Fourier restriction for smooth hyperbolic 2-surfaces

Abstract

We prove Fourier restriction estimates by means of the polynomial partitioning method for compact subsets of any sufficiently smooth hyperbolic hypersurface in $\mathbb {R}^3.$ Our approach exploits in a crucial way the underlying hyperbolic geometry, which leads to a novel notion of strong transversality and corresponding “exceptional” sets. For the division of these exceptional sets we make crucial and perhaps surprising use of a lemma on level sets for sufficiently smooth one-variate functions from a previous article of ours.

1 Introduction

Let $S\subset \mathbb {R}^n$ be a sufficiently smooth hypersurface. The Fourier restriction problem, introduced by E. M. Stein in the seventies (for general submanifolds), asks for the range of exponents ${\tilde{p}}$ and ${\tilde{q}}$ for which an a priori estimate of the form

$$\begin{aligned} \bigg (\int _{S}|\widehat{f}|^{{\tilde{q}}}\,d\sigma \bigg )^{1/\tilde{q}}\le C\Vert f\Vert _{L^{{\tilde{p}}}(\mathbb {R}^n)} \end{aligned}$$

holds true for every Schwartz function $f\in {\mathcal {S}}(\mathbb {R}^n),$ with a constant C independent of f. Here, $d\sigma$ denotes the Riemannian surface measure on S.

The sharp range in dimension $n=2$ for curves with non-vanishing curvature was determined through work by Fefferman et al. [15, 42]. In higher dimension, the sharp $L^{{\tilde{p}}}-L^2$ result for hypersurfaces with non-vanishing Gaussian curvature was obtained by Stein and Tomas [30, 38] (see also Strichartz [32]). Some more general classes of surfaces were treated by Greenleaf [16]. In work by Ikromov et al. [21] and Ikromov and Müller [22, 23], the sharp range of Stein–Tomas type $L^{{\tilde{p}}}-L^2$ restriction estimates has been determined for a large class of smooth, finite-type hypersurfaces, including all analytic hypersurfaces.

The question about general $L^{{\tilde{p}}}-L^{{\tilde{q}}}$ restriction estimates is nevertheless still wide open. Fourier restriction to hypersurfaces with non-negative principal curvatures has been studied intensively by many authors. Major progress was due to Bourgain in the nineties [3,4,5]. At the end of that decade the bilinear method was introduced [26,27,28, 33, 35,36,37, 41]. A new impulse to the problem has been given with the multilinear method [2, 6]. The best results up to date have been obtained with the polynomial partitioning method, developed by Guth [18, 19] (see also [20, 40] for recent improvements).

For the case of hypersurfaces of non-vanishing Gaussian curvature but principal curvatures of different signs, besides Tomas–Stein type Fourier restriction estimates, until a few years ago the only case which had been studied successfully was the case of the hyperbolic paraboloid (or “saddle”) in $\mathbb {R}^3$: in 2005, independently Lee [25] and Vargas [39] established results analogous to Tao’s theorem [33] on elliptic surfaces (such as the 2-sphere), with the exception of the end-point, by means of the bilinear method.

First results based on the bilinear approach for particular one-variate perturbations of the saddle were eventually proved by the authors in [10,11,12]. Furthermore, Stovall [31] was able to include also the end-point case for the hyperbolic paraboloid. Building on the papers [25, 31, 39], and by strongly making use of Lorentzian symmetries, even global restriction estimates for one-sheeted hyperboloids have been established recently by Bruce et al. [9], with extensions to higher dimensions by Bruce [8]. Results on higher dimensional hyperbolic paraboloids have been reported by Barron [1]. All these results are in the bilinear range given by [33].

Improvements over the results for the saddle by means of an adaptation of the polynomial partitioning method from Guth’s articles [18] were achieved by Cho and Lee [14], and Kim [24]. Moreover, for a particular class of one-variate perturbations of the hyperbolic paraboloid, an analogue of Guth’s result had been proved by the authors in [13], and more lately by making use of Lorentzian symmetries, Bruce [7] has established analogous results for compact subsets of the one-sheeted hyperboloid.

In this article, we shall obtain the analogous result to [18] for compact subsets of any sufficiently smooth hyperbolic surface.

More precisely, we shall study embedded $C^m$- hypersurfaces S in $\mathbb {R}^3$ of sufficiently high degree of regularity $m\ge 3$ which are hyperbolic in the sense that the Gaussian curvature is strictly negative at every point, i.e., that at every point of S one principal curvature is strictly positive, and the other one is strictly negative.

A result comparable to the one of the authors was reported by Guo and Oh [17], though the initial approach is different and is based on approximation of arbitrary compact hypersurfaces with negative curvature in $\mathbb {R}^3$ by polynomial surfaces.

As usual, it will be more convenient to use duality and work in the adjoint setting. If ${{\mathcal {R}}}$ denotes the Fourier restriction operator $g\mapsto {{\mathcal {R}}}g:={\hat{g}}|_{S}$ to the surface S,  its adjoint operator ${{\mathcal {R}}}^*$ is given by ${{\mathcal {R}}}^*f(\xi )={{\mathcal {E}}}f(-\xi ),$ where ${{\mathcal {E}}}={{\mathcal {E}}}_{S}$ denotes the “Fourier extension” operator given by

$$\begin{aligned} {{\mathcal {E}}}f(\xi ):=\widehat{f\,d\sigma }(\xi )= \int _{S} f(x)e^{-i\xi \cdot x}\,d\sigma (x), \end{aligned}$$

with $f\in L^q(S,\sigma ).$ The restriction problem is therefore equivalent to the question of finding the appropriate range of exponents for which the estimate

$$\begin{aligned} \Vert {\mathcal {E}} f\Vert _{L^p(\mathbb {R}^3)}\le C\Vert f\Vert _{L^q(S,d\sigma )} \end{aligned}$$

holds true with a constant C independent of the function $f\in L^q(S,d\sigma ).$ We shall here concentrate on local estimates of this form, where S is replaced by a sufficiently small neighborhood of a given point on S. Such estimates then allow for estimates of the form

$$\begin{aligned} \Vert {\mathcal {E}}_{S_c} f\Vert _{L^p(\mathbb {R}^3)}\le C_{S_c,p,q}\Vert f\Vert _{L^q(S,d\sigma )}, \end{aligned}$$

(1.1)

for any compact subset $S_c$ of S,  where we have put

$$\begin{aligned} {{\mathcal {E}}}_{S_c} f(\xi ):= \int _{S_c} f(x)e^{-i\xi \cdot x}\,d\sigma (x). \end{aligned}$$

Our main result will be

Theorem 1.1

Assume that $p>3.25$ and $p>2q'.$ Then there is some sufficiently large $M(p,q)\in {\mathbb {N}}$ such that for any embedded hyperbolic hypersurface $S\subset \mathbb {R}^3$ of class $C^{M(p,q)}$ the estimate (1.1) holds true, i.e., for any compact subset $S_c$ of S,  we have

$$\begin{aligned} \Vert {\mathcal {E}}_{S_c} f\Vert _{L^p(\mathbb {R}^3)}\le C_{S_c,p,q}\Vert f\Vert _{L^q(S,d\sigma )}. \end{aligned}$$

For the proof of this result, we shall consider the following classes of functions: Let $\Sigma :=[-1,1]\times [-1,1].$ For $M\in \mathbb {N}, M\ge 3,$ we denote by $\mathrm {Hyp}^{M}=\mathrm {Hyp}^{M}$ $({\Sigma })$ $\subset C^{M}$ (${\Sigma }$) the set of all functions $\phi$ on $\Sigma$ satisfying the following properties:

$\phi$ extends from $\Sigma$ to a $C^M$-function on $2\Sigma ,$ also denoted by $\phi ,$ which satisfies the following conditions (1.2), (1.3) on $2 \Sigma :$

$$\begin{aligned} \phi (0)=0, \ \nabla \phi (0)=0, \ D^2\phi (0)=\left( \begin{array}{cc} 0 &{} 1 \\ 1 &{} 0 \\ \end{array} \right) , \end{aligned}$$

(1.2)

and

$$\begin{aligned} \Vert {\partial }_x^\alpha {\partial }_y^\beta \phi \Vert _\infty \le 10^{-5} \quad \text{ for } \quad 3\le \alpha +\beta \le M. \end{aligned}$$

(1.3)

Note: (i) $\phi \in C^M(\Sigma )$ lies in $\mathrm {Hyp}^M$ if and only if $\phi (x,y)$ is a small perturbation of xy,  in the sense that $\phi (x,y)=xy+\psi (x,y),$ where $\psi (0)=0, \ \nabla \psi (0)=0,$ and $D^2\psi (0)=0,$ and $\psi$ satisfies the estimates (1.3) (even on $2\Sigma$), with $\phi$ replaced by $\psi .$

(ii) If $\phi \in \mathrm {Hyp}^M,$ then

$$\begin{aligned} \max \big \{\Vert \phi _{xx}\Vert _\infty , \Vert \phi _{yy}\Vert _\infty , \Vert \phi _{xy}-1\Vert _\infty \big \}\le 2\cdot 10^{-5}. \end{aligned}$$

(1.4)

Our key result then is

Theorem 1.2

Assume that $p>3.25$ and $p>2q'.$ Then there is some sufficiently large $M(p,q)\in \mathbb {N}$ such that for any $\phi \in \mathrm {Hyp}^{M(p,q)}$ the Fourier extensions operator

$$\begin{aligned} {{\mathcal {E}}}_\phi f(\xi ):=\int _{\Sigma } f(x,y) e^{-i(\xi _1 x+\xi _2 y+\xi _3\phi (x,y))} \, dx dy \end{aligned}$$

associated to the graph $S_\phi$ of $\phi$ satisfies the estimate

$$\begin{aligned} \Vert {{\mathcal {E}}}_\phi f\Vert _{L^p(\mathbb {R}^3)}\le C_{p,q}\,\Vert f\Vert _{L^q(\Sigma )} \qquad \text{ for } \text{ every } \quad f\in {{\mathcal {S}}}(\mathbb {R}^2), \end{aligned}$$

(1.5)

with a constant which is independent of $\phi$ and f.

Note that Theorem 1.1 follows easily from Theorem 1.2. Indeed, if S is an embedded hyperbolic hypersurface $S\subset \mathbb {R}^3$ of class $C^{M(p,q)},$ with M(p, q) as in Theorem 1.2, and if $S_c$ is a compact subset, then by compactness we can localized to sufficiently small neighborhoods of any points $x^0$ in $S_c.$ So, after permuting coordinates, we may assume that near such a point S is given as the graph of a $C^{M(p,q)}$ function $\phi ,$ and after translation and linear change of coordinates, that $x^0=0\,$ and that S is the graph of $\phi$ over some sufficiently small neighborhood U of the origin in $\mathbb {R}^2,$ where $\phi$ satisfies (1.2). Finally, after applying a suitable isotropic scaling by putting $\tilde{\phi }(z):=\frac{1}{r^2} \phi (rz),$ where $0<r\ll 1,$ assuming that U was sufficiently small, we see that we can reduce to a function ${\tilde{\phi }}$ in $\mathrm {Hyp}^{M(p,q)}$ $(\Sigma ).$

Denote by $B_R$ the cube $B_R:=[-R,R]^3, \, R\ge 0.$ In a similar way as in [13], Theorem 1.2 will be a consequence of the following local Fourier extension estimate:

Theorem 1.3

Assume that $3.25\ge q >2.6.$ Then, for every $\varepsilon >0,$ there is a sufficiently large $M(\varepsilon )\in \mathbb {N}$ such that for any $\phi \in \mathrm{Hyp}^{M(\varepsilon )}$ the following holds true: there is a constant $C_\varepsilon$ such that for any $R\ge 1$

$$\begin{aligned} \Vert {{\mathcal {E}}_\phi }f\Vert _{L^{3.25}(B_R)} \le C_\varepsilon R^\varepsilon \Vert f\Vert _{L^2(\Sigma )}^{2/q}\, \Vert f\Vert _{L^\infty (\Sigma )}^{1-2/q}, \end{aligned}$$

(1.6)

for all $f\in L^\infty (\Sigma )$.

Indeed, a simple interpolation argument as in [13] shows that the estimate in Theorem 1.3 implies the following:

If $p>3.25$ and $p>2q',$ then

$$\begin{aligned} \Vert {{\mathcal {E}}}_\phi f\Vert _{L^p(B_R)}\le C_{p,q,\varepsilon } R^\varepsilon \,\Vert f\Vert _{L^q(\Sigma )}, \end{aligned}$$

(1.7)

with a constant which is independent of $\phi$ and f.

Finally, as usual, we can invoke an $\varepsilon$-removal theorem to pass to Theorem 1.2, but we have to be a bit more precise here than usually:

From [24, Theorem 5.3] (which, as Kim observes, is an immediate extension of Tao’s $\varepsilon$-removal theorem [34, Theorem 1.2]), applied to the adjoints of the restriction operators, we see that the estimate in (1.7) implies the following: there is a constant $C>0$ such that if

$$\begin{aligned} \frac{1}{p}>\frac{1}{{\tilde{p}}}+\frac{C}{-\log \varepsilon }, \end{aligned}$$

(1.8)

then

$$\begin{aligned} \Vert {{\mathcal {E}}}_\phi f\Vert _{L^{{\tilde{p}}}(\mathbb {R}^3)}\le C_{ \tilde{p},q}\,\Vert f\Vert _{L^q(\Sigma )}. \end{aligned}$$

(1.9)

Thus, given ${\tilde{p}}$ with ${\tilde{p}}>3.25$ and ${\tilde{p}}>2q',$ we can first choose an appropriate p such that ${\tilde{p}}>p>3.25$ and ${\tilde{p}}>p>2q',$ and then an appropriate $\varepsilon =\varepsilon ({\tilde{p}},q)>0$ so that (1.8) holds true, hence also (1.9). This proves Theorem 1.2.

Later in our proofs we shall always assume without further mentioning that $\varepsilon$ is sufficiently small.

Outline of the paper: Since Guth’s polynomial partitioning method has been discussed in various papers by now, we shall be brief in many parts and refer to Guth [18] and our previous paper [13] whenever possible. Instead, we will focus on the novelties of our approach, so that some familiarity with the polynomial partitioning method is recommended.

A crucial property in restriction estimates is “strong” transversality (which we shall define precisely at a later stage). A major task in our paper will then be to understand the “exceptional” sets which lack this kind of transversality. In case of the unperturbed hyperbolic paraboloid, i.e., the graph of xy,  any two small caps that are not strongly transversal have to be contained in the part of the surface lying over a vertical or horizontal strip in the (x, y)-plane. For our general hyperbolic surfaces, the “exceptional” sets are again certain rectangles, but their size and slope strongly depend on the geometry of the surface. Rescaling arguments for functions restricted to a rectangle are here a priori difficult, since, unlike for the unperturbed hyperbolic paraboloid, our class of phase functions $\mathrm {Hyp}^m$ is not closed under anisotropic scalings, nor any other suitably large group of symmetries, such as the Lorentz group in case of the one-sheeted hyperboloid. However, we will show that the lack of strong transversality eventually still does allow for a rescaling argument.

The article is organized as follows: in Sect. 2 we present two important auxiliary results: on the one hand our “sublevel lemma” (Corollary 2.2) that will allow us to control certain sub-level sets, on the other hand Lemma 2.4 which will allow to control certain derivatives in the rescaling argument.

In Sect. 3, we relate the hyperbolic geometry of our surfaces to the strong transversality property which is needed to establish the required bilinear estimates. In particular, we shall derive a ”hyperbolic factorization” of the crucial transversality function $\Gamma _{z}(z_1,z_2)$ in (3.14), which will be of central importance to our approach and will lead to our notation of “strong separation” of caps. Our definition of “exceptional” sets will be based on this notion. In Sects. 3.3 and 3.4 of Sect. 3 we shall show how to move such exceptional rectangles into vertical position and prepare for the subsequent rescaling argument.

Motivated by these “exceptional” sets, we will devise a notion of $\alpha$-broadness adapted to our class of surfaces in Sect. 4, and prove our crucial “Geometric Lemma”, and we shall show that any two small caps that are not strongly separated have to be contained in the part of the surface lying over some rectangle (of possibly quite arbitrary direction) or be contained in a larger cap, thus ensuring that a family of pairwise not strongly separated caps is in some sense sparse.

In Sect. 5, we reduce our main result to estimates for the broad part of the extension operator, and in Sect. 6 we outline the actual polynomial partitioning argument and indicate which changes will be required compared to previous work, in particular to [13, 18]. Here, in particular, we will be brief and only highlight the steps that differ from previous work.

Convention: Unless stated otherwise, $C > 0$ will stand for an absolute constant whose value may vary from occurrence to occurrence. We will use the notation $A\sim _C B$ to express that $\frac{1}{C}A\le B \le C A$. In some contexts where the size of C is irrelevant we shall drop the index C and simply write $A\sim B.$ Similarly, $A\lesssim B$ will express the fact that there is a constant C (which does not depend on the relevant quantities in the estimate) such that $A\le C B,$ and we write $A\ll B,$ if the constant C is sufficiently small.

