\Sigma \). Given this definition, a more accurate rendering of our separation requirement would be that \(\mathrm{dist}_{\mathrm{L}}(\xi ,\zeta ) \gtrsim 1\) for all \(\xi \in \tau _1\) and \(\zeta \in \tau _2\). Near the end of this section, we will prove Lemma 2.2, which relates \(\mathrm{dist}_{\mathrm{L}}\) to some other geometric quantities.
Having described the geometry of the hyperboloid, we turn to defining broad points. Our first step is to divide each surface \(\Sigma _r\) into caps that lie above special sets which we call tiles. Consider the map \(\Phi : \mathbb {R}^2 \rightarrow \mathbb {R}^2\) given by
and, for each \(r \in (0,1]\), let \(\Phi _r(\xi ) := r^{-1}\Phi (r\xi )\). Recall the constant \(\delta _0\) used to define U, and assume henceforth that \(\delta _0\) is dyadic. Given two dyadic numbers \( \delta ,\delta ' \in (0,\delta _0]\), a \((\delta ,\delta ',r)\)-tile is any nonempty set of the form
where \(I_\delta \) and \(I_{\delta '}\) are dyadic intervals contained in \([-\delta _0,\delta _0)\) of length \(\delta \) and \(\delta '\), respectively. We denote the set of \((\delta , \delta ',r)\)-tiles by \(\mathcal {T}_{\delta ,\delta ',r}\). Observe that \(\Phi \) is a diffeomorphism near the origin. (Indeed, \(\Phi \) can be viewed as a perturbation of the map \(\xi \mapsto \frac{1}{2}(\xi _1+\xi _2, \xi _2-\xi _1)\) for \(\xi \) small.) Taking \(\delta _0\) sufficiently small, it is straightforward to check that \(\Vert \Phi _r^{-1}\Vert _{C^1(U)} \lesssim 1\) uniformly in r, and consequently that \(U \subseteq \Phi _r([-\delta _0,\delta _0)^2)\) for every r. We also note that for fixed \(\delta ,\delta ',r\), the \((\delta ,\delta ',r)\)-tiles are pairwise disjoint and satisfy
Let us briefly mention the geometry underlying these definitions. The map \(\Phi \) was created with the following property in mind: If \(\ell \subset \mathbb {R}^2\) is a vertical or horizontal line that intersects \(\Phi _r^{-1}(U)\), then \(\Phi _r(\ell )\) is a line that lifts to a line contained in \(\Sigma _r\). Thus, each tile lifts to a quadrilateral (in fact, nearly rectangular) cap bounded by four lines. We can think of the collection \(\{\mathcal {T}_{\delta ,\delta ',r}\}_{\delta ,\delta '}\) as a dyadic grid adapted to \(\Sigma _r\). A more precise geometric description of \(\Phi \) will appear in Lemma 2.3 at the end of this section.
Now, let \(K \ge \delta _0^{-1}\) be a large dyadic constant. As suggested above, we will analyze contributions to \(\mathcal {E}_r\) from square-like sets \(\tau \). The \((K^{-1},K^{-1},r)\)-tiles will function as these basic pieces. However, controlling contributions from longer rectangle-like sets will also be essential. (As we will see, a collection of non-separated squares \(\tau \) must cluster around a line.) For each dyadic number \(\delta \in [K^{-1},\delta _0]\), let
and also set
Elements of \(\mathcal {R}_{\delta ,r}\) resemble rectangles of dimensions \(K^{-1} \times \delta \) and slope approximately 1 or \(-1\). We are now ready to define broad points. Given \(f \in L^1(U)\) and \(\alpha \in (0,1]\), we say that \((t,x) \in \mathbb {R}\times \mathbb {R}^2\) is \(\alpha \)-broad for \(\mathcal {E}_rf\) if
where \(f_\rho := f\chi _\rho \). The \(\alpha \)-broad part of \(\mathcal {E}_rf\) is defined as
In the next section, we will reduce Theorem 1.1 to the following estimate on the broad part:
Theorem 2.1
For every \(0 < \varepsilon \ll 1\), there exists a constant \(C_\varepsilon \), depending only on \(\varepsilon \), such that if \(K = 2^{\lceil \varepsilon ^{-10}\rceil }\), then
for all \( r \in (0,1]\), \(R \ge 1\), and balls \(B_R\) of radius R.
To conclude this section, we present two geometric lemmas. We will need the following notation: For \(\xi \in \Omega \), let \(\ell _\xi ^\pm \) denote the lines in \(\mathbb {R}^2\) parametrized by
Geometrically, \(\ell _\xi ^\pm \) are the projections to the spatial coordinates of the lines \(\ell _{(\phi (\xi ),\xi )}^\pm \) defined in (2.6).
Lemma 2.2
For all \(\xi ,\zeta \in U\), we have
-
(a)
\({\text {dist}}(\xi , \ell _\zeta ^+ \cup \ell _\zeta ^-) \lesssim \mathrm{dist}_{\mathrm{L}}(\xi ,\zeta ) \lesssim |\xi -\zeta |\);
-
(b)
\(\mathrm{dist}_{\mathrm{L}}(\xi ,\zeta )^2 \sim |\langle (\nabla ^2\phi (\xi ))^{-1}(\nabla \phi (\xi )-\nabla \phi (\zeta )),\nabla \phi (\xi )-\nabla \phi (\zeta )\rangle |\).
Proof
(a) Let \(\xi '\) be the intersection of \(\ell _\xi ^-\) and \(\ell _\zeta ^+\). An easy calculation shows that \(\angle (\ell _\eta ^+,\ell _{\eta '}^-) \gtrsim 1\) for all \(\eta ,\eta ' \in U\). (In fact, the lines are nearly orthogonal.) In particular, the law of sines implies that \(\xi ' \in CU\) for some constant C. Let \(L := B_\nu R_\omega \), as defined in (2.1) and (2.2), with
Then \(L(\phi (\zeta ),\zeta ) = (1,0)\). Let \(\eta := \overline{L}(\xi )\) and \(\eta ' := \overline{L}(\xi ')\), using the notation from (2.5). Since \(\nu \) and \(\omega _2\) are very small, \(\overline{L}\) is essentially a perturbation of the identity. It is easy to check that \(L(\phi (\xi ),\xi ) \in \Sigma \) for all \(\xi \in CU\), provided \(\delta _0\) is sufficiently small, and thus \(\overline{L}\) is invertible on CU. Additionally, we have the bound \(\Vert \overline{L}^{-1}\Vert _{C^1(\overline{L}(CU))} \lesssim 1\). Combining these facts, we see that
Since L preserves the hyperboloid and is linear, it must permute the lines contained in the surface. Therefore, since L is close to the identity,
which implies that \(\{\eta '\} = \mathbb {R}(1,1) \cap \ell _\eta ^-\). Then, since \(\eta \in \ell _\eta ^-\) and \(\angle (\mathbb {R}(1,1),\ell _\eta ^-) \gtrsim 1\), it follows that \(|\eta -\eta '| \lesssim {\text {dist}}(\eta ,\mathbb {R}(1,1))\). Thus, by (2.8), we have \({\text {dist}}(\xi ,\ell _\zeta ^+) \lesssim {\text {dist}}(\eta ,\mathbb {R}(1,1)) \le |\eta _1-\eta _2|\). A similar argument shows that \({\text {dist}}(\xi ,\ell _\zeta ^-) \lesssim |\eta _1+\eta _2|\). Hence,
where the last step used the Lorentz invariance of the Lorentz norm. The second inequality in (a) can be proved in a similar (but easier) fashion. (It also follows from part (b), using the Cauchy–Schwarz inequality and bounds on the derivatives of \(\phi \).)
