2.1 Basic notation

Let A and B be two nonnegative quantities. We write $A\lesssim B$ and $B\gtrsim A$ if $A\leq C B$ holds for some (unimportant) finite positive constant C. We write $A\sim B$ if $c B\leq A\leq C B$ holds for finite positive constants c and C. Throughout the paper, it will be understood that any of these constants $c,C$ are allowed to depend on the dimension of the ambient Euclidean space, the number n, the parameters from (1.2), and (in Section 3) also on the unit vectors from (1.3), but are independent of all other parameters or variables.

The open Euclidean ball with radius r centered at x will be denoted $\textrm {B}(x,r)$ . The (standard) inner product and the Euclidean norm on $\mathbb{R}^d$ will be written as $(x,y)\mapsto x\cdot y$ and $x\mapsto |x|$ , respectively. The distance from a point $x\in \mathbb{R}^d$ to a set $S\subseteq \mathbb{R}^d$ will be denoted $\mathop {\textrm {dist}}(x,S)$ . The linear span of a set of vectors $S\subseteq \mathbb{R}^d$ will be written as $\mathop {\textrm {span}}(S)$ . We will write for the indicator function of a set $A\subseteq \mathbb{R}^d$ . The floor function is denoted $x\mapsto \lfloor x\rfloor $ , i.e., $\lfloor x\rfloor $ is the largest integer less than or equal to $x\in \mathbb{R}$ . The complex imaginary unit will be written as . The logarithm function will be written “ $\log $ ,” and it will be understood that its base is the number e.

We will always specify the measure with respect to which the integrals are evaluated, unless we are working with the Lebesgue measure. Similarly, we will simply write $|A|$ for the Lebesgue measure of A. By , we denote the average of a locally integrable complex function f over a bounded measurable set $A\subseteq \mathbb{R}^d$ . We will write $f\mapsto \|f\|_{\textrm {L}^p}$ for the norm of $\textrm {L}^p(\mathbb{R}^d)$ , $p\in [1,\infty ]$ , and $(f,g)\mapsto \langle f,g\rangle _{\textrm {L}^2}$ for the inner product in $\textrm {L}^2(\mathbb{R}^d)$ .

Now, we come to dilates and convolutions of functions and measures. For an integrable function $f\colon \mathbb{R}^d\to \mathbb{C}$ and a number $\lambda \in \mathbb{R}\setminus \{0\}$ , we define

$$ \begin{align*} f_{\lambda}(x) := |\lambda|^{-d} f(\lambda^{-1}x) \quad \text{for }x\in\mathbb{R}^d. \end{align*} $$

More generally, for a finite measure $\nu $ on Borel subsets $A\subseteq \mathbb{R}^d$ , we write

$$ \begin{align*} \nu_{\lambda}(A) := \nu(\lambda^{-1}A) = \nu(\{\lambda^{-1}x : x\in A\}). \end{align*} $$

Note that the normalizations of $f_{\lambda }$ and $\nu _{\lambda }$ are consistent with each other: if $\nu $ happens to be an absolutely continuous measure with density f, then $\nu _{\lambda }$ will have $f_{\lambda }$ for its density. In fact, for a bounded measurable function $h\colon \mathbb{R}^d\to \mathbb{C}$ , we have

$$ \begin{align*} \int_{\mathbb{R}^d} h(x) \,\textrm{d}\nu_{\lambda}(x) & = \int_{\mathbb{R}^d} h(\lambda x) \,\textrm{d}\nu(x), \\ \int_{\mathbb{R}^d} h(x) f_{\lambda}(x) \,\textrm{d}x & = \int_{\mathbb{R}^d} h(\lambda x) f(x) \,\textrm{d}x. \end{align*} $$

If $g\colon \mathbb{R}^d\to \mathbb{C}$ is another integrable function, then it makes sense to define

$$ \begin{align*} (f\ast g)(x) := \int_{\mathbb{R}^d} f(y) g(x-y) \,\textrm{d}y \quad \text{for a.e. }x\in\mathbb{R}^d. \end{align*} $$

It is well known that the operation of convolution is commutative. More generally, the convolution of a finite measure $\nu $ and an integrable function g is defined as

$$ \begin{align*} (\nu\ast g)(x) := \int_{\mathbb{R}^d} g(x-y) \,\textrm{d}\nu(y) \quad \text{for a.e. }x\in\mathbb{R}^d. \end{align*} $$

Even at this level of generality, the operation of convolution is associative, i.e., $({\nu \ast f})\ast g=\nu \ast (f\ast g)$ , while the Dirac delta measure at the origin, denoted $\delta _{\textbf {0}}$ , serves as the identity element.