2 Auxiliary results

In this section, we prove two auxiliary results which will be crucial for our analysis, but which may also be of independent interest. We begin by recalling the following theorem from our previous paper [12]:

Theorem 2.1

Let $I=[a,b]$ be a compact interval and $g\in C^r(I, \mathbb {R})$, $r\ge 1,$ and put $C_r:=\Vert g^{(r)}\Vert _{L^\infty (I)}.$ Then there exists a decomposition of $\{g\ne 0\}$ into pairwise disjoint intervals $J_{\lambda ,\iota }$, where $\lambda$ ranges over the set of all positive dyadic numbers $\lambda \le \Vert g\Vert _\infty ,$ and where for any given $\lambda ,$ the index $\iota$ is from some index set ${\mathcal {J}}_\lambda$, such that the following hold true:

• (i) $|{{\mathcal {J}}}_\lambda |\le 10r\big (1+|I|C_r^{1/r} \lambda ^{-1/r}\big ) \lesssim 1+\lambda ^{-1/r}$.

• (ii) For any $\lambda$ dyadic, $\iota \in {{\mathcal {J}}}_\lambda$ and any $t\in J_{\lambda ,\iota }$ we have $\frac{1}{2}\lambda<|g(t)|<4\lambda .$

The following corollary will later allow us to control certain sub-level sets which will be important for our definition of broad points.

Corollary 2.2

Let $I=[a,b]$ be a compact interval and $g\in C^r(I, \mathbb {R})$, $r\ge 1,$ and put $C_r:=\Vert g^{(r)}\Vert _{L^\infty (I)}.$ Then for any $0< \lambda \le \Vert g\Vert _\infty ,$ there is a finite family of pairwise disjoint intervals $I_{\lambda ,i}$, where the index i is from some index set ${\mathcal {I}}_\lambda$, so that the following hold true:

• (i) $|{{\mathcal {I}}}_\lambda |\le 30 r\big (1+|I|C_r^{1/r} \lambda ^{-1/r}\big ) \lesssim 1+\lambda ^{-1/r}$.

• (ii) If we denote by $V_{\lambda }$ the union $\bigcup \nolimits _{i\in {\mathcal {I}}_{\lambda }}I_{{\lambda },i}$ of all the intervals $I_{{\lambda },i},$ then

  $$\begin{aligned} \{|g|< {\lambda }\}\subset V_{\lambda }\subset \{|g|<8{\lambda }\}. \end{aligned}$$

Proof

It is sufficient to prove this for any dyadic number ${\lambda },$ however, with the slightly stronger estimates

$$\begin{aligned} \{|g|< {\lambda }\}\subset V_{\lambda }\subset \{|g|<4{\lambda }\}. \end{aligned}$$

(2.1)

We shall consider I to be endowed with its relative topology from $\mathbb {R}.$ Consider then the open subset $U_{\lambda }:=\{|g|<{\lambda }\}$ of I. We decompose it into its connected components $U_{{\lambda },\nu },$ where $\nu$ will be from an at most countable index set. The $U_{{\lambda },\nu }$ are then open subintervals of I. Observe that if such an interval $U_{{\lambda },\nu }$ has endpoints $\alpha <\beta ,$ then $|g(\alpha )|={\lambda }$ if $\alpha >a,$ and $|g(\beta )|={\lambda }$ if $\beta <b.$ Moreover, if $\alpha =a$ and $\beta =b,$ the case where $|g(\alpha )|<{\lambda }$ and $|g(\beta )|<{\lambda }$ cannot arise, since then we would have $U_{{\lambda },\nu }=I$ and $\Vert g\Vert _\infty <{\lambda },$ contradicting our assumptions. Thus we see that every $U_{{\lambda },\nu }$ will have at least one endpoint, say $t_{{\lambda },\nu },$ such that $|g(t_{{\lambda },\nu })|={\lambda }.$

Next, according to Theorem 2.1, the point $t_{{\lambda },\nu }$ must be contained in either one of the intervals $J_{{\lambda },\iota },$ or in one of the intervals $J_{{\lambda }/2,\iota }.$ This interval is unique, and we denote it by $J_{{\lambda }}^\nu .$ Observe that then also $U_{{\lambda },\nu }\cup J_{\lambda }^\nu$ is an interval, and that $|g| <4{\lambda }$ on $U_{{\lambda },\nu }\cup J_{\lambda }^\nu .$

Let us finally put

$$\begin{aligned} V_{\lambda }:=\bigcup \limits _{\nu }U_{{\lambda },\nu }\cup J_{\lambda }^\nu . \end{aligned}$$

Then clearly $\{|g|< {\lambda }\}=U_{\lambda }\subset V_{\lambda },$ and $|g|< 4{\lambda }$ on $V_{\lambda }.$ Moreover, if we decompose $V_{\lambda }=\bigcup \nolimits _{i\in {\mathcal {I}}_{\lambda }}I_{{\lambda },i}$ into its connected components $I_{{\lambda },i},$ then each $I _{{\lambda },i}$ is an interval. And, if t is any point in $I _{{\lambda },i},$ then there is some $\nu _i$ so that $t\in U_{{\lambda },\nu _i}\cup J_{\lambda }^{\nu _i},$ so that $U_{{\lambda },\nu _i}\cup J_{\lambda }^{\nu _i}\subset I _{{\lambda },i}.$ The mapping $I _{{\lambda },i}\mapsto J_{\lambda }^{\nu _i}$ is clearly injective, since the intervals $I _{{\lambda },i}$ are pairwise disjoint, so that $|{{\mathcal {I}}}_\lambda |\le |{{\mathcal {J}}}_{{\lambda }/2}|+|{{\mathcal {J}}}_\lambda |.$ The estimate in (i) follows thus from estimate (i) in Theorem 2.1. $\square$

The following remark shows that we may even assume that the intervals from Corollary 2.2 are not too short. We shall not directly make use of this remark, but the same idea will be used later in Sect. 4 to show that we may choose the rectangles in our definition of $\alpha$-broadness sufficiently long.

Remark 2.3

If $\Vert g'\Vert _\infty \le 1,$ we may even assume that all the intervals $I_{{\lambda },i}$ in Corollary 2.2 have length $|I_{{\lambda },i}|\ge C{\lambda }.$

More precisely, assume that $g\in C^r(I, \mathbb {R})$ satisfies the assumptions of Corollary 2.2, that $C>0$ is given, and that in addition $\Vert g'\Vert _\infty \le 1.$ Then for any $0< \lambda \le \Vert g\Vert _\infty$ such that $C{\lambda }\le |I|,$ there is a finite family of pairwise disjoint open intervals $I_{\lambda ,i}$ of length $|I_{{\lambda },i}|\ge C{\lambda },$ where the index i is from some index set ${\mathcal {I}}_\lambda$, so that the following hold true:

• (i) $|{{\mathcal {I}}}_\lambda |\le 30 r\big (1+|I|C_r^{1/r} \lambda ^{-1/r}\big ) \lesssim 1+\lambda ^{-1/r}$.

• (ii) If we denote by $V_{\lambda }$ the union $\bigcup \nolimits _{i\in {\mathcal {I}}_{\lambda }}I_{{\lambda },i}$ of all the intervals $I_{{\lambda },i},$ then

  $$\begin{aligned} \{|g|< {\lambda }\}\subset V_{\lambda }\subset \{|g|<(8+C){\lambda }\}. \end{aligned}$$

Proof

Let us denote for ${\delta }>0$ by $A^{\delta }:=(A+(-{\delta },{\delta }))\cap I$ the ${\delta }$-thickening within the interval I of any subset A of I. For this proof, let us endow the quantities appearing in Corollary 2.2 with a superscript $\tilde{},$ so that for instance ${\tilde{I}}_{{\lambda },{\tilde{i}}}, {\tilde{i}}\in \tilde{ \mathcal I_\lambda }$ denote the intervals devised in this corollary. Then clearly, with ${\delta }:=C{\lambda },$

$$\begin{aligned} \{|g|< {\lambda }\}\subset \{|g|< {\lambda }\}^ {\delta }\subset V_{\lambda }^{\delta }=\bigcup \limits _{{\tilde{i}}\in \tilde{{\mathcal {I}}_{\lambda }}}(\tilde{I}_{{\lambda },{\tilde{i}}})^{\delta }\subset \{|g|<8{\lambda }\}^{\delta }\subset \{|g|<(8+C){\lambda }\}. \end{aligned}$$

Let us again decompose the set $V_{\lambda }^{\delta }=\bigcup \nolimits _{i\in {\mathcal {I}}_{\lambda }}I_{{\lambda },i}$ into its connected components $I_{{\lambda },i}.$ Since $V_{\lambda }^{\delta }$ is open, the $I_{{\lambda },i}$ are open intervals. Moreover, clearly any $I_{{\lambda },i}$ must contain at least one of the intervals $({\tilde{I}}_{{\lambda },{\tilde{i}}})^{\delta },$ so that its length must at least be ${\delta }=C {\lambda },$ and since the mapping from i to the chosen index ${\tilde{i}}$ is injective, we see that $|{{\mathcal {I}}}_\lambda | \le |\tilde{{{\mathcal {I}}}_\lambda }|.$ $\square$

The next lemma will become important for certain induction on scales arguments.

Lemma 2.4

Let $g\in C^k(I,\mathbb {R}),$ where I is an interval of length $|I|=b,$ and $k\in \mathbb {N},$ and assume that

$$\begin{aligned} \Vert g^{(m)}\Vert _\infty \le c_m,\quad m=0, \dots , k, \quad \text { where} \quad \ c_0<1. \end{aligned}$$

(2.2)

Let $\varepsilon >0,$ and assume that $k\ge 1/\varepsilon ,$ and that $b\le (c_0)^\varepsilon .$ Then there are constants ${\tilde{C}}_m(\varepsilon )\ge 0$ depending only on $\varepsilon$ and the constants $c_1, \dots , c_k$ (increasing with the values of the $c_j$), but not on b and $c_0,$ so that

$$\begin{aligned} \Vert g^{(m)}\Vert _\infty \le c_0\, {\tilde{C}}_m(\varepsilon ) b^{-m}, \qquad m=0, \dots , k. \end{aligned}$$

(2.3)

Remark 2.5

In our later application, we shall have $c_1,\dots ,c_k\sim 1$ and $c_0\ll 1$, so that for $m\ge 1$ and $c_0$ sufficiently small the estimates in (2.3) are stronger than the a priori estimates from (2.2), at least when m is not too large.

Proof

We scale by setting $h(x):= g(bx).$ Then, after translation, we may assume that $I=[0,1],$ and that

$$\begin{aligned} \Vert h^{(j)}\Vert _\infty \le c_jb^j,\quad j=0, \dots , k, \end{aligned}$$

(2.4)

and what we need to show is that there are constants $\tilde{C}_m(\varepsilon )\ge 0$ as above such that

$$\begin{aligned} \Vert h^{(m)}\Vert _\infty \le c_0\, {\tilde{C}}_m(\varepsilon ), \qquad m=0, \dots , k. \end{aligned}$$

(2.5)

To this end, choose $M=M(\varepsilon )\in \mathbb {N}$ minimal so that $M\ge 1/\varepsilon .$ Then $M\le k.$ Assume $m\le k.$

a) If $m\ge M,$ then

$$\begin{aligned} c_m b^m\le c_mb^M\le c_mc_0^{M\varepsilon }\le c_m c_0, \end{aligned}$$

so we may choose ${\tilde{C}}_m(\varepsilon ):= C_m.$ Notice that in particular ${\tilde{C}}_M(\varepsilon ):=c_{M(\varepsilon )}.$

b) Assume next that $0\le m\le M=M(\varepsilon ).$ We know already that

$$\begin{aligned} \Vert h\Vert _\infty\le & {} c_0, \end{aligned}$$

(2.6)

$$\begin{aligned} \Vert h^{(M)}\Vert _\infty\le & {} c_0\, {\tilde{C}}_M(\varepsilon ). \end{aligned}$$

(2.7)

The estimates in (2.5) for $0< m<M$ then follow from these two estimates by interpolation. Let us give an elementary argument for this claim. Fix m with $0\le m\le M-1.$

We first claim that (2.6) implies that for any $j=0,\dots ,m$ there are points

$$\begin{aligned} t^j_0<t^j_1<\cdots <t^j_{2^{m-j}-1} \end{aligned}$$

in [0, 1] such that $t^j_{i+1}-t^j_{i}\ge 2^{-m}$ and

$$\begin{aligned} |h^{(j)}(t^j_i)|\le c_0 2^{(m+1)j} \end{aligned}$$

(2.8)

for every i. This is easily proved by induction on j.

If $j=0,$ we may choose $t^0_i:= i 2^{-m}, \ i=0,\dots , 2^m-1.$ And, assuming that the claim holds for j,  by the induction hypothesis and the mean value theorem we can find points $t^{j+1}_i\in (t^j_{2i}, t^j_{2i+1})$ so that

$$\begin{aligned} |h^{(j+1)}(t^{j+1}_i)|=\Big |\frac{h^{(j)}(t^{j}_{2i+1})-h^{(j)}(t^{j}_{2i})}{t^j_{2i+1}-t^j_{2i}}\Big |\le \frac{2\cdot c_02^{(m+1)j}}{2^{-m}}=c_02^{(m+1)(j+1)}. \end{aligned}$$

Moreover, since

$$\begin{aligned} t^{j+1}_i<t^j_{2i+1}<t^j_{2i+2}<t^{j+1}_{i+1}, \end{aligned}$$

where $t^j_{2i+2}-t^{j}_{2i+1}\ge 2^{-m},$ we also have that $t^{j+1}_{i+1}-t^{j+1}_{i}\ge 2^{-m}.$

In particular, for $j=m,$ we find a $t^m:=t^m_0$ so that $|h^{(m)}(t^m)|\le c_0 2^{(m+1)m}.$ Then, for any $t\in [0,1],$ we have

$$\begin{aligned} |h^{(m)}(t)|\le |h^{(m)}(t^m)|+|h^{(m)}(t)-h^{(m)}(t^m)|\le c_0 2^{(m+1)m}+\Vert h^{(m+1)}\Vert _\infty \cdot 1, \end{aligned}$$

so that

$$\begin{aligned} \Vert h^{(m)}\Vert _\infty \le c_0 2^{(m+1)m}+\Vert h^{(m+1)}\Vert _\infty . \end{aligned}$$

(2.9)

Now we can use “downward induction” on m,  starting with $m=M-1,$ to prove (2.5). Indeed, when $m=M-1,$ then by (2.9) we have

$$\begin{aligned} \Vert h^{(M-1)}\Vert _\infty \le c_0 2^{M(M-1)}+\Vert h^{(M)}\Vert _\infty \le c_0 (2^{M(M-1)} +{\tilde{C}}_M(\varepsilon ))=: c_0 {\tilde{C}}_{M-1}(\varepsilon ). \end{aligned}$$

Finally, we can pass from m to $m-1$ by means of our induction hypothesis on m and (2.9):

$$\begin{aligned} \Vert h^{(m-1)}\Vert _\infty \le c_0 2^{m(m-1)}+\Vert h^{(m)}\Vert _\infty \le c_0 2^{m(m-1)}+c_0 {\tilde{C}}_{m}(\varepsilon )=: c_0{\tilde{C}}_{m-1}(\varepsilon ). \end{aligned}$$

$\square$

3 Geometric background on strong transversality

Assume that $\phi \in \mathrm{Hyp}^M(\Sigma ), M\ge 3,$ and recall that S is the graph of $\phi .$

3.1 Strong transversality for bilinear estimates

We begin by recalling some facts about what kind of “strong transversality” is required for establishing suitable bilinear estimates.

Following [25], given two open subsets $U_1,U_2\subset \Sigma ,$ we consider the quantity

$$\begin{aligned} \Gamma _{z}(z_1,z_2,z_1',z_2'):= \left\langle (D^2\phi (z))^{-1}(\nabla \phi (z_2)-\nabla \phi (z_1)),\nabla \phi (z_2')-\nabla \phi (z_1')\right\rangle \nonumber \\ \end{aligned}$$

(3.1)

for $z_i=(x_i,y_i),\, z'_i=(x'_i,y'_i)\in U_i\, , i=1,2$, and $z=(x,y)\in U_1\cup U_2.$ Bilinear estimates have constants depending only on upper bounds for the derivatives of $\phi$ and on lower bounds of (the modulus of) (3.1). As in [13], for our estimates it will be enough to have lower bounds only for $z\in U_2$ (or only for $z\in U_1$). If $U_1$ and $U_2$ are sufficiently small (with sizes depending on upper bounds of the first and second order derivatives of $\phi$ and a lower bound for the Hessian determinant of $\phi$) this condition reduces to the estimate

$$\begin{aligned} |\Gamma _{z}(z_1,z_2)|\ge c>0, \end{aligned}$$

(3.2)

for $z_i=(x_i,y_i)\in U_i$, $i=1,2$, $z=(x,y)\in U_2$, where

$$\begin{aligned} \Gamma _{z}(z_1,z_2):= \left\langle (D^2\phi (z))^{-1}(\nabla \phi (z_2)-\nabla \phi (z_1)),\nabla \phi (z_2)-\nabla \phi (z_1)\right\rangle . \end{aligned}$$

(3.3)

In contrast to [10,11,12], where we had to devise quite specific “admissible pairs” of sets $U_1, U_2$ for our bilinear estimates, as in [13] we shall here only have to consider “caps” (cf. Sect. 3.1.3) $\tau _1, \tau _2$ for $U_1,U_2,$ and the required bilinear estimates will be a of somewhat different nature. Nevertheless, the geometric transversality conditions that we need here will be the same.