(b) A straightforward computation shows that the right-hand side of (b) is equal to
The expression inside absolute value signs is equal to
which, by the relations \(\phi (\xi )^2 = 1 + \xi _1^2 - \xi _2^2\) and \(\phi (\zeta )^2 = 1 +\zeta _1^2 - \zeta _2^2\), simplifies to
Thus, by a bit more algebra, the right-hand side of (b) factors as
We also compute that
and (b) follows. \(\square \)
Let us briefly interpret Lemma 2.2. Part (a) says that points with small Lorentz separation lie near a common line, while points with large Lorentz separation are genuinely separated. Part (b) relates Lorentz distance to a measure of ‘transversality’ that naturally arises in bilinear restriction theory (see [7, Theorem 1.1]). Crucially, whenever \(\xi \) and \(\zeta \) belong to separated squares (as discussed above), the right-hand side of (b) will be bounded below.
Lemma 2.3
If \(\xi \in U\) and \(\zeta = \Phi ^{-1}(\xi )\), then \(\ell _\xi ^+ \cap 2U = \Phi (\{\zeta _1\} \times \mathbb {R}) \cap 2U\) and \(\ell _\xi ^- \cap 2U = \Phi (\mathbb {R}\times \{\zeta _2\}) \cap 2U\).
Proof
We will only prove the first equality; the second follows in a similar manner. The proof rests on two claims.
Claim 1. If \(|\xi |,|\xi '| \le 1/2\) and \(\xi ' \in \ell _\xi ^+\), then \(\ell _{\xi '}^+ = \ell _\xi ^+\). Consider the lines \(\ell _\xi ^+\), \(\ell _{\xi '}^+\), and \(\ell _{\xi '}^-\). Each one contains \(\xi '\) and lifts to a line contained in \(\Sigma \). By elementary geometry, no three lines in the hyperboloid intersect at a common point. Thus, two of \(\ell _\xi ^+\), \(\ell _{\xi '}^+\), and \(\ell _{\xi '}^-\) must be identical. Since \(\ell _{\xi '}^+ \ne \ell _{\xi '}^-\) and \(\ell _{\xi }^+ \ne \ell _{\xi '}^-\), as is easy to check, we conclude that \(\ell _{\xi '}^+ = \ell _\xi ^+\).
Claim 2. For every \(\xi \in \mathbb {R}^2\), we have \(\{\Phi (\xi )\} = \ell _{(\xi _1,0)}^+ \cap \ell _{(\xi _2,0)}^-\). This relation can be checked directly, using (2.7). It is helpful to reparametrize (2.7) so that the second coordinates of \(\ell _{(\xi _1,0)}^+(t)\) and \(\ell _{(\xi _2,0)}^-(t)\) are identically t and \(-t\), respectively.
Now, fix \(\xi \in U\) and let \(\zeta := \Phi ^{-1}(\xi )\). Let \(\xi ' \in \ell _\xi ^+ \cap 2U\) and \(\zeta ' := \Phi ^{-1}(\xi ')\). Claim 2 implies that \(\xi \in \ell _{(\zeta _1,0)}^+\) and \(\xi ' \in \ell _{(\zeta _1',0)}^+\). Hence, by claim 1, we have \(\ell _{(\zeta _1,0)}^+ = \ell _\xi ^+ = \ell _{\xi '}^+ = \ell _{(\zeta _1',0)}^+\), and it follows that \(\zeta _1 = \zeta _1'\). Since \(\xi '\) was arbitrary, we conclude that \(\ell _\xi ^+ \cap 2U \subseteq \Phi (\{\zeta _1\} \times \mathbb {R}) \cap U\). The other direction is similar: Let \(\xi ' \in \Phi (\{\zeta _1\} \times \mathbb {R}) \cap 2U\), so that \(\xi ' = \Phi (\zeta _1,t)\) with \((\zeta _1,t) \in \Phi ^{-1}(2U) \subseteq \Omega \). Claim 2 implies that \(\xi ,\xi ' \in \ell _{(\zeta _1,0)}^+\). Hence, \(\xi ' \in \ell _\xi ^+\) by claim 1, and it follows that \(\Phi (\{\zeta _1\} \times \mathbb {R}) \cap 2U \subseteq \ell _\xi ^+ \cap 2U\). \(\square \)
3 Reduction to Theorem 2.1
In this section, we adapt the argument of Kim in [6] to show that Theorem 2.1 implies Theorem 1.1. The following parabolic rescaling lemma is the main tool required for this reduction.
Lemma 3.1
Let \(r \in (0,1]\) be dyadic and let \(\theta \in [0,1]\). If \(\Vert \mathcal {E}_s g\Vert _{L^q(B_{R/2})} \le M\Vert g\Vert _2^{1-\theta }\Vert g\Vert _\infty ^\theta \) for all \(s \in (0,1]\), balls \(B_{R/2}\) of radius R/2, and \(g \in L^\infty (U)\), then there exists an absolute constant C such that \(\Vert \mathcal {E}_r h\Vert _{L^q(B_R)} \le CM(\delta \delta ')^{\frac{1+\theta }{2} -\frac{2}{q}}\Vert h\Vert _2^{1-\theta }\Vert h\Vert _\infty ^\theta \) for all bounded functions h supported in \(\rho \in \mathcal {T}_{\delta ,\delta ',r}\), provided \(\delta ,\delta '\) are sufficiently small.
Proof
Fix \(h \in L^\infty (U)\) supported in \(\rho \in \mathcal {T}_{\delta ,\delta ',r}\). There exists \(\rho _1 \in \mathcal {T}_{r\delta ,r\delta ',1}\) such that \(r\rho \subseteq \rho _1\). By parabolic rescaling, we have
where \(h_{\rho _1} := h(r^{-1}\cdot )\) is supported in \(\rho _1\). We assume without loss of generality that \(\delta \le \delta '\) and fix \(\eta \in \rho _1\). We claim that \(\rho _1\) lies in the intersection of an \(O(r\delta )\)-neighborhood of \(\ell _\eta ^+\) and an \(O(r\delta ')\)-neighborhood of \(\ell _\eta ^-\). Indeed, let \(\eta ' \in \rho _1\) and set \(\zeta = \Phi ^{-1}(\eta )\) and \(\zeta ' = \Phi ^{-1}(\eta ')\). By the definition of \((r\delta ,r\delta ',1)\)-tile, we have
Thus, by Lemma 2.3 and the boundedness of \(\Vert \nabla \Phi \Vert \) near the origin, it follows that
proving the claim.
Now, let \(L := (D_\lambda B_\nu R_\omega )^{-1}\) with
using the notation from (2.1)–(2.3). As in the proof of Lemma 2.2, the map \(\overline{B_\nu R_\omega }\) sends \(\eta \) to the origin and \(\ell _\eta ^\pm \) to \(\ell _0^\pm = \mathbb {R}(1,\pm 1)\) and satisfies \(\Vert \overline{B_\nu R_\omega }\Vert _{C^1(U)} \lesssim 1\). Thus, by the claim, \(\overline{B_\nu R_\omega }(\rho _1)\) lies in an \(O(r\delta ) \times O(r\delta ')\) rectangle with slope 1 centered at the origin, and consequently \(\overline{D_\lambda }(\overline{B_\nu R_\omega }(\rho _1))\) is contained in sU for some \(s \lesssim r\sqrt{\delta \delta '}\). It is easy to check that \(B_\nu R_\omega (\phi (\xi ),\xi ) \in \Sigma \) for all \(\xi \in U\), and thus by the discussion following (2.5),
We claim that
Indeed, given a set \(V \subseteq \Omega \), let \(V^\pm := \{(\pm \phi (\xi ),\xi ) : \xi \in V\}\). Then
It is easy to check that \(e_1 \cdot L^{-1}(\phi (\zeta ),\zeta ) > 0\) for every \(\zeta \in U\). Thus, since \(-\rho _1 \subseteq U\) and \(\phi \ge 0\), we have \((\phi (\xi ),\xi ) \notin -L^{-1}((-\rho _1)^+)\) for every \(\xi \). Hence,
proving the claim.