The Fourier transform of an integrable function $f\colon \mathbb{R}^d\to \mathbb{C}$ is $\widehat {f}\colon \mathbb{R}^d\to \mathbb{C}$ defined as

while the Fourier transform of a finite Borel measure $\nu $ is $\widehat {\nu }\colon \mathbb{R}^d\to \mathbb{C}$ defined as

Basic properties of the Fourier transform can be found in any introductory textbook on the harmonic analysis, such as [Reference Stein and Weiss54]. For instance, it is useful to know that, for any $\lambda \in \mathbb{R}\setminus \{0\}$ and $f,\nu $ as above, one has

$$ \begin{align*} \widehat{f_{\lambda}}(\xi) = \widehat{f}(\lambda \xi), \quad \widehat{\nu_{\lambda}}(\xi) = \widehat{\nu}(\lambda \xi) \quad \text{for } \xi\in\mathbb{R}^d. \end{align*} $$

Important instances of Borel measures are spherical measures. If $\sigma $ is the normalized surface measure of the $(d-1)$ -dimensional standard unit sphere $\mathbb{S}^{d-1}$ in $\mathbb{R}^{d}$ , $d\geq 2$ , then its Fourier transform satisfies the well-known decay

(2.1) $$ \begin{align} |\widehat{\sigma}(\xi)| \lesssim \min\big\{1,|\xi|^{-(d-1)/2}\big\} \quad \text{for } \xi\in\mathbb{R}^d; \end{align} $$

see [Reference Stein52, Section VIII.5.B]. If $\sigma $ is a normalized $(k-1)$ -dimensional spherical measure supported on a sphere of radius $r>0$ in a k-dimensional plane in $\mathbb{R}^{d}$ orthogonal to some $(d-k)$ -dimensional linear subspace $H\subset \mathbb{R}^{d}$ , then

(2.2) $$ \begin{align} |\widehat{\sigma}(\xi)| \lesssim \min\left\{1,\left(r\mathop{\textrm{dist}}(\xi,H)\right)^{-(k-1)/2}\right\} \quad \text{for } \xi\in\mathbb{R}^d. \end{align} $$

2.2 Gaussian identities

We write

for the standard d-dimensional Gaussian function,

We also reserve special letters for its partial derivatives,

and for its Laplacian

Basic properties of the Fourier transform easily yield

and

where $\xi =(\xi _1,\xi _2,\ldots ,\xi _d)\in \mathbb{R}^d$ is arbitrary. These formulae (and the fact that the Fourier transform interchanges convolutions and pointwise products) imply the following convolution identities:

(2.3)

(2.4)

and

(2.5)

for any $\alpha ,\beta \in (0,\infty )$ . Identities (2.3)–(2.5) will be useful for splitting

or

into a convolution of one term with a desired scale and a uniquely determined remaining term; see Sections 3.2 and 4.2.

The above Gaussian functions are easily seen to satisfy the heat equation:

(2.6)

on $(t,x)\in (0,\infty )\times \mathbb{R}^d$ . By a simple chain rule, (2.6) generalizes to

(2.7)

where $a,b\in (0,\infty )$ can be arbitrary. The heat equation will govern the smoothing dynamics; see the beginning of Section 3.2 and the beginning of Section 4.2.

On the one hand, because Gaussian tails decay faster than any polynomial, we trivially have

(2.8)

for $x\in \mathbb{R}^d$ . On the other hand,

(2.9)

for $x\in \mathbb{R}^d$ . In words, Schwartz tails are dominated by a superposition of dilated Gaussians. Formula (2.9) was first used in a similar context by Durcik [Reference Durcik12]. It can be shown easily by investigating the asymptotic behavior of the right-hand side as $|x|\to \infty $ ; see the details in [Reference Durcik12, Section 3] or [Reference Durcik, Kovač, Škreb and Thiele18, Section 3]. A combination of (2.8) and (2.9) will be used to bound a convolution of a Gaussian and an arbitrary probability measure with bounded support by a superposition of (nontranslated) Gaussians; see the computations leading to (3.11) and (3.13) below.