We shall next exploit the hyperbolicity assumption on S in order to gain a better understanding of $\Gamma _{z}(z_1,z_2,z_1',z_2')$ for such surfaces. In particular, we shall derive the “hyperbolic factorization” (3.14) which will be of central importance.

3.1.1 Null vectors for $D^2\phi$

Recall that $\phi \in \mathrm{Hyp}^M(\Sigma ), M\ge 3.$ We put $H:=\phi _{xy}^2-\phi _{xx}\phi _{yy},$ so that $-H$ is the Hessian determinant of $\phi .$ From (1.4) we easily deduce that $|H(z)-1|\le 10^{-4}$ for every $z\in \Sigma .$

It is easy to check that we then explicitly have

$$\begin{aligned} -H(z)\Gamma _{z}(z_1,z_2)= & {} \phi _{yy}(z)\big (\phi _x(z_2)-\phi _x(z_1)\big )^2 +\phi _{xx}(z)\big (\phi _y(z_2)-\phi _y(z_1)\big )^2 \nonumber \\&- 2\phi _{xy}(z)\big (\phi _x(z_2)-\phi _x(z_1)\big )\big (\phi _y(z_2)-\phi _y(z_1)\big ). \end{aligned}$$

(3.4)

Let us further introduce the functions on $\Sigma$ defined by

$$\begin{aligned} A(z):=\frac{\phi _{yy}}{\phi _{xy}+\sqrt{H}}(z), \quad B(z):=\frac{\phi _{xx}}{\phi _{xy}+\sqrt{H}}(z). \end{aligned}$$

Note that

$$\begin{aligned} \begin{aligned} 1+AB&=2\frac{\phi _{xy}}{\phi _{xy}+\sqrt{H}}, \ \ 1-AB=2\frac{\sqrt{H}}{\phi _{xy}+\sqrt{H}}, \\ \frac{A}{1+AB}&=\frac{\phi _{yy}}{2\phi _{xy}},\ \quad \frac{B}{1+AB}=\frac{\phi _{xx}}{2\phi _{xy}}, \end{aligned} \end{aligned}$$

(3.5)

and that (1.4) implies that

$$\begin{aligned} |\phi _{xy}+\sqrt{H}-2|\le 10^{-4}, \ |\phi _{xx}|, |\phi _{yy}|\le 10^{-4}\quad \text {on} \ \Sigma , \end{aligned}$$

(3.6)

so that $|A(z)|, |B(z)|\le 10^{-3}.$

A and B are in fact closely linked with the geometry of the surface S. Indeed, consider the vectors ${\omega }:= (-A(z),1)$ and $\nu :=(1,-B(z)).$ Then one checks easily that these two vectors form a basis of null vectors of the Hessian matrix $D^2\phi (z),$ i.e., for every $z\in \Sigma ,$ we have

$$\begin{aligned} (-A(z),1) D^2\phi (z) {\,}^t(-A(z),1)=0 \quad \text {and}\quad (1, -B(z)) D^2\phi (z) {\,}^t(1,-B(z))=0.\nonumber \\ \end{aligned}$$

(3.7)

For fixed z,  let us therefore set

$$\begin{aligned} T:= T_{z}:=\left( \begin{array}{cc} 1 &{} -A(z) \\ -B(z) &{} 1 \end{array} \right) . \end{aligned}$$

(3.8)

Then clearly

$$\begin{aligned} (\xi _1,\xi _2) {\,}^tT D^2\phi (z)\, T \, {\,}^t(\eta _1,\eta _2)= & {} (\xi _1\nu +\xi _2{\omega })D^2\phi (z) {\,}^t(\eta _1\nu +\eta _2{\omega })\\= & {} q(z) (\xi _1\eta _2+\xi _2\eta _1), \end{aligned}$$

where $q(z):={\omega }D^2\phi (z) {\,}^t\nu .$ This shows that

$$\begin{aligned} {\,}^tTD^2\phi (z) T=q(z) \left( \begin{array}{cc} 0 &{} 1 \\ 1 &{}0 \end{array} \right) . \end{aligned}$$

(3.9)

Moreover, we have

$$\begin{aligned} q(z)=(-A(z),1) D^2\phi (z){\,}^t(1,-B(z))=\big (-A\phi _{xx}+(1+AB) \phi _{xy} -B\phi _{yy}\big )(z). \end{aligned}$$

And, by our definitions of A and B,  and (3.5), we easily see that

$$\begin{aligned} -A\phi _{xx}+(1+AB) \phi _{xy} -B\phi _{yy}=2 \frac{H}{\phi _{xy}+\sqrt{H}}, \end{aligned}$$

so that

$$\begin{aligned} q(z)=2 \frac{H}{\phi _{xy}+\sqrt{H}}(z). \end{aligned}$$

(3.10)

This implies in particular that $|q(z)-1|\le 10^{-3}.$ Note also that by (3.5)

$$\begin{aligned} \det T_{z}=1-A(z)B(z)= 2 \frac{\sqrt{H}}{\phi _{xy}+\sqrt{H}}(z)= \frac{q(z)}{\sqrt{H(z)}}\sim q(z)\sim 1. \end{aligned}$$

(3.11)

3.1.2 Back to $\Gamma _{z}(z_1,z_2,z_1',z_2')$

Observe next that (3.9) implies that

$$\begin{aligned} (D^2\phi (z))^{-1} =\frac{1}{q(z)}T \left( \begin{array}{cc} 0 &{} 1 \\ 1 &{}0 \end{array} \right) {\,}^tT. \end{aligned}$$

Thus,

$$\begin{aligned}&(\xi _1,\xi _2) ( D^2\phi (z))^{-1} {\,}^t(\eta _1,\eta _2)\nonumber \\&\quad =\frac{1}{q(z)}\Big [ (\xi _1-B(z) \xi _2)(\eta _2-A(z) \eta _1)+(\eta _1-B(z) \eta _2)(\xi _2-A(z) \xi _1)\Big ]. \nonumber \\ \end{aligned}$$

(3.12)

If we accordingly define the functions

$$\begin{aligned} t^1_{z}(z_1,z_2):= & {} \phi _x(z_2)-\phi _x(z_1)-B(z)(\phi _y(z_2)-\phi _y(z_1)),\\ t^2_{z}(z_1,z_2):= & {} \phi _y(z_2)-\phi _y(z_1)-A(z)(\phi _x(z_2)-\phi _x(z_1)), \end{aligned}$$

then the identity (3.12) shows that we may re-write

$$\begin{aligned} \Gamma _{z}(z_1,z_2,z_1',z_2')=\frac{1}{q(z)}\Big [ t^1_{z}(z_1,z_2)\cdot t^2_{z}(z'_1,z'_2)+t^1_{z}(z'_1,z'_2)\cdot t^2_{z}(z_1,z_2)\Big ].\nonumber \\ \end{aligned}$$

(3.13)

In particular, we obtain the following “hyperbolic factorization”:

$$\begin{aligned} \Gamma _{z}(z_1,z_2)=\tfrac{2}{q(z)}\cdot t^1_{z}(z_1,z_2)\cdot t^2_{z}(z_1,z_2), \end{aligned}$$

(3.14)

where the first factor 2/q(z) is of size 2, more precisely $|2/q(z)-2|\le 10^{-3},$ so that it is irrelevant.

Note also that, e.g.,

$$\begin{aligned} t^2_{z_1}(z_1,z_2)-t^2_{z_2}(z_1,z_2)=(A(z_1)-A(z_2))(\phi _x(z_2)-\phi _x(z_1)), \end{aligned}$$

(3.15)

and that

$$\begin{aligned} {t}^i_{z}(z_1,z_2)=-{ t}^i_{z}(z_2,z_1), \quad i=1,2. \end{aligned}$$

(3.16)

3.1.3 Caps and the basic decomposition of S

Definition 3.1

Fix $K\gg 1$ to be a large dyadic number, and $\mu \ge 1$ real (the reasons for this notation will be clarified in Sect. 6).

Given K and $\mu ,$ following [18] we shall consider a given covering of $\Sigma =[-1,1]\times [-1,1]$ by $K^2$ disjoint squares (called caps) $\tau$ of side length $\mu ^{1/2}K^{-1},$ whose centers are $K^{-1}$ separated. It can then happen that such a cap $\tau$ is no longer contained in $\Sigma ;$ in that case, we truncate it by replacing it with its intersection with $\Sigma .$ Note that one usually envisions caps to be subsets of the hypersurface S;  for our purposes, however, it is more convenient to work with caps $\tau \subset \Sigma ,$ which then can be identified with the corresponding caps $\{(z,\phi (z)): z\in \tau \}$ on S.

Observe that for $\mu =1,$ this includes in particular the case of the covering of $\Sigma$ by caps $\tau$ which are pairwise disjoint in measure - this is what we had called the basic decomposition of S into caps in [13]. If f is a given function on $\Sigma ,$ we had then defined $f_\tau :=f\chi _\tau .$

For general $\mu \ge 1,$ motivated by Guth’s inductive argument in [18], we shall, however, only assume that $f_\tau$ is a function such that $\mathrm{supp\,}f_\tau \subset \tau .$ Actually, Guth assumes more generally that $\tau$ is a cap of side length $r_\tau$ with $K^{-1}\le r_\tau \le \mu ^{1/2} K^{-1},$ but since we are only assuming that $f_\tau$ is supported in $\tau ,$ we can then as well replace $\tau$ by a larger cap of side length $\mu ^{1/2}K^{-1},$ as we did.

Assume now that $\tau _1\ne \tau _2$ are two different caps, with centers $z^c_1=(x^c_1,y^c_1),$ respectively $z^c_2=(x^c_2,y^c_2),$ of side length $\mu ^{1/2} K^{-1}.$ The previous definition of strong transversality motivates the following:

Definition 3.2

We say that $\tau _1\ne \tau _2$ are strongly separated if

$$\begin{aligned} \max \{\min \{|t^1_{z^c_1}(z^c_1,z^c_2)|, |t^2_{z^c_1}(z^c_1,z^c_2)|\},\min \{|t^1_{z^c_2}(z^c_1,z^c_2)|,|t^2_{z^c_2}(z^c_1,z^c_2)|\}\}\ge 50\mu ^{1/2}K^{-1}. \end{aligned}$$

We shall distinguish between the cases where $|y^c_2-y^c_1|\ge |x^c_2-x^c_1|,$ and where $|x^c_2-x^c_1|\ge |y^c_2-y^c_1|.$ Let us mostly concentrate on the first case; the other case can be treated in the same way be interchanging the roles of x and y. So, for the rest of this section, let us make the following assumption:

Assumption 1

Assume that $|y^c_2-y^c_1|\ge |x^c_2-x^c_1|.$

Remark 3.3

If the caps $\tau _1$ and $\tau _2$ are strongly separated, so that, say, $|t^1_{z^c_2}(z^c_1,z^c_2)|\ge 50\mu ^{1/2}K^{-1}$ and $|t^2_{z^c_2}(z^c_1,z^c_2)|\ge 50\mu ^{1/2}K^{-1},$ then

$$\begin{aligned} |\Gamma _{z}(z_1,z_2,z_1',z_2')|\ge 4 \mu K^{-2} \quad \text {for all} \quad z_1,z'_1\in \tau _1, \, z, z_2,z'_2\in \tau _2. \end{aligned}$$

(3.17)

This result generalizes the corresponding result in Remark 4.8 of [13], whose proof easily extends to our present situation by means of the identity (3.13). It will allow us to establish favorable bilinear estimates later on for the contributions by the tangential terms associated to the cells arising in Guth’s cell decomposition.

3.2 Not strongly separated caps

Assume now that $\tau _1$ and $\tau _2$ are not strongly separated.

Case A. Assume that $|y^c_2-y^c_1|\le 100 \mu ^{1/2}K^{-1}.$ Then, by Assumption 1, also $|x^c_2-x^c_1|\le 100\mu ^{1/2}K^{-1}.$ Both caps are then contained in a cap of slightly bigger size $100\mu ^{1/2}K^{-1}\le 100\mu ^{1/2}K^{-1/4}.$

Let us therefore assume from here on that $|y^c_2-y^c_1|> 100\mu ^{1/2}K^{-1}.$

Observe that our assumptions on $\phi$ in combination with Assumption 1 then easily imply that $|t^1_{z}(z_1,z_2)|\sim |y_2-y_1|$ for every $z_1\in \tau _1, z_2\in \tau _2$ and $z\in \tau _1\cup \tau _2,$ and thus we may assume that

$$\begin{aligned} \min \{|y^c_2-y^c_1|,|t^2_{z^c_1}(z^c_1,z^c_2)|\}\le & {} 100\mu ^{1/2}K^{-1}\text { and } \min \{|y^c_2-y^c_1|,|t^2_{z^c_2}(z^c_1,z^c_2)|\}\\\le & {} 100\mu ^{1/2}K^{-1}. \end{aligned}$$

In particular, we have

$$\begin{aligned} |t^2_{z^c_1}(z^c_1,z^c_2)| \le 100\mu ^{1/2}K^{-1} \text { and } |t^2_{z^c_2}(z^c_1,z^c_2)|\le 100\mu ^{1/2}K^{-1}. \end{aligned}$$

(3.18)

Note also that by (3.15)

$$\begin{aligned} |t^2_{z^c_1}(z^c_1,z^c_2)-t^2_{z^c_2}(z^c_1,z^c_2)|\sim |A(z^c_1)-A(z^c_2)||y^c_2-y^c_1|, \end{aligned}$$

(3.19)

with constants very close to 1.

We shall therefore distinguish two further cases.

Case B. Assume that $|y^c_2-y^c_1|> 100 \mu ^{1/2}K^{-1}$ and $|A(z^c_1)-A(z^c_2)|> \mu ^{1/2}K^{-3/4}.$

Then, by (3.18) and (3.19), $|y^c_2-y^c_1|\le 300 K^{-1/4}\le 300 \mu ^{1/2}K^{-1/4},$ and arguing as before we see that both caps are contained in a cap of size $400 \mu ^{1/2}K^{-1/4}.$

This leaves us with the case where $|y^c_2-y^c_1|> 100 \mu ^{1/2}K^{-1}$ and $|A(z^c_1)-A(z^c_2)|\le \mu ^{1/2}K^{-3/4}.$ Actually, in what follows, we shall not really make use of the condition $|y^c_2-y^c_1|> 100 \mu ^{1/2}K^{-1}$ and shall therefore henceforth concentrate on

Case C. Assume that

$$\begin{aligned} |t^2_{z^c_1}(z^c_1,z^c_2)|\}\le 100\mu ^{1/2}K^{-1} \quad \text {and} \quad |A(z^c_1)-A(z^c_2)|\le \mu ^{1/2}K^{-3/4}\ . \end{aligned}$$

(3.20)

Notation

We fix a point $z_1$ (which would be the point $z^c_1$ in Case C), and set $A_1:=A(z_1).$ Then, we define

$$\begin{aligned} R_I:= & {} \{z\in \Sigma : |t_{z_1}^2 (z_1,z)|\le 100 \mu ^{1/2}K^{-1}\}, \end{aligned}$$

(3.21)

$$\begin{aligned} R_{II}:= & {} \{ z\in \Sigma : |A(z)-A_1|\le \mu ^{1/2}K^{-3/4}\}. \end{aligned}$$

(3.22)

3.2.1 Level curves of $t_{z_1}^2 (z_1,\cdot )$

Let us fix $z_1\in \Sigma ,$ and let us abbreviate $\mathbf{t}_{z_1}(x,y):= t_{z_1}^2 (z_1,(x,y)).$ From our definition of $\mathbf{t}_{z_1}$ we compute that

$$\begin{aligned} \nabla \mathbf{t}_{z_1}(z)= & {} ( \phi _{xy}-A_1\phi _{xx}, \phi _{yy}-A_1 \phi _{xy})(z)=(1,0)+O(10^{-5}). \end{aligned}$$

(3.23)

Lemma 3.4

Let $\alpha :=\min \nolimits _{z\in \Sigma } t_{z_1}^2 (z_1,z), \, \beta :=\max \nolimits _{z\in \Sigma } t_{z_1}^2 (z_1,z).$ There exists a $C^M$-function $h:[\alpha , \beta ]\times [-1,1]\rightarrow \mathbb {R}$ such that the curves $\gamma _{I,v}(y):=(h(v,y),y), y\in [-1,1],$ with $v\in [\alpha , \beta ],$ are level curves of the function $t_{z_1}^2 (z_1,\cdot )$ which fibre the set $\Sigma _{z_1}:=\{(h(v,y),y):[(v,y)\in [\alpha , \beta ]\times [-1,1]\}$ into (“almost vertical”) curves, and $\Sigma \subset \Sigma _{z_1} \subset 2\Sigma .$ Moreover, the mapping ${\mathcal {H}}:[\alpha , \beta ]\times [-1,1]\rightarrow \Sigma _{z_1}, (v,y)\mapsto (h(v,y),y),$ is a $C^M$-diffeomorphism, and more precisely we have that $h_y(y,v)=O(10^{-4}), \ h_v(y,v)=1+O(10^{-4}).$

Proof

Consider the mapping ${\mathcal {G}}: (x,y)\mapsto (\mathbf{t}_{z_1}(x,y), y).$ Then, by (3.23)

$$\begin{aligned} D{\mathcal {G}}(x,y)=\left( \begin{array}{cc} \phi _{xy}-A_1\phi _{xx} &{} \phi _{yy}-A_1 \phi _{xy}) \\ 0 &{} 1 \end{array} \right) (z) =\left( \begin{array}{cc} 1&{} 0 \\ 0 &{} 1 \end{array} \right) +O(10^{-5}). \end{aligned}$$

Therefore, the results follow in a straight-forward manner from the inverse function theorem, by patching together local inverse functions. Note that the inverse function ${\mathcal {H}}$ to $\mathcal G$ must be of the form ${\mathcal {H}}(v,y)=(h(v,y),y),$ and that $t_{z_1}^2 (z_1,(h(y,v),y))=\mathbf{t}_{z_1}(h(y,v),y)=v.$ This also implies that

$$\begin{aligned} 0={\partial }_x\mathbf{t}_{z_1}\cdot h_y+{\partial }_y\mathbf{t}_{z_1}; \qquad 1={\partial }_x\mathbf{t}_{z_1}\cdot h_v, \end{aligned}$$

(3.24)

so that, in view of (3.23), $h_y(y,v)=O(10^{-4})$ and $h_v(y,v)=1+O(10^{-4}).$ $\square$

Note that this result also implies that horizontal sections of $R_I$ have length $O(\mu ^{1/2}K^{-1}).$

3.2.2 On the direction of level curves of $t_{z_1}^2 (z_1,\cdot )$ within $R_I\cap R_{II}$

The following lemma gives us an important geometric information.