Now, define \(F : \Sigma \rightarrow {\mathbb {C}}\) by \(F(\tau ,\xi ) := h_{\rho _1}(\xi )\phi (\xi )\) and assume that \(\delta ,\delta '\) are small enough that \(s \le 1\). Then, using (3.2) and (3.1), it is straightforward to check that \(L^{-1}({\text {supp}}F) \subseteq \Sigma \). Thus,
where \(d\mu \) is the Lorentz-invariant measure given by (2.4). Hence, for \(H(\xi ) := h_{\rho _1}(\overline{L}(\xi ))\frac{\phi (\overline{L}(\xi ))}{\phi (\xi )}\), we have
Noting that \(|\det L| = 1\), we obtain the relation
and parabolic rescaling then gives
where \(H(s\cdot )\) is supported in U by (3.2) and (3.1).
We claim that \(P_{s^{-1}}L^* P_r(B_R)\) is covered by a bounded number of balls of radius R/2. Assuming the claim is true, the hypothesis of the lemma implies that
To prove the claim, we may assume by translation invariance that \(B_R\) is centered at the origin. Let Q(a, b, c) denote any rectangular box centered at zero with sides of length O(a), O(b), O(c) parallel to (1, 0, 0), (0, 1, 1), \((0,1,-1)\), respectively. Thus, for example, \(B_R \subseteq Q(R,R,R)\) and
We have \(L^* = D_\lambda ^{-*}B_\nu ^{-*}R_\omega ^{-*}\), where \(S^{-*} := (S^{-1})^*\). Since \(R_\omega ^{-*}\) and \(B_\nu ^{-*}\) have bounded norm, we can ignore their contribution. Thus, from the definition of \(D_\lambda \), we have
The definition of s then implies that
which proves claim.
Finally, to finish the proof, we need to undo the changes of variable we have used. Using (3.2) and (3.1), we have \(L(\phi (\xi ),\xi ) \in \Sigma \) for all \(\xi \in \overline{L}^{-1}(\rho _1)\). Thus, \(\overline{L}\) is invertible on \({\text {supp}}H\) with \(\overline{L}^{-1}(\zeta ) = \overline{L^{-1}}(\zeta )\) for \(\zeta \in \overline{L}({\text {supp}}H) \subseteq U\). Moreover, \(\overline{L^{-1}} = \overline{D_\lambda }\circ \overline{B_\nu } \circ \overline{R_\omega }\) on U, so a straightforward calculation shows that \(|\det \nabla \overline{L}^{-1}| \lesssim 1\) on U. Using these observations, we find that
Plugging these bounds into (3.3) completes the proof. \(\square \)
Proposition 3.2
Assume that Theorem 2.1 holds. Then for every \(\theta \in (3/13, 1]\) and \(0 < \varepsilon \ll _\theta 1\), there exists a constant \(C_{\varepsilon ,\theta }\), depending only on \(\varepsilon \) and \(\theta \), such that
for all \(r \in (0,1]\), \(R \ge 1\), and balls \(B_R\) of radius R.
Proof
We will induct on R. The base case, that \(R \sim 1\), holds trivially. We assume as our induction hypothesis that the proposition holds with R/2 in place of R. Additionally, we may assume that \(2C_\varepsilon ^{13/4} \le C_{\varepsilon ,\theta }^{13/4}\), where \(C_\varepsilon \) is the constant from Theorem 2.1. The definition of \(K^{-\varepsilon }\)-broad implies that
for every \((t,x) \in \mathbb {R}\times \mathbb {R}^2\). It follows that
To bound the first term, we use Theorem 2.1 and Hölder’s inequality to get
To bound the second term, we will use Lemma 3.1. We may assume that r is dyadic by parabolic rescaling, and the other hypothesis of the lemma holds by our inductive assumption. Additionally, by Hölder’s inequality, we may assume that \(\theta \) is close to 3/13; in particular, that \(\theta \le 5/13\). Then
where the last step used the inclusion \(\ell ^2 \hookrightarrow \ell ^{\frac{13}{4}(1-\theta )}\) and that \(\mathcal {R}_{\delta ,r}\) covers U with overlap of multiplicity 2. Since \(\theta > 3/13\), the sum over \(\delta \) is bounded and the power of K is negative for \(\varepsilon \) sufficiently small. Thus, since \(K \rightarrow \infty \) as \(\varepsilon \rightarrow 0\) by the hypothesis of Theorem 2.1, the expression in square brackets is at most 1/2 for \(\varepsilon \) sufficiently small, and the induction closes. \(\square \)
Assuming Theorem 2.1 holds, Proposition 3.2 implies the restricted strong type bounds
for all \(p > 13/5\), measurable sets \(E \subseteq U\), and \(|f_E| \lesssim \chi _E\). Then, by real interpolation with the trivial \(L^1 \rightarrow L^\infty \) estimate, we obtain the strong type bounds
for all \(q > 13/4\) and \(p > (q/2)'\). Tao’s epsilon removal lemma, in the form of Theorem 5.3 in [6], consequently gives the global strong type bounds
for the same range of p, q, completing the proof of Theorem 1.1.
4 Proof of Theorem 2.1
We are left to prove Theorem 2.1; this will occupy the rest of the article. To enable an inductive argument, we will actually need to prove a slightly stronger theorem, as in [5]. In Sect. 2, we defined broad points by considering the contribution to \(\mathcal {E}_rf\) from each \(f_\rho \), where \(f_\rho := f\chi _\rho \) and \(\rho \in \mathcal {R}_r\). Soon we will work with wave packets of the form \(\mathcal {E}_rf_{\rho ,T}\), where \(f_{\rho ,T}\) is supported not in \(\rho \) but in a slight enlargement of it. Thus, we need a more general definition of broad points in which the functions \(f_\rho \) may have larger, overlapping supports. Given \(\rho = \Phi _r(I_\delta \times I_{\delta '}) \cap U \in \mathcal {T}_{\delta ,\delta ',r}\) and \(m \ge 1\), we define
where \(m(I_\delta \times I_{\delta '})\) is the m-fold dilate of the rectangle \(I_\delta \times I_{\delta '}\) with respect to its center. Let
elements of \(\mathcal {S}_r\) are essentially \(K^{-1} \times K^{-1}\) squares. Now, given \(f \in L^1(U)\), suppose that \(f = \sum _{\tau \in \mathcal {S}_r} f_\tau \), with each \(f_\tau \) supported in \(m \tau \), for some \(m \ge 1\). In our modified definition, \((t,x) \in \mathbb {R}\times \mathbb {R}^2\) is \(\alpha \)-broad for \(\mathcal {E}_r f\) if
We define the \(\alpha \)-broad part of \(\mathcal {E}_rf\), still denoted by \({\text {Br}}_{\alpha }\mathcal {E}_rf\), as in Sect. 2. These definitions depend on the particular decomposition \(f = \sum _\tau f_\tau \).