We claim a simple identity:

(2.10)

for any compactly supported $f\in \textrm {L}^2(\mathbb{R}^d)$ and $a,b\in (0,\infty )$ . Indeed, using (2.4), (2.7), and (2.3), respectively, the left-hand side of (2.10) can be rewritten as

Next, for fixed parameters (1.2), for any choice of $\gamma _1,\ldots ,\gamma _n\in (0,\infty )$ , and for a compactly supported real-valued $f\in \textrm {L}^{2^n}((\mathbb{R}^d)^n)$ , we define

(2.11)

Here, we denote

(2.12) $$ \begin{align} \mathcal{F}(\textbf{x}) := \prod_{(r_1,\ldots,r_n)\in\{0,1\}^n} f(x_1^{r_1}, \ldots, x_n^{r_n}), \end{align} $$

so that this is a function of $\textbf {x}=(x_1^0,x_1^1,\ldots ,x_n^0,x_n^1)\in (\mathbb{R}^d)^{2n}$ , and write formally

(2.13) $$ \begin{align} \textrm{d}\textbf{x} := \textrm{d}x_1^0\,\textrm{d}x_1^1 \,\textrm{d}x_2^0\,\textrm{d}x_2^1 \cdots \textrm{d}x_n^0\,\textrm{d}x_n^1. \end{align} $$

We will also denote

(2.14) $$ \begin{align} \mathcal{F}^{(m)}(x) := \prod_{(r_1,\ldots,r_{m-1},r_{m+1}\ldots,r_n)\in\{0,1\}^{n-1}} f(x_1^{r_1}, \ldots, x_{m-1}^{r_{m-1}}, x, x_{m+1}^{r_{m+1}}, \ldots, x_{n}^{r_{n}}), \end{align} $$

for $x\in \mathbb{R}^d$ and $m\in \{1,\ldots ,n\}$ , keeping in mind that $\mathcal {F}^{(m)}(x)$ also depends on other variables than just x.

Formula (2.4) allows us to rewrite

which reveals that $\Theta ^{n,m}_{\gamma _1,\ldots ,\gamma _n}(f)$ is nonnegative and well-defined. This time, we claim the identity

(2.15) $$ \begin{align} \sum_{m=1}^{n} a_m \Theta^{n,m}_{\gamma_1,\ldots,\gamma_n}(f) = 2\pi \|f\|_{\textrm{L}^{2^n}(\mathbb{R}^d)}^{2^n}. \end{align} $$

First, by the product rule for differentiation and the generalized heat equation (2.7), we can write

(2.16)

A consequence of the last display and the fundamental theorem of calculus is

The first limit above equals (allowing a slightly informal usage of $\delta _{\textbf {0}}$ ):

$$ \begin{align*} & \int_{(\mathbb{R}^d)^{2n}} \mathcal{F}(\textbf{x}) \Bigg(\prod_{k=1}^{n}\delta_{\textbf{0}}(x_k^0-x_k^1)\Bigg) \,\textrm{d}\textbf{x} \\ & = \int_{(\mathbb{R}^d)^{n}} f(x_1,\ldots,x_n)^{2^n} \,\textrm{d}x_1 \cdots \textrm{d}x_n = \|f\|_{\textrm{L}^{2^n}(\mathbb{R}^d)}^{2^n}, \end{align*} $$

while the second one is $0$ . This proves (2.15).

Every summand on the left-hand side of (2.15) is nonnegative, so the $\Theta ^{n,m}_{\gamma _1,\ldots ,\gamma _n}(f)$ is clearly bounded by $(2\pi /a_m)\|f\|_{\textrm {L}^{2^n}(\mathbb{R}^d)}^{2^n}$ . Note that this bound is uniform in the parameters $\gamma _1,\ldots ,\gamma _n$ . Identities (2.10) and (2.15) will bound multiscale expressions coming from the study of the error parts in the decomposition (1.4). They can be viewed, respectively, as cheap substitutes for square function estimates and bounds for entangled multilinear singular integrals (mentioned in Section 6.1).