Lemma 3.5

The set $R_I$ fibers into the level curves $\gamma _{I,v}(y):=(h(v,y),y), y\in [-1,1],$ with $|v|\le 100 \mu ^{1/2}K^{-1}.$ If $z=(h(v,y),y)$ lies on such a curve, denote by $X_{z}:=(h_y(v,y),1)$ the corresponding tangent vector at the point z. Then, if z lies also in $R_{II},$ i.e., if $z\in R_I\cap R_{II},$ we have that

$$\begin{aligned} |X_z-(-A_1,1)|\le 3\mu ^{1/2} K^{-3/4}. \end{aligned}$$

Thus, up to an error of order $O(\mu ^{1/2}K^{-3/4}),$ for all points z in $R_I\cap R_{II}$ the tangent vectors to the level curves of $t_{z_1}^2 (z_1,\cdot )$ point in the same direction given by $(-A_1,1).$

Proof

By Lemma 3.4 the set $R_I$ fibers into the level curves $\gamma _{I,v}(y):=(h(v,y),y), y\in [-1,1],$ with $|v|\le 100 \mu ^{1/2}K^{-1}.$ Fix any such v. Then $X_{z}=(h_y(v,y),1)$ is a tangent vector of length $1+O(10^{-4})$ to the corresponding curve, if $z:=(h(v,y),y).$ Note that, by (3.23) and (3.24),

$$\begin{aligned} h_y(v,y)=-\frac{{\partial }_y\mathbf{t}_{z_1}}{{\partial }_x\mathbf{t}_{z_1}}(z)=-\frac{\phi _{yy}(z)-A_1 \phi _{xy}(z)}{\phi _{xy}(z)-A_1\phi _{xx} (z)}. \end{aligned}$$

Let us compare this quantity with the one where $A_1$ is replaced by A(z),  i.e., with

$$\begin{aligned} -\frac{\phi _{yy}(z)-A(z) \phi _{xy}(z)}{\phi _{xy}(z)-A(z)\phi _{xx} (z)}. \end{aligned}$$

Our definition of A(z) implies that $\phi _{yy}-A \phi _{xy}=A\sqrt{H}$ and that

$$\begin{aligned} \phi _{xy}(z)-A\phi _{xx} =\frac{H+\phi _{xy} \sqrt{H}}{\phi _{xy}+\sqrt{H}}=\sqrt{H}, \end{aligned}$$

so that

$$\begin{aligned} -\frac{\phi _{yy}(z)-A \phi _{xy}(z)}{\phi _{xy}(z)-A\phi _{xx} (z)}=-A(z). \end{aligned}$$

(3.25)

Therefore, if $z=(h(v,y),y)\in R_{II},$ i.e., if $|A(z)-A_1|\le \mu ^{1/2}K^{-3/4}$, by (1.2), (1.3), we see that $|h_y(v,y)-(-A(z))|\le 2 \mu ^{1/2} K^{-3/4},$ hence $|h_y(v,y)-(-A_1)|\le 3 \mu ^{1/2}K^{-3/4},$ if K is supposed to be sufficiently large. This implies that $|X_z-(-A_1,1)|\le \mu ^{1/2}3K^{-3/4}.$ $\square$

3.3 Moving rectangular boxes into vertical position at the origin

Suppose that I is a subinterval of $[-1,1]$ of length $b:=|I|$ so that for any $y\in I$ there is some $x_y$ so that the point $z_y:=(x_y,y)$ lies in $R_I\cap R_{II}.$ Lemma 3.5 then shows that the set $(R_I\cap R_{II})\cap ([-1,1]\times I)$ is essentially contained in a rectangular box L of dimension $100 \mu ^{1/2}K^{-3/4}\times b,$ pointing in the direction of the vector ${\omega }:=(-A_1,1).$ Moreover, up to an error of order $O(\mu ^{1/2}K^{-3/4}),$ we may replace $A_1$ by A(z),  for any choice of point $z\in L.$

Indeed, in Sect. 4, based on Lemma 3.5 and Corollary 2.2, we shall devise in a systematic way such kind of rectangular boxes L,  whose lengths will in addition satisfy the following condition:

$$\begin{aligned} 100 \mu ^{1/2}K^{-3/4}\le b\le K^{-\varepsilon '}, \end{aligned}$$

(3.26)

where $\varepsilon '\in (0,\varepsilon )$ will be a fixed, but sufficiently small constant to be defined later. Moreover, in view of Remark 6.2, we may also assume that $\mu \le K^{\varepsilon },$ so that $100 \mu ^{1/2}K^{-3/4}\le K^{-\varepsilon } \le K^{-\varepsilon '}$.

Here comes another crucial observation: let $z=z_y$ for $y\in I.$ Then, by (3.7), we know that $(-A(z),1) D^2\phi (z) {\,}^t(-A(z),1)=0.$ But, since $z\in R_{II},$ we also have that $|A(z)-A_1|\le \mu ^{1/2} K^{-3/4},$ and thus ${\omega }D^2\phi (z_y) {\,}^t{\omega }= (-A_1,1) D^2\phi (z_y) {\,}^t(-A_1,1)=O(\mu ^{1/2}K^{-3/4}).$ And, since the box L is of horizontal width $100 \mu ^{1/2}K^{-3/4},$ the same estimate holds throughout L. Consequently, we see that we may assume (with ${\omega }=(-A_1,1)$) that, say,

$$\begin{aligned} |\left\langle {\omega },\nabla \right\rangle ^2 \phi (z)|\le C \mu ^{1/2}K^{-3/4} \qquad \text {for all} \ z\in L. \end{aligned}$$

(3.27)

Moreover, by our assumptions on $\phi ,$ clearly we also have

$$\begin{aligned} |\left\langle {\omega },\nabla \right\rangle ^m \phi (z)|\le c_m \qquad \text {for all} \ z\in L, \end{aligned}$$

(3.28)

for all $m=3, \dots ,M,$ with constants $c_m\lesssim 10^{-5}.$ Restricting these estimates to lines parallel to $\mathbb {R}{\omega },$ and applying Lemma 2.4 to $\left\langle {\omega },\nabla \right\rangle ^2 \phi$ along these lines, we see that these two estimates imply that

$$\begin{aligned} |\left\langle {\omega },\nabla \right\rangle ^m \phi (z)|\le \mu ^{1/2}{\tilde{C}}_m(\varepsilon ) K^{-3/4} b^{2-m}\qquad \text {for all} \ z\in L, \,m=2, \dots , M.\nonumber \\ \end{aligned}$$

(3.29)

Let us now denote by $z_0=(x_0,y_0)$ the center of our rectangular box L. For simplicity, we may and shall assume that $A_1=A(z_0).$

Our goal will be to find an affine-linear transformation $z=z_0+T{\tilde{z}}$ so that for the accordingly transformed function ${\tilde{\phi }}({\tilde{z}}):=\phi (z_0+T{\tilde{z}})$ we have that

$$\begin{aligned} D^2{\tilde{\phi }}(0)=q(z_0) \left( \begin{array}{cc} 0 &{} 1 \\ 1 &{}0 \end{array} \right) , \end{aligned}$$

where $q(z_0)\in \mathbb {R}.$ Note that in the coordinates ${\tilde{z}},$ the point $z_0$ then corresponds to ${\tilde{z}}_0=0.$

To this end, let us put $B_1:=B(z_0),$ and choose for T the matrix $T_{z_0}$ defined by (3.8), i.e.,

$$\begin{aligned} T:= T_{z_0}:=\left( \begin{array}{cc} 1 &{} -A_1 \\ -B_1 &{} 1 \end{array} \right) . \end{aligned}$$

Then, by (3.9), we have indeed that

$$\begin{aligned} D^2{\tilde{\phi }}(0)={\,}^tTD^2\phi (z_0) T=q(z_0) \left( \begin{array}{cc} 0 &{} 1 \\ 1 &{}0 \end{array} \right) , \end{aligned}$$

with q(z) defined as in Sect. 3.1.1. Note also that by (3.11) the Jacobian determinant of our change of coordinates is given by

$$\begin{aligned} \det T_{z_0}=\frac{q(z_0)}{\sqrt{H(z_0)} }\sim q(z_0)\sim 1. \end{aligned}$$

In the new affine-linear coordinates ${\tilde{z}},$ denote quantities like A, L,  etc., by ${\tilde{A}},$ ${\tilde{L}},$ etc.. Then $\tilde{A}({\tilde{z}}_0)={\tilde{A}}(0)=0,$ so that ${\tilde{{\omega }}}=(0,1).$ This corresponds to the following observation: we have ${\tilde{\phi }}(\tilde{x},{\tilde{y}})=\phi (x_0+{\tilde{x}}-A_1{\tilde{y}},y_0-B_1{\tilde{x}}+\tilde{y}),$ so that $\frac{{\partial }}{{\partial }{\tilde{y}}}{\tilde{\phi }}(\tilde{z})=(\left\langle {\omega },\nabla \right\rangle \phi )(z_0+T{\tilde{z}}).$ This shows that indeed the directional derivative $\left\langle {\omega },\nabla \right\rangle$ corresponds to the partial derivative with respect to ${\tilde{y}}$ in the coordinates ${\tilde{z}}=(\tilde{x},{\tilde{y}}).$ Thus, by (3.29), we have that

$$\begin{aligned} |{\partial }_{{\tilde{y}}}^m{\tilde{\phi }}({\tilde{z}})|\le {\tilde{C}}_m(\varepsilon ) \mu ^{1/2}K^{-3/4} b^{2-m}\qquad \text {for all} \ {\tilde{z}}\in \tilde{L}, \,m=2, \dots , M, \end{aligned}$$

(3.30)

if ${\tilde{L}}$ corresponds to L in the ${\tilde{z}}$-coordinates, i.e., ${\tilde{L}}=T^{-1}(-z_0+L).$ Note that ${\tilde{L}}$ is essentially again a rectangular box of dimension $100 \mu ^{1/2}K^{-3/4}\times b,$ but centered at the origin and vertical, so that we may assume that ${\tilde{L}}$ is contained in $\tilde{{\tilde{L}}}:=[-100 \mu ^{1/2}K^{-3/4},100 \mu ^{1/2}K^{-3/4}]\times [-b,b].$ We may and shall assume that (3.30) holds even on $\tilde{{\tilde{L}}}.$

From (1.3), we also get the following estimates on the box $\tilde{{\tilde{L}}}:$

$$\begin{aligned} \Vert {\partial }_{{\tilde{x}}}^\alpha {\partial }_{{\tilde{y}}}^\beta {\tilde{\phi }}\Vert _\infty \le C_m 10^{-5} \quad \text{ for } \quad 3\le \alpha +\beta \le M, \end{aligned}$$

(3.31)

for constants $C_m>0$ which may possibly be much bigger than 1 if m is very large.

Finally we note that by replacing ${\tilde{\phi }}$ with $q(z_0)^{-1} {\tilde{\phi }},$ and subtracting the first order Taylor polynomial at the origin from ${\tilde{\phi }},$ we may even assume that

$$\begin{aligned} {\tilde{\phi }}(0)=0, \ \nabla {\tilde{\phi }}(0)=0, \ D^2{\tilde{\phi }}(0)=\left( \begin{array}{cc} 0 &{} 1 \\ 1 &{} 0 \\ \end{array} \right) . \end{aligned}$$

(3.32)

We remark that our affine-linear coordinate change for passing to the ${\tilde{z}}$-coordinates, as well as the further adjustments to ${\tilde{\phi }}$ that we just have explained, have no essential effect on the associated Fourier extension estimates, except that the operator norms may be bigger by a factor $C\sim 1.$

Last, but not least, observe also that we may assume that the same type of estimates (3.29) and (3.31) will also hold on the double $2{\tilde{L}}$ of ${\tilde{L}},$ with constants ${\tilde{C}}_m(\varepsilon )$ respectively $C_m$ that possibly increase by yet another factor $C\sim 1.$

In the next subsection, we shall show how our previous results allow also the carry out later the induction on scales step.

3.4 The induction on scales step

For this subsection, we may and shall assume that $\mu =1.$ Moreover, to simplify the subsequent discussions, let us drop the factor 100 from the horizontal length of our box ${\tilde{L}},$ i.e., let us assume that ${\tilde{L}}$ is contained in $\tilde{{\tilde{L}}}$ given by $[-K^{-3/4},K^{-3/4}]\times [-b,b],$ and let us correspondingly drop this factor also from (3.26).

In a second step, let us then scale the ${\tilde{z}}$-coordinates by writing ${\tilde{x}}=K^{-3/4} x', \ {\tilde{y}}=by',$ $z'=(x',y'),$ and

$$\begin{aligned} \phi ^s(z'):=\frac{ K^{3/4}}{b}{\tilde{\phi }}(K^{-3/4} x', by'). \end{aligned}$$

Note that in the new coordinates $z',$ $\tilde{{\tilde{L}}}$ corresponds to our standard square $\Sigma .$ Then, on $\Sigma ,$ we have

$$\begin{aligned} {\partial }_{x'}^\alpha {\partial }_{y'}^\beta \phi ^s(x',y')=(K^{-3/4})^{\alpha -1} b^{\beta -1} {\partial }_{{\tilde{x}}}^\alpha {\partial }_{{\tilde{y}}}^\beta {\tilde{\phi }}(K^{-3/4} x', by'), \end{aligned}$$

so that in view of (3.32)

$$\begin{aligned} \phi ^s(0)=0, \ \nabla \phi ^s(0)=0, \ D^2\phi ^s(0)=\left( \begin{array}{cc} 0 &{} 1 \\ 1 &{} 0 \\ \end{array} \right) . \end{aligned}$$

(3.33)

And, if $\alpha \ge 1$ and $\alpha +\beta \ge 3$, by (3.31) and (3.26), we have

$$\begin{aligned} \Vert {\partial }_{x'}^\alpha {\partial }_{y'}^\beta \phi ^s\Vert _\infty \le (K^{-3/4})^{\alpha -1} b^{\beta -1} C_m 10^{-5} \le b^{\alpha -1} b^{\beta -1} C_m 10^{-5} \le C_m 10^{-5} b \le 10^{-5}. \end{aligned}$$

For the remaining case $\alpha =0$, we use the improved estimate (3.29), which implies that

$$\begin{aligned} \Vert {\partial }_{y'}^\beta \phi ^s\Vert _\infty \le (K^{-3/4})^{-1} b^{\beta -1} {\tilde{C}}_\beta (\varepsilon ) K^{-3/4} b^{2-\beta } = {\tilde{C}}_\beta (\varepsilon ) b \le 10^{-5}, \end{aligned}$$

since $b\le K^{-\varepsilon '}$ and we can choose K sufficiently large.

We thus see that

$$\begin{aligned} \Vert {\partial }_{x'}^\alpha {\partial }_{y'}^\beta \phi ^s\Vert _\infty \le 10^{-5} \quad \text{ for } \quad 3\le |\alpha |+|\beta |\le M, \end{aligned}$$

if we assume $K\gg 1$ to be sufficiently large.