Theorem 4.1
For every \(0 < \varepsilon \ll 1\), there exists a constant \(C_\varepsilon '\), depending only on \(\varepsilon \), such that if \(K = 2^{\lceil \varepsilon ^{-10}\rceil }\), then the following holds: If \(f = \sum _{\tau \in \mathcal {S}_r} f_\tau \) with each \(f_\tau \) supported in \(m \tau \), for some \(m \ge 1\), and if additionally f satisfies
for all \(\xi \in U\) and \(\tau \in \mathcal {S}_r\), then
for all \(r \in (0,1]\), \(R \gg _\varepsilon 1\), balls \(B_R\) of radius R, and \(\alpha \in [K^{-\varepsilon }, 1]\).
A couple of remarks may be helpful. Firstly, the dyadic structure of our tiles, as defined in Sect. 2, implies that if \(\tau \in \mathcal {S}_r\) and \(\rho \in \mathcal {R}_r\), then either \(\tau \cap \rho = \emptyset \) or \(\tau \subseteq \rho \). More generally, if \(\rho _1 \in \mathcal {T}_{\delta _1,\delta _1',r}\) and \(\rho _2 \in \mathcal {T}_{\delta _2,\delta _2',r}\), then either \(\rho _1 \cap \rho _2 = \emptyset \) or \(\rho _1 \cap \rho _2 \in \mathcal {T}_{\min _i \delta _i, \min _i\delta _i',r}\). Secondly, Theorem 4.1 is indeed stronger than Theorem 2.1. We can derive the latter from the former as follows: If \(R \sim _\varepsilon 1\), then the estimate in Theorem 2.1 is trivial, so we may assume that \(R \gg _\varepsilon 1\). By scaling, we also may assume that \(\Vert f\Vert _\infty = 1\). Thus, the condition (4.1) holds automatically. We now apply Theorem 4.1 with \(\alpha = K^{-\varepsilon }\) and \(m=1\) to get
and then raising both sides to the power 4/13 finishes the proof.
4.1 Preliminaries
Before beginning the proof of Theorem 4.1, we lay some groundwork. For the remainder of the article, \(\varepsilon \), m, r, R, \(B_R\), and \(\alpha \) are fixed. Implicit constants will be allowed to depend on \(\varepsilon \). The propositions and lemma we record in this subsection are by now quite standard.
We begin with the wave packet decomposition. Let \(\Theta \) be a collection of discs \(\theta \) of radius \(R^{-1/2}\) which cover U with bounded overlap. We denote by \(c_\theta \) the center of \(\theta \), and we let \(v_\theta \) be the unit normal vector to \(\Sigma _r\) at \((\phi _r(c_\theta ),c_\theta )\). We may assume that \(c_\theta \in U\) for every \(\theta \). Let \(\delta := \varepsilon ^2\), and for each \(\theta \), let \(\mathbb {T}(\theta )\) be a collection of tubes parallel to \(v_\theta \) with radius \(R^{\delta +1/2}\) and length R and which cover \(B_R\) with bounded overlap. If \(T \in \mathbb {T}(\theta )\), then \(v(T) := v_\theta \) denotes the direction of T. Finally, we set \(\mathbb {T}:= \bigcup _{\theta \in \Theta }\mathbb {T}(\theta )\). The following wave packet decomposition resembles Proposition 2.6 in [5]:
Proposition 4.2
For each \(T \in \mathbb {T}\), there exists a function \(f_T \in L^2(\mathbb {R}^2)\) such that:
-
(i)
If \(T \in \mathbb {T}(\theta )\), then \(f_T\) is supported in \(3\theta \);
-
(ii)
If \((t,x) \in B_R \setminus T\), then \(|\mathcal {E}_rf_T(t,x)| \le R^{-1000}\Vert f\Vert _2\);
-
(iii)
\(|\mathcal {E}_rf(t,x) - \sum _{T \in \mathbb {T}}\mathcal {E}_rf_T(t,x)| \le R^{-1000}\Vert f\Vert _2\) for every \((t,x) \in B_R\);
-
(iv)
If \(T_1,T_2 \in \mathbb {T}(\theta )\) and \(T_1 \cap T_2 = \emptyset \), then \(|\int f_{T_1}\overline{f_{T_2}}| \le R^{-1000}\Vert f\Vert _{L^2(\theta )}^2\);
-
(v)
\(\sum _{T \in \mathbb {T}(\theta )}\Vert f_T\Vert _2^2 \lesssim \Vert f\Vert _{L^2(\theta )}^2\).
Proof
Adapting Guth’s argument in [5] is straightforward. The fact that the derivatives of \(\phi _r\) are bounded in r (i.e. \(\sup _{\xi \in U}|\nabla ^k\phi _r(\xi )| \lesssim _k 1\)) ensures that all constants arising in the argument can be made uniform in r. We note, in particular, that the crucial derivative estimates appearing in line (17) of [5] hold uniformly in r when adapted to our setting. \(\square \)
Next, we record an orthogonality lemma from [5]. The special case \(N =1\) will be of particular use.
Lemma 4.3
Let \(\mathbb {T}_1,\ldots ,\mathbb {T}_N\) be subsets of \(\mathbb {T}\). Suppose that each tube in \(\mathbb {T}\) belongs to at most M of the \(\mathbb {T}_i\), and for each \(\tau \in \mathcal {S}_r\), let
where the functions \(f_{\tau ,T}\) come from applying Proposition 4.2 to \(f_\tau \). Then
for every \(\theta \in \Theta \), and
Finally, we turn to polynomial partitioning. Let P be a polynomial on \(\mathbb {R}^d\). We denote the zero set of P by Z(P) and say that \(z \in Z(P)\) is nonsingular if \(\nabla P(z) \ne 0\). If z is nonsingular, then Z(P) is a smooth hypersurface near z. If every point of Z(P) is nonsingular, then we say that P is nonsingular.
Proposition 4.4
(Guth [5]) Given \(g \in L^1(\mathbb {R}^d)\) and \(D \ge 1\), there exists a polynomial P of degree at most D such that P is a product of nonsingular polynomials and each connected component O of \(\mathbb {R}^d \setminus Z(P)\) satisfies
We note that a product of nonsingular polynomials may have singular points. However, by a perturbation argument using Sard’s theorem, one can ensure that nonsingular points are dense in the zero set of the partitioning polynomial.
4.2 Main Proof
We are now ready to prove Theorem 4.1 in earnest. We will induct on R and \(\sum _{\tau \in \mathcal {S}_r}\Vert f_\tau \Vert _2^2\). The base cases, that \(R \sim 1\) or \(\sum _{\tau }\Vert f_\tau \Vert _2^2 \le R^{-1000}\), are easy to check, and our induction hypotheses are that Theorem 4.1 holds with: (i) R/2 in place of R, or (ii) g in place of f whenever \(\sum _{\tau }\Vert g_\tau \Vert _2^2 \le \frac{1}{2}\sum _{\tau }\Vert f_\tau \Vert _2^2\). Throughout the proof, we will assume that \(\varepsilon \) is sufficiently small and that R is sufficiently large in relation to \(\varepsilon \).
We begin by setting \(D := R^{\varepsilon ^4}\) and applying Proposition 4.4 to the function \(|{\text {Br}}_\alpha \mathcal {E}_r f|^{13/4}\chi _{B_R}\) to produce a polynomial P of degree at most D such that
where the ‘cells’ \(O_i\) are connected, pairwise disjoint, and satisfy
In particular, the number of cells is \(\#I \sim D^3\). We define the ‘wall’ W as the \(R^{1/2+\delta }\)-neighborhood of Z(P), and we set \(O_i' := O_i \setminus W\). Thus,
We now argue by cases, according to which term on the right-hand side of (4.3) dominates.