2.3 Conditional expectations

A few concepts from probability theory will be useful in the proofs below. Even though the dyadic setting would be sufficient, the general probabilistic notation makes arguments elegant and concise.

We are working in a fixed probability space $(\Omega ,\mathcal {F},\mathbb{P})$ . The associated expectation is the operator $\mathbb{E}\colon f\mapsto \int _{\Omega } f\,\textrm {d}\mathbb{P}$ defined on $\textrm {L}^1(\Omega ,\mathcal {F},\mathbb{P})$ . Moreover, for any $\sigma $ -algebra $\mathcal {G}\subseteq \mathcal {F}$ , one can construct the operator of conditional expectation with respect to $\mathcal {G}$ as the map

$$ \begin{align*} \textrm{L}^1(\Omega,\mathcal{F},\mathbb{P}) \to \textrm{L}^1(\Omega,\mathcal{G},\mathbb{P}), \quad f\mapsto \mathbb{E}(f|\mathcal{G}) \end{align*} $$

such that

$$ \begin{align*} \int_{A} \mathbb{E}(f|\mathcal{G})\,\textrm{d}\mathbb{P} = \int_{A} f\,\textrm{d}\mathbb{P} \end{align*} $$

for every $f\in \textrm {L}^1(\Omega ,\mathcal {F},\mathbb{P})$ and every $A\in \mathcal {G}$ . The proof of its existence (and uniqueness) can be found in many textbooks on introductory probability theory, such as the one by Durrett [Reference Durrett22]. If $\mathcal {G}$ is generated by a finite partition of $\Omega $ into atoms and if, among these atoms, $A_1,\ldots ,A_N$ have nonzero probability, then we have a simple formula

(2.17)

In words, on each of the atoms, $\mathbb{E}(f|\mathcal {G})$ is constantly equal to the average value of f over that atom. As a very special case of only one atom, we get

(2.18) $$ \begin{align} \mathbb{E}\big(f\big|\{\emptyset,\Omega\}\big) = \mathbb{E}f \quad \textrm{a.s.} \end{align} $$

The following properties of conditional expectations are standard; see [Reference Durrett22, Section 4.1]. The operator $f\mapsto \mathbb{E}(f|\mathcal {G})$ is linear and monotone. If $f\in \textrm {L}^1(\Omega ,\mathcal {F},\mathbb{P})$ and $g\in \textrm {L}^{\infty }(\Omega ,\mathcal {G},\mathbb{P})$ , $\mathcal {G}\subseteq \mathcal {F}$ , then

(2.19) $$ \begin{align} \mathbb{E}(fg|\mathcal{G}) = \mathbb{E}(f|\mathcal{G}) g \quad \textrm{a.s.} \end{align} $$

Next, for $p\in [1,\infty )$ and $f\in \textrm {L}^p(\Omega ,\mathcal {F},\mathbb{P})$ , we have

(2.20) $$ \begin{align} \big|\mathbb{E}(f|\mathcal{G})\big|^p \leq \mathbb{E}\big(|f|^p\big|\mathcal{G}\big) \quad \textrm{a.s.} \end{align} $$

In particular, the conditional expectation is a (not necessarily strict) contraction in the $\textrm {L}^p$ norms. Finally, if $\mathcal {G}$ and $\mathcal {H}$ are two $\sigma $ -algebras such that $\mathcal {F}\supseteq \mathcal {G}\supseteq \mathcal {H}$ , then

(2.21) $$ \begin{align} \mathbb{E}\big(\mathbb{E}(f|\mathcal{G})\big|\mathcal{H}\big) = \mathbb{E}(f|\mathcal{H}) = \mathbb{E}\big(\mathbb{E}(f|\mathcal{H})\big|\mathcal{G}\big) \quad \textrm{a.s.} \end{align} $$

for any integrable function f.