Actually, if we denote by 2L the doubling of L which keeps the center of L fixed, we may even assume that the estimates (3.28), (3.29) hold true on 2L,  with constants bigger by some fixed factor only, and so the same arguments used before show that we may even assume that $\phi ^s$ is defined on $2\Sigma ,$ and that the previous estimates hold true even on $2\Sigma .$

Thus we see that the function $\phi ^s$ lies again in $\mathrm{Hyp}^M(\Sigma ).$

3.4.1 Final rescaling step in the induction on scales argument

Recall that we assume that $\mu =1.$ Explicitly, our construction of $\phi ^s$ shows that

$$\begin{aligned} \phi ^s(x',y')= & {} \frac{1}{ q(z_0)}bK^{-3/4} \Big [ \phi (x_0+K^{-3/4} x'-A_1by', y_0-B_1K^{-3/4}x'+by')-\phi _x(z_0)\nonumber \\&\times (K^{-3/4}x'-A_1by') -\phi _y(z_0) (-B_1K^{-3/4}x'+by') \Big ]+\text {constant}. \end{aligned}$$

(3.34)

Thus, if $f_L:=f\chi _L,$ and if $f^L$ denotes the corresponding function in the $z'$-coordinates, then changing coordinates we obtain

$$\begin{aligned} {{\mathcal {E}}}_\phi f_L(\xi )= & {} \int f_L(x,y) e^{-i(\xi _1 x+\xi _2 y+\xi _3\phi (x,y))} \, dx dy\\= & {} C\, bK^{-3/4}\int f^L(x',y') e^{-i\Phi (x',y'; \xi )} dx'dy', \end{aligned}$$

$C\sim 1,$ and where the phase is given by

$$\begin{aligned} \Phi (x',y'; \xi ):= & {} \xi _3\Big [ b K^{-3/4}\phi ^s(x',y') +\phi _x(z_0)( K^{-3/4} x'-A_1by')\\&+\phi _y(z_0)(-B_1K^{-3/4} x'+by') \Big ]+ \xi _1(x_0+K^{-3/4}x'-A_1 by') \\&+\xi _2(y_0-B_1K^{-3/4}x'+by') \nonumber \\&+\text {constant} \cdot \xi _3. \end{aligned}$$

Thus, up to a fixed linear function in $\xi ,$ which is irrelevant, we may assume that

$$\begin{aligned} \Phi (x',y'; \xi )= & {} \xi _3 b K^{-3/4}\phi ^s(x',y') +x'K^{-3/4} \big (\xi _1-B_1\xi _2+(\phi _x(z_0)-B_1\phi _y(z_0)) \xi _3\big )\\&+ y'b\big ( \xi _2-A_1\xi _1+(\phi _y(z_0) -A_1\phi _x(z_0))\xi _3\big ). \end{aligned}$$

This implies that

$$\begin{aligned} |{{\mathcal {E}}}_\phi f_L(\xi )|= b K^{-3/4} |{{\mathcal {E}}}_{\phi ^s} f^L(S\xi )|, \end{aligned}$$

(3.35)

where $S\xi$ is defines by

$$\begin{aligned}&S\xi :=\Big ( K^{-3/4}(\xi _1-B_1\xi _2+(\phi _x(z_0)-B_1\phi _y(z_0)) \xi _3, \\&\quad b(\xi _2-A_1\xi _1+(\phi _y(z_0) -A_1\phi _x(z_0))\xi _3), b K^{-3/4} \xi _3\Big ). \end{aligned}$$

We shall be interested in estimating $\Vert {{\mathcal {E}}}_\phi f_L\Vert _{L^p(B_R)},$ where $B_R$ denotes the Euclidean ball of radius R centered at the origin. Note that if $\xi \in B_R,$ then $\xi '=S\xi$ lies in the set $B'_R$ defined by

$$\begin{aligned} |\xi '_1|\le 3 K^{-3/4} R, \quad |\xi '_2|\le 2b R, \quad |\xi '_3|\le b K^{-3/4} R. \end{aligned}$$

What is important to us is the estimate for the third component $\xi ',$ which is bounded by $R':=K^{-3/4}R\ll R.$ This will allow to go from scale $R'$ to scale R as in [13].

Indeed, observe the following estimates, which follow easily from our definition of $f^L$ and (3.35):

$$\begin{aligned} \begin{aligned} \Vert f^L\Vert _2&\le (b K^{-3/4})^{-1/2} \Vert f_L\Vert _2, \quad \Vert f^L\Vert _\infty \le \Vert f\Vert _\infty , \\ \Vert {{\mathcal {E}}}_\phi f_L\Vert _{L^p(B_R)}&\le (b K^{-3/4})^{1-\frac{2}{p}}\Vert {{\mathcal {E}}}_\phi ^sf^L\Vert _{L^p(B'_R)}. \end{aligned} \end{aligned}$$

(3.36)

Now assume by induction hypothesis that

$$\begin{aligned} \Vert {{\mathcal {E}}}_{\phi ^s} f^L\Vert _{L^{p}(B_{R'})} \le C_\varepsilon R'^\varepsilon \Vert f^L\Vert _{L^2(\Sigma )}^{2/q}\,\Vert f^L\Vert _{L^\infty (\Sigma )}^{1-2/q}, \end{aligned}$$

where $\varepsilon >0$ is as in Theorem 1.3. By means of Lemma 5.1 in [13] we can replace the ball $B_{R'}$ on the left-hand side by $\mathbb {R}^2\times [-R',R']$ and keep the same estimate, with a possibly slightly larger constant $C'_\varepsilon .$ In particular, we see that

$$\begin{aligned} \Vert {{\mathcal {E}}}_{\phi ^s} f^L\Vert _{L^{p}(B'_{R})} \le C'_\varepsilon R'^\varepsilon \Vert f^L\Vert _{L^2(\Sigma )}^{2/q}\,\Vert f^L\Vert _{L^\infty (\Sigma )}^{1-2/q}. \end{aligned}$$

Combining this with (3.36), we see that

$$\begin{aligned} \Vert {{\mathcal {E}}}_\phi f_L\Vert _{L^p(B_R)}&\le (b K^{-3/4})^{1-\frac{2}{p}}C'_\varepsilon R'^\varepsilon \Vert f^L\Vert _{L^2(\Sigma )}^{2/q}\,\Vert f^L\Vert _{L^\infty (\Sigma )}^{1-2/q}\nonumber \\&\le C'_\varepsilon R'^{\varepsilon }(b K^{-3/4})^{\frac{1}{q'}-\frac{2}{p}}\Vert f_L\Vert _2^{2/q}\Vert f\Vert _\infty ^{1-2/q}\nonumber \\&\le C'_\varepsilon R^\varepsilon (K^{-3/4})^{\frac{1}{q'}-\frac{2}{p}+\varepsilon }\Vert f_L\Vert _2^{2/q}\Vert f\Vert _\infty ^{1-2/q}, \end{aligned}$$

(3.37)

since we assume that $p>2q'.$ It is important that the last estimate does not depend on the length b of L,  which may vary with L. By means of (3.37), we can now proceed similarly as in Section 1 of [13] and sum these estimates over all boxes L in an appropriate way—for the details of this, we refer to Sect. 5.

4 Broad points

4.1 Definition of broad points and the underlying family of rectangles

For the unperturbed hyperbolic paraboloid, the definition of broadness is based on horizontal and vertical strips, since these are the sets which lack strong transversality. Here, we will devise a family ${\bar{{{\mathcal {L}}}}}$ of rectangles (and their intersections) adapted to our perturbed hyperbolic paraboloid.

Let us assume as in Definition 3.1 that $K\gg 1.$ Moreover, in view of Remark 6.2, let us also assume that $1\le \mu \le K^{\varepsilon /2},$ where $\varepsilon >0$ is as in Theorem 1.3. Let us further fix an according family of caps $\tau$ of side length $\mu ^{1/2}K^{-1}.$ Note that as in [18, Theorem 2.4] and [13, Theorem 2.1], our main goal later will be to prove Theorem 5.1, in which we are indeed assuming that the caps $\tau$ form the basic decomposition of $\Sigma ,$ with $\mu =1.$ However, for the inductive argument which is used to prove this theorem, we shall be forced to consider also cases where $\mu >1.$

As usual, for $0<\alpha <1$, we define a point $\xi$ to be $\alpha$-broad for ${{\mathcal {E}}_\phi }f$, if

$$\begin{aligned} \max _{L\in {\bar{{{\mathcal {L}}}}}} |{{\mathcal {E}}_\phi }f_L(\xi )|\le \alpha |{{\mathcal {E}}_\phi }f(\xi )|, \end{aligned}$$

where $f_L:=\sum \nolimits _{\tau \subset L} f_\tau$. We define $Br_\alpha {{\mathcal {E}}_\phi }f(\xi )$ to be $|{{\mathcal {E}}_\phi }f(\xi )|$ if $\xi$ is $\alpha$-broad, and zero otherwise.

We explain now how to construct the family ${\bar{{{\mathcal {L}}}}}$.

As we saw in Sect. 3.2, Case C, the most troublesome sets lacking strong transversality are the intersection of the sets $R_I$ and $R_{II}$ from (3.213.22). We recall the functions

$$\begin{aligned} A(z)=\frac{\phi _{yy}}{\phi _{xy}+\sqrt{H}}(z), \quad B(z)=\frac{\phi _{xx}}{\phi _{xy}+\sqrt{H}}(z). \end{aligned}$$

Note that for $\phi \in \text {Hyp}^3$, we have

$$\begin{aligned} |\nabla A(z)|,\ |\nabla B(z)| \le 1 \end{aligned}$$

(4.1)

for all $z\in \Sigma$.

We shall focus the discussion on A;  there is an analogous construction with B. Let $\{A_k\}_k$ be a equidistant decomposition of the interval $A(\Sigma )$ of distance $C\mu ^{1/2}K^{-3/4}$, that is, $A_{k+1}=A_k+C\mu ^{1/2}K^{-3/4}$, where we will choose the constant C later, and put

$$\begin{aligned} R_{II}^k:=\{z\in \Sigma :|A(z)-A_k|< C \mu ^{1/2}K^{-3/4}\}. \end{aligned}$$

The sets $R_{II}^k$ are not pairwise disjoint, but $R_{II}^k$ and $R_{II}^{k'}$ may only overlap if $|k-k'|\le 1$. As we saw in Lemma 3.5, on $R_I\cap R_{II}$, the tangents to $R_I$ point essentially in a fixed direction. This suggests to denote for any given k by ${\tilde{\omega }}_k$ the unit vector pointing in the direction of $\omega _k:=(-A_k,1),$ and decompose $\Sigma$ into strips

$$\begin{aligned} S_{k,j}:=\big [(j-1)C'\mu ^{1/2}K^{-3/4},(j+1)C'\mu ^{1/2}K^{-3/4}\big ]{\tilde{\omega }}_k^\perp +\mathbb {R}{\tilde{\omega }}_k \end{aligned}$$

of thickness $2C'\mu ^{1/2}K^{-3/4}$ and direction $\omega _k,$ indexed by suitable integers $j\in \mathbb {Z}.$ Here, ${\tilde{\omega }}_k^\perp$ denotes a unit vector orthogonal to $\omega _k,$ and $C'$ denotes yet another suitable constant. For fixed k, $S_{k,j}$ and $S_{k,j'}$ do not overlap unless $|j-j'|\le 1$. Of course they can overlap quite a lot for different values of k, but we want to consider only the part of $S_{k,j}$ that intersects $R_{II}^k$. More precise, we define the following subset $S_{k,j}^0$ of $S_{k,j}:$

$$\begin{aligned} S_{k,j}^0:=\{z\in S_{k,j}: (z+\mathbb {R}{\tilde{\omega }}_k^\perp )\cap S_{k,j}\cap R_{II}^k\ne \emptyset \}. \end{aligned}$$

It is clear that the connected components of $S_{k,j}^0$ are all rectangles inside $S_{k,j}$ of full width $2C'\mu ^{1/2}K^{-3/4},$ but unknown length. Since for technical reasons that will become clear later, we do not want the rectangles to be too short, we set

$$\begin{aligned} S_{k,j}^1:=S_{k,j}^0+[-\mu ^{1/2}K^{-3/4},\mu ^{1/2}K^{-3/4}]{\tilde{\omega }}_k. \end{aligned}$$

Then the connected components of $S_{k,j}^1$ are rectangles inside $S_{k,j}$ of full width and length at least $2\mu ^{1/2}K^{-3/4}.$

On the other hand, we want the lengths of these rectangles not to be too long either, and therefore divide $S_{k,j}^1$ into rectangles ${\tilde{L}}_{k,j}^i$ of lengths at most $\frac{1}{2} K^{-\varepsilon '},$ but at least $\mu ^{1/2}K^{-3/4}$ ($\varepsilon '\ll \varepsilon$ to be determined later), by artificially chopping any connected component that is too long. Finally, since that artificial chopping may split a small cap $\tau$ of size $\mu ^{1/2}K^{-3/4}$ into two, we set

$$\begin{aligned} L_{k,j}^i:={\tilde{L}}_{k,j}^i+[-\mu ^{1/2}K^{-3/4},\mu ^{1/2}K^{-3/4}]{\tilde{\omega }}_k, \end{aligned}$$

so that for fixed k and j, the sets $L_{k,j}^i$ may intersect, but only two can overlap at any given point. Since $2\mu ^{1/2}K^{-3/4}<\frac{1}{2} K^{-\varepsilon '}$ for sufficiently small $\varepsilon$, every $L_{k,j}^i$ has length between $\mu ^{1/2}K^{-3/4}$ and $K^{-\varepsilon '}$.

Let ${{\mathcal {L}}}_1:=\{L_{k,j}^i\}_{k,j,i}$ denote the set of all these rectangles (which we often simply shall call “strips”).

By construction, $\mathrm{dist\,}(z,R_{II}^k)\le 2C'\mu ^{1/2}K^{-3/4}$ for all $z\in S_{k,j}^0$; for $z\in L^i_{k,j}$, we still have

$$\begin{aligned} \mathrm{dist\,}(z,R_{II}^k)\le (2C'+2)\mu ^{1/2}K^{-3/4}, \end{aligned}$$

so that

$$\begin{aligned} |A(z)-A_k|\le C\mu ^{1/2}K^{-3/4}+ \Vert \nabla A\Vert _\infty (2C'+2)\mu ^{1/2}K^{-3/4}\lesssim \mu ^{1/2}K^{-3/4}\nonumber \\ \end{aligned}$$

(4.2)

for all $z\in L_{k,j}^i$.

We summarize the most important properties of our family of rectangles, which follow immediately from their definition:

Remarks 4.1

There exist absolute constants $C_1,N_1\ge 1$ such that the following hold true:

• (i) For all k, j, i and all $z\in L^i_{k,j},$

  $$\begin{aligned} |A(z)-A_k|\le C_1\mu ^{1/2}K^{-3/4}. \end{aligned}$$

• (ii) At any point, at most $N_1$ sets from ${{\mathcal {L}}}_1$ overlap.

• (iii) Every $L\in {{\mathcal {L}}}_1$ has length between $\mu ^{1/2}K^{-3/4}$ and $K^{-\varepsilon '}$.

Of course there is the symmetric situation, where the coordinates are interchanged, and A is replaced by B, which will give us a similar set of rectangles ${{\mathcal {L}}}_2$.

To cover also the Cases A and B from Sect. 3.2, we furthermore need caps of size $\sim \mu ^{1/2}K^{-1/4}$. Let ${{\mathcal {L}}}_3$ be a collection of squares of side length $C\mu ^{1/2}K^{-1/4}$ covering $\Sigma$, whose centers are $\frac{C}{2}\mu ^{1/2}K^{-1/4}$ separated.

Finally, we set ${{\mathcal {L}}}:={{\mathcal {L}}}_1\cup {{\mathcal {L}}}_2\cup {{\mathcal {L}}}_3$. This family is already well suited for proving our geometric Lemma 4.3. However, later on the application of polynomial partitioning method will even require a family which is closed under intersections. We therefore define

$$\begin{aligned} {\bar{{{\mathcal {L}}}}}:=\{L_1\cap \ldots \cap L_m:L_1,\ldots ,L_m\in {{\mathcal {L}}}_1\cup {{\mathcal {L}}}_2\cup {{\mathcal {L}}}_3,\ m\in \mathbb {N}\}. \end{aligned}$$

Note that due to Remark 4.1(ii), the number m of possible intersections is uniformly bounded, and each $\Delta \in \bar{{{\mathcal {L}}}}$ is either contained in a strip $L\in {{\mathcal {L}}}_1\cup {{\mathcal {L}}}_2$ of dimensions $\mu ^{1/2}K^{-3/4}\times b$ (respectively $b\times \mu ^{1/2}K^{-3/4}$) for some b with $\mu ^{1/2}K^{-3/4}\le b\le K^{-\varepsilon '},$ or is contained in a large cap $L\in {{\mathcal {L}}}_3$ of side length $C\mu ^{1/2} K^{-1/4}.$ It is then easy to verify the following remark.

Remark 4.2

There exists an absolute constant ${\bar{N}}\in \mathbb {N}$ such that at most ${\bar{N}}$ sets $\Delta \in {\bar{{{\mathcal {L}}}}}$ overlap at any given point $z\in \Sigma .$

4.2 The geometric lemma

A key observation which is important to adapt the polynomial partitioning method is that caps which are mutually not strongly transversal are somewhat sparse.