4.3 Cellular Case
Suppose that the total contribution from the shrunken cells \(O_i'\) dominates. In this case, we have
Using (4.2), we then see that the contribution from any single \(O_i'\) is controlled by the average of all such contributions. Thus, ‘most’ cells should contribute close to the average, and it is straightforward to show that there exists \(J \subseteq I\) such that \(\#J \sim D^3\) and
for all \(i \in J\). The lower bound on \(\#J\) will be the basis for a pigeonholing argument shortly.
First, some definitions are needed. For each \(i \in I\) and \(\tau \in \mathcal {S}_r\), we set
and
where the functions \(f_{\tau ,T}\) come from applying Proposition 4.2 to \(f_\tau \). We also set
Since \(f_\tau \) is supported in \(m \tau \), property (i) in Proposition 4.2 implies that \(f_{\tau ,i}\) is supported in an \(O(R^{-1/2})\)-neighborhood of \(m\tau \). Let \(\overline{f}_i := \chi _U f_i\) and \(\overline{f}_{i,\tau } := \chi _U f_{i,\tau }\). If R is sufficiently large, then \({\text {supp}}\overline{f}_{\tau ,i} \subseteq 2m\tau \). Consequently, \(\overline{f}_i\) has a well defined broad part with respect to these larger squares. Soon we will apply our induction hypothesis to \(\overline{f}_i\) (for some special i) with m replaced by 2m.
Lemma 4.5
If \((t,x) \in O_i'\) and \(\alpha \le 1/2\), then
Proof
First, we may assume that
otherwise, the required inequality is trivial. Since \((t,x) \in O_i'\), properties (iii) and (ii) in Proposition 4.2 imply that
for each \(\tau \). Summing over \(\tau \), we get
Now it suffices to show that if (t, x) is \(\alpha \)-broad for f, then (t, x) is \(2\alpha \)-broad for \(\overline{f}_i\). Assume the former and fix \(\rho \in \mathcal {R}_r\). Using Proposition 4.2 again, we have
Using (4.5), (4.6), and the fact that \(\alpha \ge K^{-\varepsilon }\), the right-hand side is at most \(2\alpha |\mathcal {E}_rf_i(t,x)| = 2\alpha |\mathcal {E}_r\overline{f}_i(t,x)|\) for R sufficiently large. \(\square \)
If \(\alpha > 1/2\), then the estimate in Theorem 4.1 holds trivially, since the power of R can then be made at least 1000 by taking \(\varepsilon \) sufficiently small. Thus, we may assume that \(\alpha \le 1/2\). Applying Lemma 4.5 to (4.4) and recalling that \(D = R^{\varepsilon ^4}\), we get
for every \(i \in J\). We will now pick \(i_0 \in J\) so that \(\sum _{\tau \in \mathcal {S}_r}\Vert \overline{f}_{\tau ,i_0}\Vert _2^2\) is small, which will allow us to apply our induction hypothesis to \(\overline{f}_{i_0}\). Because Z(P) is the zero set of a polynomial of degree at most D, any line is either contained in Z(P) or intersects Z(P) at most D times. Thus, each tube in \(\mathbb {T}\) belongs to at most \(D+1\) of the sets \(\mathbb {T}_i\). Now, applying Lemma 4.3 and the bound \(\#J \gtrsim D^3\), we must have
for some constant C. Consequently, there exists \(i_0 \in J\) such that
for R sufficiently large. We can apply Theorem 4.1 to \(\overline{f}_{i_0}\) with 2m in place of m, provided (4.1) holds. Since (4.1) holds for f, Lemma 4.3 gives
Thus, after multiplying \(\overline{f}_{i_0}\) by a constant, we can apply Theorem 4.1 to (4.7) with \(i = i_0\) to get
for some C. If the big O term dominates, then the desired estimate follows easily. Assuming it does not, then by (4.8) and the definition of D, we have altogether
and the induction closes if \(\varepsilon \) is sufficiently small and R sufficiently large.
4.4 Algebraic Case
Next, suppose that the contribution from W dominates in (4.3), so that
Following Guth [5], we distinguish between tubes that intersect W transversely and those essentially tangent to W. Let \(\mathcal {B}\) be a collection of balls B of radius \(R^{1-\delta }\) that cover \(B_R\) with bounded overlap.
Definition 4.6
Fix \(B \in \mathcal {B}\). Let \(\mathbb {T}_B^\flat \) be the set of tubes T satisfying \(T \cap W \cap B \ne \emptyset \) and \(\angle (v(T),T_z Z(P)) \le R^{2\delta - 1/2}\) for every nonsingular point \(z \in Z(P) \cap 2B \cap 10T\). Let \(\mathbb {T}_B^\sharp \) be the set of tubes T satisfying \(T \cap W \cap B \ne \emptyset \) and \(T \notin \mathbb {T}_B^\flat \).
Observe that if T intersects \(W \cap B\), then T belongs to exactly one of \(\mathbb {T}_B^\flat \) and \(\mathbb {T}_B^\sharp \). (The definition of \(\mathbb {T}_B^\flat \) would be vacuous if \(Z(P) \cap 2B \cap 10T\) contained only singular points; however, as noted above, we can arrange for nonsingular points to be dense in Z(P).) Thus, on \(W \cap B\), each \(\mathcal {E}_rf_\tau \) is well approximated by the sum of the ‘tangent’ and ‘transverse’ wave packets, \(\{\mathcal {E}_rf_{\tau ,T}\}_{T \in \mathbb {T}_B^\flat }\) and \(\{\mathcal {E}_rf_{\tau ,B}\}_{T \in \mathbb {T}_B^\sharp }\), respectively. Roughly speaking, our desired bound for \(\Vert {\text {Br}}_\alpha \mathcal {E}_rf\Vert _{L^{13/4}(B \cap W)}\) will soon be reduced to a broad part estimate on the transverse contribution and a bilinear estimate on the tangent contribution. The following geometric lemma, due to Guth [5], will be critical for establishing those bounds:
Lemma 4.7
(a) Each \(T \in \mathbb {T}\) belongs to \(\mathbb {T}_B^\sharp \) for at most \(D^{O(1)}\) balls \(B \in \mathcal {B}\). (b) For each \(B \in \mathcal {B}\), the number of discs \(\theta \in \Theta \) such that \(\mathbb {T}_B^\flat \cap \mathbb {T}(\theta ) \ne \emptyset \) is at most \(R^{O(\delta ) + 1/2}\).
To carry out the bilinear argument, we need to define the separation condition mentioned in Sect. 2. Recall how we defined the Lorentz separation \(\mathrm{dist}_{\mathrm{L}}(\xi ,\zeta )\) of \(\xi ,\zeta \in \Omega \). We say that two squares \(\tau _1,\tau _2 \in \mathcal {S}_r\) are separated if
for all \(\xi \in 2m\tau _1\) and \(\zeta \in 2m\tau _2\), where \(C_0 \ge 1\) is a constant to be chosen later. Part (a) of Lemma 2.2 implies that points having small Lorentz separation must lie near a common line. The next lemma extends this property to collections of non-separated squares.
Lemma 4.8
Let \(\mathcal {I}\subseteq \mathcal {S}_r\) be a collection of pairwise non-separated squares. Then there exist \(\sigma _1,\sigma _2,\sigma _3,\sigma _4 \in \mathcal {T}_{\delta ,\delta _0,r} \cup \mathcal {T}_{\delta _0,\delta ,r}\), with \(K^{-1} \le \delta \lesssim mK^{-1}\), such that \(\tau \subseteq \bigcup _{i=1}^4\sigma _i\) for every \(\tau \in \mathcal {I}\).