Now, suppose that we are given a filtration $(\mathcal {G}_m)_{m=0}^{\infty }$ of the probability space $(\Omega ,\mathcal {F},\mathbb{P})$ , i.e., a sequence of $\sigma $ -algebras satisfying $\mathcal {G}_0\subseteq \mathcal {G}_1\subseteq \mathcal {G}_2\subseteq \cdots \subseteq \mathcal {F}$ . For shortness, let us denote the conditional expectation operator $f\mapsto \mathbb{E}(f|\mathcal {G}_m)$ simply by $\mathbb{E}_m$ , for each index m. Conditional expectations with respect to filtrations are widely studied in the literature on martingales; see, for instance, [Reference Durrett22, Chapter 4]. We will need the following slightly nonstandard inequality in Sections 3.1 and 5.1. We claim that for any nonnegative bounded measurable function f and for nonnegative integers $m,m_1,m_2,\ldots ,m_n$ satisfying $m\leq \min \{m_1,\ldots ,m_n\}$ , we have

(2.22) $$ \begin{align} \mathbb{E}_m \big( f (\mathbb{E}_{m_1}f) (\mathbb{E}_{m_2}f) \cdots (\mathbb{E}_{m_n}f) \big) \geq (\mathbb{E}_m f)^{n+1} \quad \textrm{a.s.} \end{align} $$

For the proof of (2.22), we can assume, without loss of generality, that $m\leq m_1\leq \cdots \leq m_n$ . We use the mathematical induction on $k=0,1,\ldots ,n-1$ to show

(2.23) $$ \begin{align} \mathbb{E}_m \bigg( \bigg( \prod_{i=1}^{k}\mathbb{E}_{m_i}f \bigg) (\mathbb{E}_{m_{k+1}}f)^{n+1-k} \bigg) \geq (\mathbb{E}_m f)^{n+1} \quad \textrm{a.s.} \end{align} $$

The case $k=n-1$ of (2.23) is precisely (2.22), because we can use (2.21) and (2.19), respectively, to equate their left-hand sides:

$$ \begin{align*} \mathbb{E}_m \bigg( f \bigg( \prod_{i=1}^{n}\mathbb{E}_{m_i}f \bigg) \bigg) = \mathbb{E}_m \mathbb{E}_{m_n} \bigg( f \bigg( \prod_{i=1}^{n}\mathbb{E}_{m_i}f \bigg) \bigg) = \mathbb{E}_m \bigg( \bigg( \prod_{i=1}^{n-1}\mathbb{E}_{m_i}f \bigg) (\mathbb{E}_{m_n}f)^2 \bigg) \quad \textrm{a.s.} \end{align*} $$

The induction basis $k=0$ of (2.23) is just a consequence of (2.20) and (2.21):

$$ \begin{align*} \mathbb{E}_m \big( (\mathbb{E}_{m_1}f)^{n+1} \big) \geq (\mathbb{E}_m \mathbb{E}_{m_1}f)^{n+1} = (\mathbb{E}_m f)^{n+1} \quad \textrm{a.s.} \end{align*} $$

For the induction step, we only need to rewrite and estimate the left-hand side of (2.23) as

$$ \begin{align*} & \mathbb{E}_m \mathbb{E}_{m_{k}} \bigg( \bigg( \prod_{i=1}^{k}\mathbb{E}_{m_i}f \bigg) (\mathbb{E}_{m_{k+1}}f)^{n+1-k} \bigg) = \mathbb{E}_m \bigg( \bigg( \prod_{i=1}^{k}\mathbb{E}_{m_i}f \bigg) \,\mathbb{E}_{m_{k}} \big( (\mathbb{E}_{m_{k+1}}f)^{n+1-k} \big) \bigg) \\ & \geq \mathbb{E}_m \bigg( \bigg( \prod_{i=1}^{k}\mathbb{E}_{m_i}f \bigg) (\mathbb{E}_{m_{k}}\mathbb{E}_{m_{k+1}}f)^{n+1-k} \bigg) = \mathbb{E}_m \bigg( \bigg( \prod_{i=1}^{k-1}\mathbb{E}_{m_i}f \bigg) (\mathbb{E}_{m_{k}}f)^{n+2-k} \bigg) \quad \textrm{a.s.}, \end{align*} $$

where we used properties (2.21), (2.19), (2.20), and (2.21) again, in that order. This completes the inductive proof of (2.23) and thus also confirms (2.22).