Let us fix a parameter $\varepsilon '>0$ depending on $\varepsilon$ and ${\bar{N}}$ and then $M=M(\varepsilon )\in \mathbb {N}$ so that

$$\begin{aligned} 10{\bar{N}}\varepsilon ' =\varepsilon ^8 \quad \text {and}\quad \frac{3}{4M(\varepsilon )} \le \varepsilon '\le \frac{3}{2M(\varepsilon )}, \end{aligned}$$

(4.3)

where ${\bar{N}}$ is the constant from Remark 4.2.

Lemma 4.3

(The Geometric Lemma). There is some constant $K_1(\varepsilon )$ such that, for every $K\ge K_1(\varepsilon )$ and every $\phi \in \mathrm {Hyp}^{M(\varepsilon )},$ the following holds true: if ${{\mathcal {F}}}$ is any family of caps of side length $\mu ^{1/2}K^{-1}$ which does not contain two strongly separated caps, then there is subcollection ${\mathcal {L}}_0\subset {\mathcal {L}}$ of cardinality $|{\mathcal {L}}_0|\le K^{3\varepsilon '},$ such that each cap in ${{\mathcal {F}}}$ is contained in at least one element of ${\mathcal {L}}_0.$

Proof

Fix a cap $\tau _1\in {{\mathcal {F}}},$ and recall that we denoted its center by $z_1^c=(x_1^c,y_1^c)$. If $\tau _2\in {{\mathcal {F}}}$, then

$$\begin{aligned} \min \{|y^c_2-y^c_1|,|t^2_{z^c_1}(z^c_1,z^c_2)|\}\le & {} 100 \mu ^{1/2}K^{-1}\text { and } \min \{|y^c_2-y^c_1|,|t^2_{z^c_2}(z^c_1,z^c_2)|\}\\\le & {} 100 \mu ^{1/2}K^{-1}. \end{aligned}$$

We will follow the discussion in Sect. 3.2, and the division into cases we devised there. In cases A and B, both caps $\tau _1,\tau _2$ are clearly contained in a cap of size $100 \mu ^{1/2}K^{-1/4},$ which is contained in a cap of size $C\mu ^{1/2}K^{-1/4}$ from our collection ${{\mathcal {L}}}_3,$ if we assume that C is sufficiently large.

This leaves us with Case C, where

$$\begin{aligned} |t^2_{z^c_1}(z^c_1,z^c_2)|\}\le 100 \mu ^{1/2}K^{-1} \ \text {and} \ |A(z^c_1)-A(z^c_2)|\le \mu ^{1/2}K^{-3/4}. \end{aligned}$$

(4.4)

Recall from (3.23) that $\partial _{x} t^2_{z^c_1}(z^c_1,z)\ge 1/2.$ In Lemma 3.4 we used (3.23) in order to show that we may parametrize the zero set of $t^2_{z^c_1}(z^c_1,\cdot )$ by a curve $\gamma =(\gamma _1,\gamma _2):[-1,1]\rightarrow \Sigma$ such that $\gamma _2(t)=t.$ In particular, $\gamma _2(y_2^c)=y_2^c$. But then $z_2^c-\gamma (y_2^c)=(x_2^c-\gamma _1(y_2^c),0),$ and thus

$$\begin{aligned} |z_2^c-\gamma (y_2^c)| \le 2|t^2_{z^c_1}(z^c_1,z^c_2)-t^2_{z^c_1}(z^c_1,\gamma (y_2^c))| =2|t^2_{z^c_1}(z^c_1,z^c_2)| \le 200\mu ^{1/2}K^{-1}.\nonumber \\ \end{aligned}$$

(4.5)

Consider the $C^{M}$ function

$$\begin{aligned} g(t) := A(\gamma (t))-A(z_1^c), \qquad t\in [-1,1]. \end{aligned}$$

Applying Corollary 2.2 to g and level $\lambda :=2 \mu ^{1/2}K^{-3/4}$, by our choice of $M=M(\varepsilon )$ there exists a family ${{\mathcal {I}}}$ of subintervals of $[-1,1]$ such that

• (i) $|{{\mathcal {I}}}|\le 60 M \big (1+\Vert g^{(M)}\Vert ^{1/M}_\infty K^{3/4M}\big ) \le G(M)\,K^{\varepsilon '},$

• (ii) for all $I\in {{\mathcal {I}}}$ and all $t\in I$ we have $|g(t)|< 16\mu ^{1/2}K^{-3/4},$

• (iii) and for all $t\in I_0\backslash \bigcup _{I\in {{\mathcal {I}}}} I$ we have $|g(t)|\ge 2 \mu ^{1/2}K^{-3/4},$

where $G(M)\ge 1$ for any integer $M\ge 3$ is a constant such that the last estimate in (i) holds uniformly for any $\phi \in \mathrm {Hyp}^M.$ Indeed, by our definition of the function g and the uniform estimates that are assumed to hold for all functions $\phi$ in $\mathrm {Hyp}^M$ in combination with Faà di Bruno’s theorem [29] on derivatives of compositions of functions we easily see that a uniform estimate

$$\begin{aligned} 60 M \big (1+\Vert g^{(M)}\Vert _\infty ^{1/M})\le G(M) \end{aligned}$$

holds true for all $\phi \in \mathrm {Hyp}^M.$

In particular, if we set $K_1(\varepsilon ):= G(M(\varepsilon ))^{8M(\varepsilon )/3},$ then in combination with (4.3) we see that

• (i’) $|{{\mathcal {I}}}|\le K^{3\varepsilon '/2}$ if $K\ge K_1(\varepsilon ).$

By (4.1), (4.4) and (4.5), we see that

$$\begin{aligned} |g(y_2^c)|\le |A(\gamma (y_2^c))-A(z_2^c)|+|A(z_2^c)-A(z_1^c)| < 2\mu ^{1/2}K^{-3/4}, \end{aligned}$$

and hence by (iii), $y_2^c\in I$ for some $I\in {{\mathcal {I}}}$. Again by (4.5), we see that $\tau _2$ is contained in the neighborhood

$$\begin{aligned} {{\mathcal {N}}}(I):=\gamma (I)+B(0,\mu ^{1/2}K^{-3/4}) \end{aligned}$$

of the curve $\gamma (I)$.

Choose k so that $|A_k-A(z^c_1)|\le \frac{1}{2}C\mu ^{1/2}K^{-3/4}$. Fix any $t_0\in I,$ and choose $S_{k,j}$ to be one of our strips of direction $\omega _k$ which contains $\gamma (t_0)$. Since these strips slightly overlap, we may and shall choose j so that $\mathrm{dist\,}(\gamma (t_0),\partial S_{k,j})\ge \frac{1}{2} C'\mu ^{1/2}K^{-3/4}$.

Recall also that by (ii), for any $t\in I$ we have

$$\begin{aligned} |A(\gamma (t))-A(z_1^c)|= |g(t)| \le 16\mu ^{1/2}K^{-3/4}, \end{aligned}$$

hence $\gamma (t)$ is in the intersections of the regions $R_I$ and $R_{II}$ as defined in (3.213.22) (modified by a harmless factor 16, which we will ignore). Using Lemma 3.5, we see that the tangential of $\gamma$ on I points essentially in direction $\omega _k$, and more precisely we obtain that

$$\begin{aligned} |\langle \gamma (t)-\gamma (t_0),{\tilde{\omega }}_k^\perp \rangle |\lesssim \mu ^{1/2}K^{-3/4}. \end{aligned}$$

This means that ${{\mathcal {N}}}(I)\subset S_{k,j}$, provided we choose the constant $C'$ from the construction of these strips big enough.

For $z\in {{\mathcal {N}}}(I)$, we find a $t\in I$ with $|\gamma (t)-z|<\mu ^{1/2}K^{-3/4}$, so that

$$\begin{aligned}&|A(z)-A_k|\le |A(z)-A(\gamma (t))|+|g(t)|+|A(z^c_1)-A_k|\\&\quad \le (1+16+ C/2 )\mu ^{1/2}K^{-3/4}<C\mu ^{1/2}K^{-3/4}, \end{aligned}$$

i.e., ${{\mathcal {N}}}(I)\subset R^k_{II},$ provided $C>34$. This shows that ${{\mathcal {N}}}(I)\subset S^1_{k,j}$, and, since ${{\mathcal {N}}}(I)$ is clearly connected, even that ${{\mathcal {N}}}(I)$ is contained in a connected component of $S^1_{k,j}$. Recall that we had divided $S_{k,j}^1$ into rectangles ${\tilde{L}}_{k,j}^i$ of lengths at most $\frac{1}{2} K^{-\varepsilon '},$ by artificially chopping any connected component that is too long. But, for any $\tau _2\subset {{\mathcal {N}}}(I)$ with $\tau _2\cap {\tilde{L}}_{k,j}^i\ne \emptyset$, $\tau _2\subset L_{k,j}^i\in {{\mathcal {L}}}_1$, and there are at most $6K^{\varepsilon '}$ sets $\tilde{L}_{k,j}^i$ in any connected component of $S^1_{k,j}$. In combination with (i’), this proves the claim of the lemma also in Case C. $\square$

5 Reduction to estimates for the broad part

In this section, we fix $\mu =1$ and consider our basic decomposition of $\Sigma$ into caps $\tau$ of side length $K^{-1}$ which have pairwise disjoint interiors. Consider the families ${\mathcal {L}}$ and $\overline{{\mathcal {L}}}$ defined in the previous section for $\mu =1.$ Recall that each $\Delta \in \overline{\mathcal L}$ is either contained in a strip of dimensions $K^{-3/4}\times b$ for some $K^{-3/4}\le b\le K^{-\varepsilon '},$ or is contained in a large cap of side length $100 K^{-1/4}.$

As in the previous section (for $\mu =1$), given the function f,  $\alpha \in (0,1)$ and K,  we say that the point $\xi \in \mathbb {R}^3$ is $\alpha$-broad for ${{\mathcal {E}}_\phi }f$ if

$$\begin{aligned} \max _{\Delta \in \overline{{\mathcal {L}}} }|{{\mathcal {E}}_\phi }f_\Delta (\xi )|\le \alpha |{{\mathcal {E}}_\phi }f(\xi )|, \end{aligned}$$

where $f_\Delta :=\sum \nolimits _{\tau \subset \Delta } f_\tau .$

We define $Br_\alpha {{\mathcal {E}}_\phi }f(\xi )$ to be $|{{\mathcal {E}}_\phi }f(\xi )|$ if $\xi$ is $\alpha$-broad, and zero otherwise.

We shall prove the following analogue to [18, Theorem 2.4] and [13, Theorem 2.1]:

Theorem 5.1

Let $0<\varepsilon < 10^{-10},$ and choose $M=M(\varepsilon )\in \mathbb {N}$ sufficiently large so that (4.3) is satisfied. Then there are constants $K=K(\varepsilon )\gg 1$ and $C_\varepsilon$ such that for any $\phi \in \mathrm {Hyp}^M$ and any radius $R\ge 1$ the following hold true:

$$\begin{aligned} \Vert Br_{K^{-\varepsilon }} {{\mathcal {E}}_\phi }f\Vert _{L^{3.25}(B_R)} \le C_\varepsilon R^\varepsilon \Vert f\Vert _{L^2(\Sigma )}^{12/13}\, \Vert f\Vert _{L^\infty (\Sigma )}^{1/13} \end{aligned}$$

for every $f\in L^\infty (\Sigma ),$ and moreover $K(\varepsilon )\rightarrow \infty$ as $\varepsilon \rightarrow 0.$

To show that Theorem 1.3 follows from this result, let us note that it is enough to consider q close to 2.6 and put $p:=3.25.$ We divide the domain of integration $B_R$ in (1.6) into three subsets:

$$\begin{aligned} A:= & {} \{\xi \in B_R:\xi \text { is }K^{-\varepsilon }-\text { broad for }{{\mathcal {E}}_\phi }f\},\\ B:= & {} \{\xi \in B_R:|{{\mathcal {E}}_\phi }f_\Delta (\xi )|> K^{-\varepsilon }|{{\mathcal {E}}_\phi }f(\xi )|\text { for some }\Delta \in \overline{{\mathcal {L}}},\\&\quad \,\text {with } \Delta \text { contained in a strip } L\in {\mathcal {L}}_1\cup {{\mathcal {L}}}_2\}, \\ C:= & {} \{\xi \in B_R{\setminus } B:|{{\mathcal {E}}_\phi }f_\Delta (\xi )|> K^{-\varepsilon }|{{\mathcal {E}}_\phi }f(\xi )|\text { for some }\Delta \in \overline{{\mathcal {L}}}, \\&\quad \text {with } \Delta \text { contained in a large cap } L\in {\mathcal {L}}_3 \}. \end{aligned}$$

If $\xi \in A$, then $|{{\mathcal {E}}_\phi }f(\xi )|=Br_{K^{-\varepsilon }}{{\mathcal {E}}_\phi }f(\xi )$, so that the contribution of A can be controlled using Theorem 5.1. Notice that

$$\begin{aligned} \Vert f\Vert _{L^2(\Sigma )}^{12/13}\, \Vert f\Vert _{L^\infty (\Sigma )}^{1/13}\le \Vert f\Vert _{L^2(\Sigma )}^{2/q}\, \Vert f\Vert _{L^\infty (\Sigma )}^{1-2/q}, \end{aligned}$$

since $q>2.6>13/6$.

If $\xi \in B,$ then there is some $\Delta \in \overline{{\mathcal {L}}}$ which is contained in a strip of dimensions $K^{-3/4}\times b$ for some $b\le K^{-\varepsilon '},$ so that $|{{\mathcal {E}}_\phi }f_\Delta (\xi )|> K^{-\varepsilon }|{{\mathcal {E}}_\phi }f(\xi )|.$ Then we may estimate

$$\begin{aligned} |{{\mathcal {E}}_\phi }f(\xi )|< K^\varepsilon \sup _\Delta |{{\mathcal {E}}_\phi }f_{\Delta }(\xi )| \le K^\varepsilon \Big (\sum _\Delta |{{\mathcal {E}}_\phi }f_\Delta (\xi )|^p\Big )^{1/p}, \end{aligned}$$

where the supremum and sum are taken over all $\Delta \in {\bar{{{\mathcal {L}}}}}$ contained in a strip $L\in {\mathcal {L}}$ of dimensions $K^{-3/4}\times b$ for some $b\le K^{-\varepsilon '}$ (which may depend on $\Delta$). Thus, we can apply to any such $f_\Delta$ the scaling (associated to the corresponding strip L) described in Sect. 3.4, more precisely estimate (3.37), and obtain

$$\begin{aligned} \Vert {{\mathcal {E}}_\phi }f_\Delta \Vert _{L^p(B_R)}\le C' _\varepsilon R^\varepsilon (K^{-3/4})^{\frac{1}{q'}-\frac{2}{p}+\varepsilon }\Vert f_\Delta \Vert _2^{2/q}\Vert f\Vert _\infty ^{1-2/q}. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert {{\mathcal {E}}_\phi }f\Vert _{L^p(B)}\le & {} K^{\varepsilon } \Big (\sum _\Delta \Vert {{\mathcal {E}}_\phi }f_\Delta \Vert ^p_{L^p}\Big )^{1/p}\\\le & {} C'_\varepsilon K^{\varepsilon } R^\varepsilon (K^{-3/4})^{\frac{1}{q'}-\frac{2}{p}+\varepsilon } \Big (\sum _\Delta \Vert f_\Delta \Vert _{2}^{2p/q}\,\Vert f\Vert _{\infty }^{p(1-2/q)}\Big )^{1/p}. \end{aligned}$$

Since $2p/q>2,$ taking into account the overlap of the elements of $\overline{{\mathcal {L}}}$ (see Remark 4.2) we estimate

$$\begin{aligned} \sum _\Delta \Vert f_\Delta \Vert _{2}^{2p/q}\le \Big (\sum _\Delta \Vert f_\Delta \Vert _{2}^{2}\big )^{p/q}\le \bar{N}^{p/q}\Vert f\Vert _{2}^{2p/q}. \end{aligned}$$

Hence,

$$\begin{aligned} \Vert {{\mathcal {E}}_\phi }f\Vert _{L^p(B)}\le & {} C'_\varepsilon {\bar{N}}^{1/q} K^{\varepsilon } R^\varepsilon (K^{-3/4})^{\frac{1}{q'}-\frac{2}{p}+\varepsilon }\Vert f\Vert _{2}^{2/q}\,\Vert f\Vert _{\infty }^{(1-2/q)} \\\le & {} \frac{1}{10}C_{\varepsilon } R^\varepsilon \Vert f\Vert _{2}^{2/q}\,\Vert f\Vert _{\infty }^{1-2/q}, \end{aligned}$$

since $p> 2q'$.

For $\xi \in C,$ isotropic scaling gives the same result.