Proof
For \(\xi \in U\), let \(\overline{\xi } := \Phi ^{-1}(r\xi )\) and also set
Fix \(\tau _1,\tau _2 \in \mathcal {I}\). By part (a) of Lemma 2.2 and the definition of (non-)separated squares, there exist \(\xi ^*\in 2m\tau _1\) and \(\zeta ^* \in 2m\tau _2\) such that
Let \(\eta \) be a point in \(\ell _{r\zeta ^*}^+ \cup \ell _{r\zeta ^*}^-\) closest to \(r\xi ^*\). By elementary geometry, \(\eta \) lies in 2U. Thus, from the bound \(\Vert \Phi ^{-1}\Vert _{C^1(2U)} \lesssim 1\) and Lemma 2.3, we have
Since \({\text {diam}}\Phi ^{-1}(r\cdot 2m\tau ) \lesssim rmK^{-1}\) for each \(\tau \), it follows that
for all \(\xi ,\zeta \in {I}\) and some \(A \lesssim rmK^{-1}\). Fix \(\zeta \in {I}\) and set
so that \(\overline{I} \subseteq S \cup T\). Additionally, define
We consider three exhaustive cases:
-
(i)
If \(\overline{I} \cap (S \setminus 3T) \ne \emptyset \), then (4.10) implies that \(\overline{I} \cap (T \setminus 3S) = \emptyset \), and consequently \(\overline{I} \subseteq 3S\).
-
(ii)
If \(\overline{I} \cap (T \setminus 3S) \ne \emptyset \), then (4.10) implies that \(\overline{I} \cap (S \setminus 3T) = \emptyset \), and consequently \(\overline{I} \subseteq 3T\).
-
(iii)
Otherwise, \(\overline{I} \subseteq 3S \cap 3T\).
Thus, by symmetry, we may assume that \(\Phi _r^{-1}({I}) = r^{-1}\overline{I} \subseteq (r^{-1}\cdot 3S) \cap [-\delta _0,\delta _0)^2\). The interval
is covered by two dyadic intervals \(I_1,I_2 \subseteq [-\delta _0,\delta _0)\) of length \(\delta \lesssim r^{-1}A \lesssim mK^{-1}\). Thus, if we set
then \(I \subseteq \bigcup _{i=1}^4 \sigma _i\) and the proof is complete. \(\square \)
As mentioned above, estimating \(\Vert {\text {Br}}_\alpha \mathcal {E}_rf\Vert _{L^{13/4}(B \cap W)}\) can be reduced to estimating certain contributions from transverse and tangent wave packets. The next lemma carries out this reduction. First, some notation is needed. For \(\tau \in \mathcal {S}_r\) and \(B \in \mathcal {B}\), we set
We also let
Given \(\mathcal {I}\subseteq \mathcal {S}_r\), we set
We note that \(f_{\mathcal {I},B}^\sharp \) (analogously \(f_{\mathcal {I},B}^\flat \)) has the natural decomposition \(f_{\mathcal {I},B}^\sharp = \sum _{\tau \in \mathcal {S}_r} f_{\tau ,\mathcal {I},B}^\sharp \), where
Let \(\overline{f}_{\mathcal {I},B}^\sharp := \chi _U f_{\mathcal {I},B}^\sharp \) and \(\overline{f}_{\tau ,B}^\sharp := \chi _Uf_{\tau ,B}^\sharp \). Then \({\text {supp}}\overline{f}_{\tau ,B}^\sharp \subseteq 2m\tau \), and thus \(\overline{f}_{\mathcal {I},B}^\sharp \) has a well defined broad part. Finally, we define
Lemma 4.9
If \((t,x) \in B \cap W\) and \(\alpha m\) is sufficiently small, then
Proof
We may assume that (t, x) is \(\alpha \)-broad for \(\mathcal {E}_rf\) and that
Let
If \(\mathcal {I}\) contains a pair of separated squares, then the bound \(|{\text {Br}}_\alpha \mathcal {E}_r f(t,x)| \le K^{100}{\text {Bil}}(\mathcal {E}_r f_B^\flat )(t,x)\) follows immediately. Thus, we may assume that \(\mathcal {I}\) contains no pair of separated squares. By Lemma 4.8, there exist \(\sigma _1,\sigma _2,\sigma _3,\sigma _4 \in \mathcal {T}_{\delta ,\delta _0,r} \cup \mathcal {T}_{\delta _0,\delta ,r}\), with \(K^{-1} \le \delta \lesssim mK^{-1}\), such that \(\tau \subseteq \bigcup _{i=1}^4\sigma _i\) for every \(\tau \in \mathcal {I}\). Let
Then
Since \(\delta \lesssim mK^{-1}\), each \(\sigma _i\) is a union of at most Cm elements of \(\mathcal {R}_{\delta _0,r}\) where C is a constant. Thus, since (t, x) is \(\alpha \)-broad for \(\mathcal {E}_rf\) and \(\alpha m\) is sufficiently small, we have
and consequently,
Since \((t,x) \in B \cap W\), properties (iii) and (ii) in Proposition 4.2 imply that
for every \(\tau \in \mathcal {S}_r\). Summing over \(\tau \in \mathcal {J}^c\), we get
Since \(\mathcal {J}^c \subseteq \mathcal {I}^c\), we have
Hence,
Using (4.11), we see that
provided \(\varepsilon \) is sufficiently small and R sufficiently large. To finish the proof, we will show that (t, x) is \(10\alpha \)-broad for \(\mathcal {E}_r \overline{f}_{\mathcal {J}^c,B}^\sharp \). It suffices to show that
for every \(\rho \in \mathcal {R}_r\). Fixing \(\rho \in \mathcal {R}_r\), we have
As above, \(\mathcal {J}^c \subseteq \mathcal {I}^c\) implies that
It is straightforward to check that \(\sigma _i \cap \rho \in \mathcal {R}_r\) for each \(i = 1,\ldots ,4\). Thus, since (t, x) is \(\alpha \)-broad for \(\mathcal {E}_rf\), we have
Using the preceding three estimates and (4.11), we arrive at (4.12). \(\square \)
If \(\alpha m \gtrsim 1\) so that Lemma 4.9 does not apply, then the estimate in Theorem 4.1 holds trivially, since the power of R can then be made at least 1000 by taking \(\varepsilon \) sufficiently small. Thus, we may assume that \(\alpha m \ll 1\). We now apply Lemma 4.9 to (4.9) to get
note that the implicit constant is allowed to depend on K, a function of \(\varepsilon \). If the last term dominates in (4.13), then the estimate in Theorem 4.1 holds trivially.
4.4.1 Transverse Subcase
Suppose that the first term dominates in (4.13), so that
Each ball \(B \in \mathcal {B}\) has radius \(R^{1-\delta }\), so by induction on R, we can apply Theorem 4.1 to each summand in (4.14), whenever (4.1) holds. Since (4.1) holds for f, Lemma 4.3 gives
Thus, after multiplying by a constant, Theorem 4.1 implies that
By Lemma 4.7, each \(T \in \mathbb {T}\) belongs to at most \(D^{O(1)}\) sets \(\mathbb {T}_B^\sharp \). Therefore, by Lemma 4.3, we have
Since \(\delta = \varepsilon ^2\), \(D = R^{\varepsilon ^4}\), and the number of subsets \(\mathcal {I}\subseteq \mathcal {S}_r\) depends only on K, we have altogether
for some C (depending on \(\varepsilon \)). The power of the first R is negative for \(\varepsilon \) sufficiently small, and then the induction closes for R sufficiently large.