6 Proof of Theorem 1.2

Following Section 3 in [18] and [13], we shall next formulate a more general statement in Theorem 5.1 which will become amenable to inductive arguments. We have to consider $\mu \ge 1.$

We assume that we are given $\mu \ge 1,$ a dyadic natural number $K\gg 1$ and a family of caps $\tau$ of side length $\mu ^{1/2}K^{-1},$ covering $\Sigma =[0,1]\times [0,1],$ such that their centers are $K^{-1}$-separated. Hence, at any point there will be at most $\mu$ of these caps which overlap at that point. Notice also that there are at most $K^2$ caps $\tau$ in the family. We also assume that we have a decomposition

$$\begin{aligned} f=\sum _\tau f_\tau , \end{aligned}$$

where $\mathrm{supp\,}f_\tau \subset \tau .$

We adapt the notion of broadness to the modified family of caps $\tau$: For each $\Delta \in {\bar{{{\mathcal {L}}}}},$ we define $f_\Delta :=\sum \nolimits _{\tau \subset \Delta } f_\tau .$

Let $\alpha \in (0,1).$ Given the function f and K,  we say that the point $\xi \in \mathbb {R}^3$ is $\alpha$-broad for ${{\mathcal {E}}_\phi }f$ if

$$\begin{aligned} \max _{\Delta \in \overline{{\mathcal {L}}}} |{{\mathcal {E}}_\phi }f_\Delta (\xi )|\le \alpha |{{\mathcal {E}}_\phi }f(\xi )|. \end{aligned}$$

We define $Br_\alpha {{\mathcal {E}}_\phi }f(\xi )$ to be $|{{\mathcal {E}}_\phi }f(\xi )|$ if $\xi$ is $\alpha$-broad, and zero otherwise.

Theorem 5.1 will be a consequence of the following

Theorem 6.1

Let $0<\varepsilon < 10^{-10}.$ Then there are constants $M=M(\varepsilon )\in \mathbb {N}, \,K=K(\varepsilon )\gg 1,$ and $C_\varepsilon$ such that for any $\phi \in \mathrm {Hyp}^M,$ any family of caps $\tau$ with multiplicity at most $\mu$ covering $\Sigma$ as above and associated functions $f_\tau$ which decompose f,  any radius $R\ge 1,$ and any $\alpha \ge K^{-\varepsilon },$ the following hold true:

If for every $\omega \in \Sigma ,$ and every cap $\tau$ as above,

$$\begin{aligned} \oint _{B(\omega ,R^{-1/2})}|f_\tau |^2\le 1, \end{aligned}$$

(6.1)

then

$$\begin{aligned} \int _{B_R}(Br_{\alpha } {{\mathcal {E}}_\phi }f)^{3.25} \le C_\varepsilon R^\varepsilon \bigg (\sum _\tau \int |f_\tau |^2\bigg )^{3/2+\varepsilon }R^{\delta _{trans}\log (K^\varepsilon \alpha \mu )}, \end{aligned}$$

(6.2)

where $\delta _{trans}:=\varepsilon ^6.$ Moreover, $K(\varepsilon )\rightarrow \infty$ as $\varepsilon \rightarrow 0.$

Here, in $\mathbb {R}^n,$ by $B(\omega ,r)$ we denote the Euclidean ball of radius $r>0$ and center $\omega ,$ and by $\oint _A f:=\frac{1}{|A|}\int _A f$ we denote the mean value of f over the measurable set A of volume $|A|>0.$

From this, as in [13, 18], Theorem 5.1 follows taking $\mu =1.$

As in these papers, let us also choose

$$\begin{aligned} \delta _{trans}:=\varepsilon ^6,\qquad \delta _{deg}:=\varepsilon ^4, \qquad \delta :=\varepsilon ^2, \end{aligned}$$

(6.3)

so that in particular

$$\begin{aligned} \delta _{trans}\ll \delta _{deg}\ll \delta \ll \varepsilon < 10^{-10}. \end{aligned}$$

We next choose $M=M(\varepsilon )\in \mathbb {N}$ and $\varepsilon '>0$ according to (4.3), i.e.,

$$\begin{aligned} 10{\bar{N}}\varepsilon ' =\varepsilon ^8 \quad \text {and}\quad \frac{3}{4M(\varepsilon )} \le \varepsilon '\le \frac{3}{2M(\varepsilon )}, \end{aligned}$$

and assume that $\varepsilon >0$ is so small that also $M\delta \gg 1000$ holds true (compare Proposition 6.3).

We also set

$$\begin{aligned} K=K(\varepsilon ):= \left\lfloor K_1(\varepsilon ) +e^{\varepsilon ^{-10}}+1 \right\rfloor , \qquad D=D(\varepsilon ):=R^{\delta _{deg}}=R^{\varepsilon ^4}, \end{aligned}$$

(6.4)

where $K_1(\varepsilon )$ is the constant from the Geometric Lemma 4.3 and $\left\lfloor x \right\rfloor$ denotes the integer part of x,  and where we are assuming that $R\ge 1$ is given. In particular, we then have $K\ge K_1(\varepsilon ),$ as assumed in the Geometric Lemma.

The following remark is similar to Remark 4.1 in [13] and holds (with the same proof) also here:

Remarks 6.2

a) It is enough to consider the case where $\alpha \mu \le K^{-\varepsilon /2},$ because in the other case, the exponent $\delta _{trans}\log (K^\varepsilon \alpha \mu )$ is very large and the estimate (6.2) trivially holds true. Note that since in Theorem 6.1 we are also assuming that $\alpha \ge K^{-\varepsilon },$ this implies that $\mu \le K^{\varepsilon /2}.$ Henceforth, we shall therefore always assume that $\alpha \mu \le K^{-\varepsilon /2}\le 10^{-5}.$

b) It is then also enough to consider the case where R is tremendously bigger than K,  say $R\ge 1000\, e^{K^{ \,e^{\varepsilon ^{-1000}}}}.$

As usual, we will work with wave packet decompositions of the functions f defined on ${S_\phi }:=\{(x,y,\phi (x,y)\;:\; (x,y)\in \Sigma \}.$ Following [18], we decompose $\Sigma$ into squares (“caps”) $\theta$ of side length $R^{-1/2}.$ By $\omega _\theta$ we shall denote the center of $\theta ,$ and by $\nu (\theta )$ the “outer” unit normal to ${S_\phi }$ at the point $(\omega _\theta ,\phi (\omega _\theta ))\in {S_\phi },$ which points into the direction of $(-\nabla \phi (\omega _\theta ),-1).$ ${{\mathbb {T}}}(\theta )$ will denote a set of $R^{1/2}$-separated tubes T of radius $R^{1/2+\delta }$ and length R,  which are all parallel to $\nu (\theta )$ and for which the corresponding thinner tubes of radius $R^{1/2}$ with the same axes cover $B_R.$ We will write $\nu (T):=\nu (\theta )$ when $T\in {{\mathbb {T}}}(\theta ).$

Note that for each $\theta ,$ every point $\xi \in B_R$ lies in $O(R^{2\delta })$ tubes $T\in {{\mathbb {T}}}(\theta ).$ We put ${{\mathbb {T}}}:=\bigcup \nolimits _{\theta } {{\mathbb {T}}}(\theta ).$ Arguing in the same way as in [18, Proposition 2.6], and observing that it is not necessary for the arguments in the proof, which are based on integrations by parts, that the phase $\phi$ is $C^\infty$, but merely $C^{M}$ for sufficiently large M (more precisely, $M\delta \gg 1000$), we arrive at the following approximate wave packet decomposition.

Proposition 6.3

Assume that R is sufficiently large (depending on $\delta$). Then, for any $\phi \in \mathrm {Hyp}^M$ (with $M=M(\varepsilon )$ as before), given $f\in L^2(\Sigma ),$ we may associate to each tube $T\in {{\mathbb {T}}}$ a function $f_T$ such that the following hold true:

• a) If $T\in {{\mathbb {T}}}(\theta ),$ then $\mathrm{supp\,}f_T\subset 3\theta .$

• b) If $\xi \in B_R{\setminus } T,$ then $|{{\mathcal {E}}_\phi }f_T(\xi )|\le R^{-1000}\Vert f\Vert _2.$

• c) For any $x\in B_R,$ we have $|{{\mathcal {E}}_\phi }f(x)-\sum _{T\in {{\mathbb {T}}}}{{\mathcal {E}}_\phi }f_T(x)|\le R^{-1000}\Vert f\Vert _2.$

• d) (Essential orthogonality) If $T_1,T_2\in {{\mathbb {T}}}(\theta )$ are disjoint, then $\big |\int f_{T_1} \overline{f_{T_2} }\big | \le R^{-1000} \int _{3\theta } |f|^2.$

• e) $\sum _{T\in {{\mathbb {T}}}(\theta )}\int _{\Sigma }|f_T|^2\le C\int _{3\theta }|f|^2.$

We next recall the version of the polynomial ham sandwich theorem with non-singular polynomials from [18]. If P is a real polynomial on $\mathbb {R}^n,$ we denote by $Z(P):=\{\xi \in \mathbb {R}^n: P(\xi )=0\}$ its null variety. P is said to be non-singular if $\nabla P(\xi )\ne 0$ for every point $\xi \in Z(P).$

Then, by in [18, Corollary 1.7], there is a non-zero polynomial P of degree at most D which is a product of non-singular polynomials such that the set $\mathbb {R}^3{\setminus } Z(P)$ is a disjoint union of $\sim D^3$ cells $O_i$ such that, for every i, 

$$\begin{aligned} \int _{O_i\cap B_R} (Br_{\alpha } {{\mathcal {E}}_\phi }f)^{3.25}\sim D^{-3}\int _{B_R} (Br_{\alpha } {{\mathcal {E}}_\phi }f)^{3.25}. \end{aligned}$$

(6.5)

We next define W as the $R^{1/2+\delta }$ neighborhood of Z(P) and put $O_i':=(O_i\cap B_R){\setminus } W.$

Moreover, note that if we apply Proposition 6.3 to $f_\tau$ in place of f (what we shall usually do), then by property (a) in Proposition 6.3, for every tube $T\in {{\mathbb {T}}}$ the function $f_{\tau ,T}$ is supported in an $O(R^{-1/2})$ neighborhood of $\tau .$ Following Guth, we define

$$\begin{aligned}&{{\mathbb {T}}}_i:=\{T\in {{\mathbb {T}}}: T\cap O_i'\ne \emptyset \},\quad f_{\tau ,i}:=\sum _{T\in {{\mathbb {T}}}_i} f_{\tau ,T},\quad f_{\Delta ,i}:=\sum _{\tau \subset \Delta } f_{\tau ,i} \\&\text {and}\quad f_i:=\sum _\tau f_{\tau ,i}. \end{aligned}$$

Then we can use the following analogue to [18, Lemma 3.2]:

Lemma 6.4

Each tube $T\in {{\mathbb {T}}}$ lies in at most $D+1$ of the sets ${{\mathbb {T}}}_i.$

We cover $B_R$ with $\sim R^{3\delta }$ balls $B_j$ of radius $R^{1-\delta }.$ Recall Definitions 3.3 and 3.4 from [18]:

Definitions 6.5

a) We define ${{\mathbb {T}}}_{j,tang}$ as the set of all tubes $T\in {{\mathbb {T}}}$ that satisfy the following conditions:

$$\begin{aligned} T\cap W\cap B_j\ne \emptyset , \end{aligned}$$

and if $\xi \in Z(P)$ is any nonsingular point (i.e., $\nabla P(\xi )\ne 0$) lying in $2B_j\cap 10T,$ then

$$\begin{aligned} \mathrm{angle}(\nu (T), T_\xi Z(P))\le R^{-1/2+2\delta }. \end{aligned}$$

Here, $T_\xi Z(P)$ denotes the tangent space to Z(P) at $\xi ,$ and we recall that $\nu (T)$ denotes the unit vector in direction of T. Accordingly, we define

$$\begin{aligned} f_{\tau ,j,tang}:=\sum _{T\in {{\mathbb {T}}}_{j,tang}}f_{\tau ,T}\quad \text {and}\quad f_{j,tang}:=\sum _\tau f_{\tau ,j,tang}. \end{aligned}$$

b) We define ${{\mathbb {T}}}_{j,trans}$ as the set of all tubes $T\in {{\mathbb {T}}}$ that satisfy the following conditions:

$$\begin{aligned} T\cap W\cap B_j\ne \emptyset , \end{aligned}$$

and there exists a nonsingular point $\zeta \in Z(P)$ lying in $2B_j\cap 10T,$ so that

$$\begin{aligned} \mathrm{angle}(\nu (T), T_\zeta Z(P))> R^{-1/2+2\delta }. \end{aligned}$$

Accordingly, we define

$$\begin{aligned} f_{\tau ,j,trans}:=\sum _{T\in {{\mathbb {T}}}_{j,trans}}f_{\tau ,T}\quad \text {and}\quad f_{j,trans}:=\sum _\tau f_{\tau ,j,trans}. \end{aligned}$$

We also recall Lemmas 3.5 and 3.6 in [18]:

Lemma 6.6

Each tube $T\in {{\mathbb {T}}}$ belongs to at most $\mathrm{Poly}(D)=R^{O(\delta _{deg})}$ different sets ${{\mathbb {T}}}_{j,trans}.$

Lemma 6.7

For each j,  the number of different $\theta$ so that ${{\mathbb {T}}}_{j,tang}\cap {{\mathbb {T}}}(\theta )\ne \emptyset$ is at most $R^{1/2+O(\delta )}.$

Note that the previous lemma makes use of the fact that the Gaussian curvature does not vanish on the surface $\Sigma ,$ so that the Gauß map is a diffeomorphism onto its image.

Lemma 6.8

Let $\xi \in O_i'.$ Then, given our assumptions on R from Remarks 6.2, we have

$$\begin{aligned} Br_\alpha {{\mathcal {E}}_\phi }f(\xi )\le Br_{2\alpha } {{\mathcal {E}}_\phi }f_i(\xi )+R^{-900}\sum _\tau \Vert f_\tau \Vert _2. \end{aligned}$$

Proof

This is analogous to [13, Lemma 4.7]. The proof of that lemma, without any changes, also gives us Lemma 6.8. $\square$

Following [18], we next define

$$\begin{aligned} \mathrm{Bil}({{\mathcal {E}}_\phi }f_{j,tang}):=\sum _{\tau _1,\tau _2\text { strongly separated}}|{{\mathcal {E}}_\phi }f_{\tau _1,j,tang}|^{1/2}|{{\mathcal {E}}_\phi }f_{\tau _2,j,tang}|^{1/2}. \end{aligned}$$

The remaining part of this subsection will be devoted to the proof of the following crucial analogue to the key Lemma 3.8 in [18]:

Lemma 6.9

If $\xi \in B_j\cap W$ and $\alpha \mu \le 10^{-5},$ then

$$\begin{aligned} Br_\alpha {{\mathcal {E}}_\phi }f(\xi )\le & {} 2\bigg (\sum _IBr_{K^{4{\bar{N}}\varepsilon '}\alpha }{{\mathcal {E}}_\phi }f_{I,j,trans}(\xi )+K^{100} \mathrm{Bil}({{\mathcal {E}}_\phi }f_{j,tang})(\xi )\nonumber \\&+R^{-900}\sum _\tau \Vert f_\tau \Vert _2\bigg ), \end{aligned}$$

(6.6)

where the first sum is over all possible subsets I of the given family of caps $\tau .$

Proof

Let $\xi \in B_j\cap W.$ We may assume that $\xi$ is $\alpha$-broad for ${{\mathcal {E}}_\phi }f$ and that $|{{\mathcal {E}}_\phi }f(\xi )|\ge R^{-900}\sum _\tau \Vert f\Vert _2.$ Let

$$\begin{aligned} I:=\{\tau : |{{\mathcal {E}}_\phi }f_{\tau ,j,tang}(\xi )|\le K^{-100} |{{\mathcal {E}}_\phi }f(\xi )|\}. \end{aligned}$$

(6.7)

We consider two possible cases:

Case 1: $I^c$ contains two strongly separated caps $\tau _1$ and $\tau _2.$ Then trivially

$$\begin{aligned} |{{\mathcal {E}}_\phi }f(\xi )|\le K^{100}|{{\mathcal {E}}_\phi }f_{\tau _1,j,tang}(\xi )|^{1/2}|{{\mathcal {E}}_\phi }f_{\tau _2,j,tang}(\xi )|^{1/2}\le K^{100}\mathrm{Bil}({{\mathcal {E}}_\phi }f_{j,tan})(\xi ), \end{aligned}$$

hence (6.6).

Case 2: $I^c$ does not contain two strongly separated caps.

We denote by ${\mathcal {L}}(\xi )\subset {\mathcal {L}}$ the family of at most $K^{3\varepsilon '}$ strips respectively large caps given by the Geometric Lemma for the family of caps ${{\mathcal {F}}}:=I^c.$ By

$$\begin{aligned} J:=\{\tau \,:\; \tau \subset L \text { for some } L\in {\mathcal {L}}(\xi )\} \end{aligned}$$

we denote the corresponding subset of caps $\tau .$ Then $I^c\subset J,$ i.e., $J^c\subset I.$ We write

$$\begin{aligned} f=\sum _{\tau \in J} f_\tau +\sum _{\tau \in J^c}f_\tau . \end{aligned}$$

Hence,

$$\begin{aligned} |{{\mathcal {E}}_\phi }f(\xi )|\le \left| \sum _{\tau \in J}{{\mathcal {E}}_\phi }f_{\tau }(\xi )\right| +\left| \sum _{\tau \in J^c}{{\mathcal {E}}_\phi }f_\tau (\xi )\right| . \end{aligned}$$

For $L\in {\mathcal {L}},$ we denote by ${\tilde{L}}:=\{\tau \,:\;\tau \subset L\}.$ Note that $f_L=\sum _{\tau \in {\tilde{L}}}f_\tau ,$ and

$$\begin{aligned} J=\bigcup _{L\in {\mathcal {L}}(\xi )} {\tilde{L}}. \end{aligned}$$

Thus, by the inclusion-exclusion principle and Remark 4.2,

$$\begin{aligned} \chi _J=\sum _{k=1}^{\overline{N}}(-1)^{k+1}\sum _{L_1,\dots ,L_k\in {\mathcal {L}}(\xi )}\chi _{{\tilde{L}}_1\cap \cdots \cap {\tilde{L}}_k}. \end{aligned}$$

(6.8)

Therefore,

$$\begin{aligned} \left| \sum _{\tau \in J}{{\mathcal {E}}_\phi }f_\tau (\xi )\right|\le & {} \sum _{k=1}^{\bar{N}}\sum _{L_1,\dots ,L_k\in {\mathcal {L}}(\xi )}\left| \sum _{\tau \in \tilde{L}_1\cap \cdots \cap {\tilde{L}}_k}{{\mathcal {E}}_\phi }f_\tau (\xi )\right| \\= & {} \sum _{k=1}^{\bar{N}}\sum _{L_1,\dots ,L_k\in {\mathcal {L}}(\xi )}|{{\mathcal {E}}_\phi }f_{L_1\cap \cdots \cap L_k}(\xi )|. \end{aligned}$$

Since $\xi$ is $\alpha$-broad, and since according to Remark 6.2 we may assume that $\alpha \le K^{-\varepsilon /2},$ this can be further estimated by

$$\begin{aligned}&\le \sum _{k=1}^{{\bar{N}}}|{\mathcal {L}}(\xi )|^k \alpha |{{\mathcal {E}}_\phi }f(\xi )| \le {\bar{N}} (K^{3\varepsilon '})^{{\bar{N}}} \alpha |{{\mathcal {E}}_\phi }f(\xi )| \\&\le {\bar{N}} K^{3\varepsilon '{\bar{N}}} K^{-\varepsilon /2} |{{\mathcal {E}}_\phi }f(\xi )| \le \frac{1}{10}|{{\mathcal {E}}_\phi }f(\xi )|, \end{aligned}$$

as one can easily see by our choices of K in (6.4) and $\varepsilon '$ in (4.3), provided $\varepsilon >0$ is assumed to be sufficiently small. Thus,

$$\begin{aligned} |{{\mathcal {E}}_\phi }f(\xi )|\le \frac{1}{10}|{{\mathcal {E}}_\phi }f(\xi )|+\left| \sum _{\tau \in J^c}{{\mathcal {E}}_\phi }f_\tau (\xi )\right| , \end{aligned}$$

and therefore

$$\begin{aligned} |{{\mathcal {E}}_\phi }f(\xi )|\le \frac{10}{9} \left| \sum _{\tau \in J^c}{{\mathcal {E}}_\phi }f_\tau (\xi )\right| . \end{aligned}$$

Since $\xi \in B_j\cap W,$ by Proposition 6.3,

$$\begin{aligned} {{\mathcal {E}}_\phi }f_\tau (\xi )={{\mathcal {E}}_\phi }f_{\tau ,j,trans}(\xi )+{{\mathcal {E}}_\phi }f_{\tau ,j,tang}(\xi )+ O(R^{-1000})\Vert f_\tau \Vert _2. \end{aligned}$$

(6.9)

Moreover, since $J^c\subset I,$ and since there are at most $K^2$ caps $\tau ,$

$$\begin{aligned} \sum _{\tau \in J^c}|{{\mathcal {E}}_\phi }f_{\tau ,j,tang}(\xi )|\le & {} \sum _{\tau \in I}|{{\mathcal {E}}_\phi }f_{\tau ,j,tang}(\xi )|\le K^{-100}\sum _{\tau \in I}|{{\mathcal {E}}_\phi }f(\xi )|\nonumber \\\le & {} K^{-98}|{{\mathcal {E}}_\phi }f(\xi )|, \end{aligned}$$

(6.10)

where the second inequality is a consequence of the definition of I. Thus,

$$\begin{aligned} \frac{9}{10}|{{\mathcal {E}}_\phi }f(\xi )|\le & {} \left| \sum _{\tau \in J^c}{{\mathcal {E}}_\phi }f_{\tau ,j,trans}(\xi )\right| +K^{-98}|{{\mathcal {E}}_\phi }f(\xi )|+\sum _\tau R^{-1000}\Vert f_\tau \Vert _2\\= & {} |{{\mathcal {E}}_\phi }f_{J^c,j,trans}(\xi )|+K^{-98}|{{\mathcal {E}}_\phi }f(\xi )|+\sum _\tau R^{-1000}\Vert f_\tau \Vert _2, \end{aligned}$$

and hence, since $|{{\mathcal {E}}_\phi }f(\xi )|\ge R^{-900}\sum \limits _{\tau }\Vert f_\tau \Vert _2,$

$$\begin{aligned} |{{\mathcal {E}}_\phi }f(\xi )|\le \frac{11}{9}|{{\mathcal {E}}_\phi }f_{J^c,j,trans}(\xi )|. \end{aligned}$$

(6.11)

It will then finally suffice to show that $\xi$ is $K^{4\bar{N}\varepsilon '}\alpha$-broad for ${{\mathcal {E}}_\phi }g,$ where $g:= f_{J^c,j,trans}.$ To this end let us set $g_\tau :=f_{\tau ,j,trans},$ if $\tau \in J^c,$ and zero otherwise, so that

$$\begin{aligned} g=\sum g_\tau . \end{aligned}$$

In what follows, we shall use the short hand notation “$\mathrm neglig$” for terms which are much smaller than $R^{-940}\sum _\tau \Vert f_\tau \Vert _2.$

Observe first that by (6.9)

$$\begin{aligned} |{{\mathcal {E}}_\phi }f_{\tau ,j,trans}(\xi )|\le |{{\mathcal {E}}_\phi }f_\tau (\xi )|+|{{\mathcal {E}}_\phi }f_{\tau ,j,tang}(\xi )|+\mathrm{neglig}, \end{aligned}$$

so that if $\tau \in J^c\subset I,$ then by the definition of I, 

$$\begin{aligned} |{{\mathcal {E}}_\phi }f_{\tau ,j,trans}(\xi )|\le |{{\mathcal {E}}_\phi }f_\tau (\xi )|+K^{-100}|{{\mathcal {E}}_\phi }f(\xi )|+\mathrm{neglig}. \end{aligned}$$

We have to show that

$$\begin{aligned} |{{\mathcal {E}}_\phi }g_{\Delta }(\xi )|\le K^{4{\bar{N}}\varepsilon '}\alpha |{{\mathcal {E}}_\phi }g(\xi )| \end{aligned}$$

for all $\Delta \in \overline{{\mathcal {L}}}.$ Write $\Delta =L_1\cap \cdots \cap L_r,$ where $L_i\in {\mathcal {L}},$ and set ${\tilde{\Delta }}:={\tilde{L}}_1\cap \cdots \cap {\tilde{L}}_r,$ so that $g_\Delta =\sum _{\tau \in {\tilde{\Delta }}}g_\tau =\sum _{\tau \in \tilde{\Delta }\cap J^c}f_{\tau ,j,trans}.$ Therefore the following two cases can arise:

• (i) There is some $i=1,\dots ,r,$ such that $L_i\in {\mathcal {L}}(\xi ).$ Then, ${\tilde{\Delta }}\subset {\tilde{L}}_i\subset J,$ hence, ${\tilde{\Delta }}\cap J^c=\emptyset .$

• (ii) For all $i=1,\dots ,r,$ $L_i\notin {\mathcal {L}}(\xi ).$

Observe first that by summing (6.9) over all $\tau \in \tilde{\Delta }\cap J^c$ we obtain

$$\begin{aligned} |{{\mathcal {E}}_\phi }g_{\Delta }(\xi )|=\left| \sum _{\tau \in {\tilde{\Delta }}}{{\mathcal {E}}_\phi }g_\tau \right| \le \left| \sum _{\tau \in {\tilde{\Delta }}\cap J^c}{{\mathcal {E}}_\phi }f_{\tau }(\xi )\right| +\sum _{\tau \in {\tilde{\Delta }}\cap J^c}|{{\mathcal {E}}_\phi }f_{\tau ,j,tang}(\xi )|+\mathrm{neglig}. \end{aligned}$$

(6.12)

By (6.10), the second term can again be estimated by

$$\begin{aligned} \sum _{\tau \in {\tilde{\Delta }} \cap J^c}|{{\mathcal {E}}_\phi }f_{\tau ,j,tang}(\xi )| \le K^{-98}|{{\mathcal {E}}_\phi }f(\xi )|. \end{aligned}$$

Case (i) is thus trivial. In case (ii), we write

$$\begin{aligned} \sum _{\tau \in {\tilde{\Delta }}\cap J^c}{{\mathcal {E}}_\phi }f_{\tau }(\xi )={{\mathcal {E}}_\phi }f_{\Delta }(\xi )-\sum _{\tau \in {\tilde{\Delta }}\cap J} {{\mathcal {E}}_\phi }f_{\tau }(\xi ). \end{aligned}$$

The first term is estimated using broadness. For the second term, again by (6.8),

$$\begin{aligned} \left| \sum _{\tau \in {\tilde{\Delta }}\cap J}{{\mathcal {E}}_\phi }f_\tau (\xi )\right|\le & {} \sum _{k=1}^{{\bar{N}}}\sum _{L'_1,\dots ,L'_k\in {\mathcal {L}}(\xi )}\left| \sum _{\tau \in {\tilde{\Delta }}\cap {\tilde{L}}'_1\cap \cdots \cap {\tilde{L}}'_k}{{\mathcal {E}}_\phi }f_\tau (\xi )\right| \\= & {} \sum _{k=1}^{{\bar{N}}}\sum _{L'_1,\dots ,L'_k\in {\mathcal {L}}(\xi )}|{{\mathcal {E}}_\phi }f_{\Delta \cap L'_1\cap \cdots \cap L'_k}(\xi )|. \end{aligned}$$

Note that $\Delta \cap L'_1\cap \cdots \cap L'_k\in \overline{\mathcal L},$ so that, since $\xi$ is $\alpha$-broad for ${{\mathcal {E}}_\phi }f,$

$$\begin{aligned} \left| \sum _{\tau \in {\tilde{\Delta }}\cap J}{{\mathcal {E}}_\phi }f_\tau (\xi )\right| \le \sum _{k=1}^{{\bar{N}}}|{\mathcal {L}}(\xi )|^k \alpha |{{\mathcal {E}}_\phi }f(\xi )| \le \bar{N}K^{3\varepsilon '{\bar{N}}}\alpha |{{\mathcal {E}}_\phi }f(\xi )|. \end{aligned}$$

Since $\alpha \ge K^{-\varepsilon }\gg 10 K^{-98},$ in combination with (6.12), and in the last step with (6.11), we conclude that

$$\begin{aligned} |{{\mathcal {E}}_\phi }g_{\Delta }(\xi )| \le ({\bar{N}} K^{3{\bar{N}}\varepsilon '}+1)\alpha |{{\mathcal {E}}_\phi }f(\xi )|+K^{-98}|{{\mathcal {E}}_\phi }f(\xi )|+\mathrm{neglig} \le K^{4{\bar{N}}\varepsilon '} \alpha |{{\mathcal {E}}_\phi }g(\xi )|. \end{aligned}$$

This completes the proof of Lemma 6.9. $\square$

The contribution by the bilinear term in (6.6) will be controlled by means of the following analogue to [13, Proposition 4.13 ] (or [18, Proposition 3.9 ]):

Proposition 6.10

We have

$$\begin{aligned} \int _{B_j\cap W}\mathrm{Bil}({{\mathcal {E}}_\phi }f_{j,tang})^{3.25}\le C_\varepsilon R^{O(\delta )+\varepsilon /2}\bigg (\sum _\tau \int |f_\tau |^2\bigg )^{3/2+\varepsilon }. \end{aligned}$$

With Proposition 6.10 at hand, the rest of the proof of Theorem 6.1, which we shall detail in Sect. 6.1, will follow the arguments in Section 4.2 in [13] (which in return are an adaptation of the arguments in pages 396–398 of [18]).

The proof of Proposition 6.10 reduces to the following analogue to [13, Lemma 4.14] and [18, Lemma 3.10]. Suppose we have covered $B_j\cap W$ with a minimal number of cubes Q of side length $R^{1/2},$ and denote by ${{\mathbb {T}}}_{j,tang,Q}$ the set of all tubes T in ${{\mathbb {T}}}_{j,tang}$ such that 10T intersects Q.

Lemma 6.11

Fix j,  i.e., a ball $B_j.$ If $\tau _1,\tau _2$ are strongly separated caps, then for any of the cubes Q we have

$$\begin{aligned}&\int _Q|{{\mathcal {E}}_\phi }f_{\tau _1,j,tang}|^2|{{\mathcal {E}}_\phi }f_{\tau _2,j,tang}|^2\\&\qquad \le R^{O(\delta )}R^{-1/2} \left( \sum _{T_1\in {{\mathbb {T}}}_{j,tang,Q}}\Vert f_{\tau _1,T_1}\Vert _2^2\right) \left( \sum _{T_2\in {{\mathbb {T}}}_{j,tang,Q}} \Vert f_{\tau _2,T_2}\Vert _2^2\right) +\mathrm{neglig}. \end{aligned}$$

Proof

Using Remark 3.3, the proof of Lemma 4.15 in [13] can be repeated word by word, giving the result. $\square$

6.1 Completing the proof of Theorem 6.1

If we compare with Subsection 4.2 in [13], which deals with the induction arguments with respect to the size of the radius R,  and the size of $\sum _\tau \int |f_\tau |^2,$ we can see that Lemma 6.9, which was the only result whose proof required substantial new arguments compared to the corresponding result in [13], is needed only for the discussion of Case 2 in this subsection, i.e., the case where the dominating term in

$$\begin{aligned} \int _{B_R} (Br_\alpha {{\mathcal {E}}_\phi }f)^{3.25}=\sum _i\int _{B_R\cap O_i'} (Br_\alpha {{\mathcal {E}}_\phi }f)^{3.25}+\int _{B_R\cap W} (Br_\alpha {{\mathcal {E}}_\phi }f)^{3.25} \end{aligned}$$

is the second term, the “wall” term. The estimation of this term could then be reduced to controlling $\sum _{j}\int _{B_j\cap W}\sum _ I (Br_{150\alpha }{{\mathcal {E}}_\phi }f_{I,j,trans})^{3.25},$ which in view of our Lemma 6.9 here has to be modified to

$$\begin{aligned} \sum _{j}\int _{B_j\cap W}\sum _ I (Br_{K^{4{\bar{N}}\varepsilon '}\alpha }{{\mathcal {E}}_\phi }f_{I,j,trans})^{3.25}. \end{aligned}$$

(6.13)

Dealing with this term by induction as in that paper, we arrive at

$$\begin{aligned}&\int _{B_R}(Br_{K^{4{\bar{N}}\varepsilon '}\alpha }E f)^{3.25}\\&\quad \le M_\varepsilon C_\varepsilon \mathrm{Poly}(D) R^{\varepsilon (1-\delta )}\bigg (\sum _\tau \int |f_\tau |^2\bigg )^{3/2+\varepsilon }R^{\delta _{trans}(1-\delta )\log (K^{4{\bar{N}}\varepsilon '}K^\varepsilon \alpha \mu )}\\&\quad \le \Big (M_\varepsilon \mathrm{Poly}(R^{\delta _{deg}}) R^{\delta _{trans}4{\bar{N}}\varepsilon '\log (K)-\varepsilon \delta }\Big )\\&\qquad \times C_\varepsilon R^{\varepsilon } \bigg (\sum _\tau \int |f_\tau |^2\bigg )^{3/2+\varepsilon } R^{\delta _{trans}\log (K^\varepsilon \alpha \mu )}\\&\quad \le C_\varepsilon R^{\varepsilon } \bigg (\sum _\tau \int |f_\tau |^2\bigg )^{3/2+\varepsilon } R^{\delta _{trans}\log (K^\varepsilon \alpha \mu )}. \end{aligned}$$

For the last inequality, note that the choices of $\delta ,\delta _{deg},\delta _{trans}$ in (6.3) and K in (6.4), in combination with (4.3), ensure that for $\varepsilon$ sufficiently small

$$\begin{aligned} C\delta _{deg}+\delta _{trans}4{\bar{N}}\varepsilon '\log (K)-\delta \varepsilon = C\varepsilon ^4+\varepsilon ^6 4 {\bar{N}}\varepsilon '\varepsilon ^{-10} - \varepsilon ^3 < -\varepsilon ^3/2. \end{aligned}$$

Here, $M_\varepsilon =2^{K^2}.$ Then, by Remark 6.2, $M_\varepsilon R^{-\varepsilon ^3/2} \ll 1.$ This completes the proof of Theorem 6.1.