4.4.2 Tangent Subcase
In the remaining case, the second term in (4.13) dominates, whence
We will bound the right-hand side directly (i.e. without induction) using basically standard bilinear restriction techniques and Lemma 4.7. Since \(\#\mathcal {B}= R^{O(\delta )} \le R^\varepsilon \), it will suffice to prove the following:
Proposition 4.10
For every \(B \in \mathcal {B}\), we have
We will need a preliminary lemma. Fix \(B \in \mathcal {B}\) and let \(\mathcal {Q}\) be a collection of cubes Q of side length \(R^{1/2}\) that cover \(B \cap W\) with bounded overlap. For each \(Q \in \mathcal {Q}\), let
Henceforth, we will write ‘negligible’ in place of any quantity of size \(O(R^{-990}\sum _{\tau \in \mathcal {S}_r}\Vert f_\tau \Vert _2)\). In particular, if \((t,x) \in Q\), then
It will suffice to bound \(\Vert {\text {Bil}}(\mathcal {E}_rf_B^\flat )\Vert _{L^{13/4}(Q)}\) for each \(Q \in \mathcal {Q}\). Informally, the tubes in \(\mathbb {T}_{B,Q}^\flat \) are tangent to W at Q and are thus coplanar. Dually, the wave packets \(\{\mathcal {E}_rf_{\tau ,T}\}_{T \in \mathbb {T}_{B,Q}^\flat }\) have Fourier support near a curve formed by the intersection of \(\Sigma _r\) and a plane. Thus, estimating \(\Vert {\text {Bil}}(\mathcal {E}_rf_B^\flat )\Vert _{L^{13/4}(Q)}\) is essentially a two-dimensional bilinear restriction problem, making the \(L^4\) argument a natural approach (as done in [5], of course).
Lemma 4.11
For all \(Q \in \mathcal {Q}\) and separated \(\tau _1, \tau _2 \in \mathcal {S}_r\), we have
Proof
Let \(\psi _Q\) be a smooth function satisfying \(\chi _Q \le \psi _Q \le \chi _{2Q}\) and
By (4.15) and Plancherel’s theorem, we have
where \(d\sigma _{\tau ,T}\) is the measure on \(\Sigma _r\) defined by
Fix \(T_1,\overline{T}_1,T_2,\overline{T}_2 \in \mathbb {T}_{B,Q}^\flat \) and let \(\xi ,\overline{\xi },\zeta ,\overline{\zeta }\) denote the centers of \(\theta (T_1),\theta (\overline{T}_1),\theta (T_2),\theta (\overline{T}_2)\), respectively. The rapid decay of \(\hat{\psi }_Q\) and the fact that \({\text {supp}}\chi _Uf_{\tau ,T} \subseteq \frac{3}{2}m\tau \) for every \(\tau \in \mathcal {S}_r\) and \(T \in \mathbb {T}\) imply that the contribution of \(T_1,\overline{T}_1,T_2,\overline{T}_2\) to (4.16) is negligible unless
and \(\xi ,\overline{\xi } \in 2m\tau _1\) and \(\zeta ,\overline{\zeta } \in 2m\tau _2\). We need to estimate the number of non-negligible terms in (4.16) involving given tubes \(T_1,T_2\).
Toward that end, we adapt some techniques of Cho–Lee [3] and Lee [7]. Assuming \(T_1,\overline{T}_1,T_2,\overline{T}_2\) contribute non-negligibly, then
We define a function \(\Psi : U \rightarrow \mathbb {R}\) by
and denote by \(Z := \Psi ^{-1}(0)\) its zero set. We claim that \(|\nabla \Psi | \gtrsim 1\) on \(2m\tau _1\). Indeed, if \(\eta \in 2m\tau _1\), then by the Cauchy–Schwarz inequality, boundedness of \(\Vert (\nabla ^2 \phi )^{-1}\Vert \) on U, part (b) of Lemma 2.2, and finally the separation of \(\tau _1\) and \(\tau _2\), we have
whence
if \(C_0\) is sufficiently large. By the claim, Z is a smooth curve near \(\xi \), and (4.18) and a Taylor approximation argument imply that
for R sufficiently large. As mentioned above, tubes in \(\mathbb {T}_{B,Q}^\flat \) are nearly coplanar. Inspecting the definition, it is straightforward to check that \(\angle (v(T),T_z Z(P)) \le R^{2\delta -1/2}\) for all \(T \in \mathbb {T}_{B,Q}^\flat \) and some (nonsingular) \(z \in 2R^\delta Q \cap Z(P)\). Thus, dually, there exists a plane \(\Pi \) through the origin such that \({\text {dist}}((-1,\nabla \phi _r(\eta )),\Pi ) \lesssim R^{2\delta -1/2}\) for each \(\eta \in \{\xi ,\overline{\xi },\zeta \}\). Consequently, there exists a line whose \(O(R^{2\delta -1/2})\)-neighborhood contains \(\nabla \phi _r(\xi )\), \(\nabla \phi _r(\overline{\xi })\), and \(\nabla \phi _r(\zeta )\). Since \(|\nabla \phi _r(\xi )-\nabla \phi _r(\zeta )| \gtrsim 1\) due to the separation of \(\tau _1\) and \(\tau _2\), it follows that \(\nabla \phi _r(\overline{\xi })\) lies in an \(O(R^{2\delta -1/2})\)-neighborhood of the line \(\ell \) containing \(\nabla \phi _r(\xi )\) and \(\nabla \phi _r(\zeta )\). We consider now the smooth curve \(\tilde{\ell } := (\nabla \phi _r)^{-1}(\ell \cap 3U)\), noting that \(\nabla \phi \) (and thus \(\nabla \phi _r\)) is invertible near the origin since \(\det \nabla ^2\phi (0) \ne 0\). This curve contains \(\xi \) by construction, and the boundedness of \(\Vert (\nabla ^2\phi )^{-1}\Vert \) implies that
Crucially, \(\tilde{\ell }\) and Z intersect transversely at \(\xi \). Indeed, parametrizing \(\tilde{\ell }\) by
the tangent line to \(\tilde{\ell }\) at \(\xi \) is parallel to
and the normal line to Z at \(\xi \) is parallel to \(\nabla \Psi (\xi ) = \nabla \phi _r(\xi ) - \nabla \phi _r(\zeta )\). Thus, the bound
which follows from part (b) of Lemma 2.2 and the separation of \(\tau _1\) and \(\tau _2\), implies the claimed transverse intersection. Consequently, by (4.19) and (4.20), we have \(|\xi - \overline{\xi }| \lesssim R^{2\delta -1/2}\). A similar argument shows that \(|\zeta - \overline{\zeta }| \lesssim R^{2\delta -1/2}\). Since \(\#(\mathbb {T}_{B,Q}^\flat \cap \mathbb {T}(\theta )) \lesssim 1\) for every \(\theta \in \Theta \), it follows that for each \(T_1,T_2 \in \mathbb {T}_{B,Q}^\flat \), there are \(O(R^{8\delta })\) pairs \(\overline{T}_1,\overline{T}_2 \in \mathbb {T}_{B,Q}^\flat \) such that \(T_1,\overline{T_1},T_2,\overline{T}_2\) contribute non-negligibly to (4.16).
Hence, by the Cauchy–Schwarz inequality (a few times) and Young’s inequality, (4.16) is at most
To estimate the convolution, we use Plancherel’s theorem and the familiar wave packet approximation
we will give a rigorous argument in Lemma 4.12, appearing at the end of the article. If \(T_1,T_2 \in \mathbb {T}\) are such that \(3\theta (T_i) \cap \tau _i \ne \emptyset \), then the separation of \(\tau _1\) and \(\tau _2\) implies that the directions \(v(T_1)\) and \(v(T_2)\) are transverse and consequently that \(|T_1 \cap T_2| \lesssim R^{3\delta + 3/2}\). Hence, by Plancherel’s theorem and (4.22), we (essentially) have
Plugging this estimate into (4.21), we obtain the lemma. \(\square \)
Given Lemma 4.11, the rest of the proof of Proposition 4.10 is identical to the corresponding part of [5]. For the convenience of the reader, we repeat the details here. We set
(cf. (4.22)). Let \(\tau _1, \tau _2 \in \mathcal {S}_r\) be separated squares. Lemma 4.11 implies that
Summing over \(Q \in \mathcal {Q}\) and exploiting the separation of \(\tau _1\) and \(\tau _2\) (as above) leads to the bound
By properties (i) and (iv) of Proposition 4.2, the functions \(f_{\tau ,T}\) are nearly orthogonal and we have
for every \(\tau \). Thus, altogether,
and consequently by Hölder’s inequality,
The well-known estimate
(which is a consequence of Plancherel’s theorem for the spatial Fourier transform), together with Hölder’s inequality, implies that
Hence, by interpolation,
for \(p \in [2,4]\). Now, on one hand, \(\Vert f_{\tau ,B}^\flat \Vert _2 \lesssim \Vert f_\tau \Vert _2\) by Lemma 4.3. On the other hand, Lemma 4.7 gives a different bound: There are at most \(R^{O(\delta )+1/2}\) discs \(\theta \in \Theta \) such that \(\mathbb {T}_B^\flat \cap \mathbb {T}(\theta ) \ne \emptyset \). By property (i) of Proposition 4.2, each \(f_{\tau ,B}^\flat \) is therefore supported in \(R^{O(\delta )+1/2}\) discs \(\theta \), on each of which we have the bound
by Lemma 4.3 and (4.1). Thus, \(\Vert f_{\tau ,B}^\flat \Vert _2 \lesssim R^{O(\delta )-1/4}\). Combining these two estimates gives \(\Vert f_{\tau ,B}^\flat \Vert _2 \lesssim \Vert f_\tau \Vert _2^{{3/p}}R^{O(\delta )-\frac{1}{4}(1-\frac{3}{p})}\) for \(p \ge 3\). Plugging this bound into (4.23) yields
and then taking \(p = 13/4\) completes the proof of Proposition 4.10.
To conclude the article, we rigorously prove the convolution estimate used in the proof of Lemma 4.11. This standard argument is sketched in [5]; we fill in the details here.
Lemma 4.12
If \(\tau _1,\tau _2 \in \mathcal {S}_r\) are separated squares and \(T_1,T_2 \in \mathbb {T}\) are such that \(3\theta (T_i) \cap \tau _i \ne \emptyset \), then
where \(d\sigma _{\tau _i,T_i}\) is given by (4.17).
Proof
Let \(\theta _i := \theta (T_i)\) and \(c_i := c_{\theta _i}\). Since \(3\theta _i \cap \tau _i \ne \emptyset \), we have \(c_i \in 2m\tau _i\), and consequently, \(|\nabla \phi _r(c_1) - \nabla \phi _r(c_2)| \gtrsim 1\) by the separation of \(\tau _1\) and \(\tau _2\). Indeed, by the Cauchy–Schwarz inequality, boundedness of \(\Vert (\nabla ^2\phi )^{-1}\Vert \) on U, and part (b) of Lemma 2.2,
It follows (from the law of sines, say) that the unit normal vectors \(n_1 := v_{\theta _1}\) and \(n_2 := v_{\theta _2}\) satisfy \(\angle (n_1,n_2) \gtrsim 1\). Using this angle bound, we will foliate \(3\theta _1\) by lines whose lifts to \(\Sigma _r\) are transverse to the tangent plane \(T_{(\phi _r(c_2),c_2)}\Sigma _r\) above \(c_2\). Define the direction set
where \(c > 0\). If c is sufficiently small relative to \(\angle (n_1,n_2)\), then V is nonempty. Choose \(\omega \in V\), let \(\overline{\omega } := (\omega _2,\omega _3)\), and let S be the rotation of \(\mathbb {R}^2\) satisfying \(S(0,1) = \overline{\omega }/|\overline{\omega }|\) (note that \(\overline{\omega } \ne 0\)). Define the lines \(\overline{\gamma }_s\) by
and note that \({\text {supp}}d\sigma _{\tau _1,T_1} \subseteq 3\theta _1 \subseteq \{\overline{\gamma }_s(t) : (s,t) \in I^2\}\), where \(I := [-3R^{-1/2},3R^{-1/2}]\). The lift of \(\overline{\gamma }_s\) to \(\Sigma _r\) is given by
for s, t small. For almost every s, the function \(t \mapsto f_{\tau _1,T_1}(\overline{\gamma }_s(t))\) is measurable and
defines a measure \(d\nu _s\) on \(\gamma _s\). Using (4.17), an easy calculation shows that \(d\sigma _{\tau _1,T_1} = d\nu _s \chi _Ids\).
Now, to prove the required convolution estimate, it suffices to show that
for all \(\psi \in C_c^\infty (\mathbb {R}^3)\); the brackets denote the pairing between distributions and test functions. We compute that
Using the definitions of \(d\sigma _{\tau _2,T_2}\) and \(d\nu _s\) and the Cauchy–Schwarz inequality, the quantity between absolute value signs is at most
Thus, if we can show that
then a simple change of variable, using the definition of \(\overline{\gamma }_s\), gives the required estimate.
Toward that end, let \(G(\zeta ,t) := (\phi _r(\zeta ),\zeta ) + \gamma _s(t)\). We claim that G is invertible on \(3\theta _2 \times I\), provided R is sufficiently large. The definition of S implies that \(\overline{\gamma }_s'(t) = \overline{\omega }/|\overline{\omega }|\) for every s, t. Thus, the Jacobian of G at \((c_2,0)\) is given by
The first two columns of this matrix are orthogonal to \(n_2\). If we replace \(\gamma _s(0)\) by \(c_1\), then the third column becomes \(\omega /|\overline{\omega }|\), since \(\omega \cdot n_1 = 0\). The angle between \(\omega \) and the orthogonal complement of \(n_2\) is bounded below, since \(|\omega \cdot n_2| \ge c\). Combining these observations, we see that
Thus, the inverse function theorem implies that G is invertible on \(3\theta _2 \times I\), if R is sufficiently large. (The meaning of ‘sufficiently large’ does not depend on r or s, since the bounds \(\Vert \nabla G(c_2,0)\Vert \sim 1\) and \(\Vert (\nabla G(c_2,0))^{-1}\Vert \sim 1\) hold uniformly in these parameters.) Additionally, the bound \(|\det \nabla G(\zeta ,t)| \gtrsim 1\) holds on \(3\theta _2 \times I\), so we obtain
completing the proof. \(\square \)
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Acknowledgements
The author is very grateful to Betsy Stovall for her advice. This project was suggested by Stovall and grew out of joint work with Stovall and Diogo Oliveira e Silva. The 2019 MSRI Summer Graduate School on the Polynomial Method provided useful discussions during the earliest stage of this project. The author was supported by NSF grant DMS-1653264.
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Bruce, B.B. Local Extension Estimates for the Hyperbolic Hyperboloid in Three Dimensions. J Fourier Anal Appl 28, 78 (2022). https://doi.org/10.1007/s00041-022-09970-2
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DOI: https://doi.org/10.1007/s00041-022-09970-2