An immediate consequence of (2.22) combined with (2.18), (2.21), and (2.20) is a scalar inequality

(2.24) $$ \begin{align} \mathbb{E} \big( f (\mathbb{E}_{m_1}f) (\mathbb{E}_{m_2}f) \cdots (\mathbb{E}_{m_n}f) \big) \geq (\mathbb{E} f)^{n+1} \end{align} $$

for any nonnegative bounded measurable f and nonnegative integers $m_1,m_2,\ldots ,m_n$ . On the other hand, an easy generalization of (2.22) is

(2.25) $$ \begin{align} \mathbb{E}_m \left(f \prod_{i=1}^{n}\mathbb{E}_{m_i}f\right) \geq \left(\overbrace{\prod_{\substack{i\\m_i<m}} \mathbb{E}_{m_i}f}^{N\text{factors}}\right)(\mathbb{E}_m f)^{n+1-N} \quad \textrm{a.s.},\end{align}$$

where m is now a completely arbitrary nonnegative integer and N is the number of indices i satisfying $m_i<m$ . One only needs to use (2.19) to factor out N terms from the left-hand side of (2.25) and then apply (2.22) to the remaining $n+1-N$ terms. Inequality (2.25) will be used to resolve nested conditional expectations when bounding them from below.

3.1 The structured part: proof of (3.2)

For $k=2,\ldots ,n$ , let us write the orthogonal projection of $u_k$ onto $\mathop {\textrm {span}}(\{u_1,\ldots ,u_{k-1}\})$ explicitly as

$$ \begin{align*} \beta_{k,1} u_1 + \beta_{k,2} u_2 + \cdots + \beta_{k,k-1} u_{k-1} \end{align*} $$

for some scalars $\beta _{k,1},\beta _{k,2},\ldots ,\beta _{k,k-1}\in \mathbb{R}$ . Moreover, denote

$$ \begin{align*} d_k := \mathop{\textrm{dist}}\big(u_k, \mathop{\textrm{span}}(\{u_1,\ldots,u_{k-1}\})\big)>0. \end{align*} $$

By the symmetry present in $\mathcal {N}^{\varepsilon }_{\lambda }$ when integrating over all systems $(y_1,\ldots ,y_n)=(\lambda ^{a_1} b_1 U u_1,\ldots ,\lambda ^{a_n} b_n U u_n)$ , we can rewrite the smoother counting form as in [Reference Bourgain4]:

Here, $\sigma $ denotes the spherical measure supported on $\mathbb{S}^n\subset \mathbb{R}^{n+1}$ , while, for $k=2,\ldots ,n$ , we write $\sigma ^{y_1,\ldots ,y_{k-1}}$ for the $(n-k+1)$ -dimensional spherical measure supported on the sphere that is centered at

$$ \begin{align*} \beta_{k,1} \frac{y_1}{|y_1|} + \beta_{k,2} \frac{y_2}{|y_2|} + \cdots + \beta_{k,k-1} \frac{y_{k-1}}{|y_{k-1}|}, \end{align*} $$

has radius $d_k$ , and belongs to an $(n-k+2)$ -dimensional plane in $\mathbb{R}^{n+1}$ orthogonal to $\mathop {\textrm {span}}(\{y_1,\ldots ,y_{k-1}\})$ . We normalize these measures in a way that each of them has its total mass equal to $1$ and we view them as being defined on all Borel subsets of $\mathbb{R}^{n+1}$ , even though their supports are lower-dimensional sets contained in the standard unit sphere $\mathbb{S}^n$ . In addition, note that the $n+1$ integrals over $\mathbb{R}^{n+1}$ are nested and the integration is performed in the order from the innermost one to the outermost one.

Here, we concentrate on the smoothest case $\varepsilon =1$ . It is easy to observe that for any probability measure $\nu $ supported inside the standard unit sphere $\mathbb{S}^n\subset \mathbb{R}^{n+1}$ , we can bound pointwise

where

The integral in $y_n$ in $\mathcal {N}^{1}_{\lambda }(f)$ can be recognized as a triple convolution,

Now, we do the same to the integral in $y_{n-1}$ , etc. Repeating this process n times, we end up with

$$ \begin{align*} \mathcal{N}^{1}_{\lambda}(f) \gtrsim \int_{(\mathbb{R}^{d})^{n+1}} f(x) \left( \prod_{k=1}^{n} (f\ast\varphi_{\lambda^{a_k}b_k})(x) \right) \,\textrm{d}x. \end{align*} $$

Once we know that

(3.5)

holds for every $t_1,t_2,\ldots ,t_n\in (0,R/2]$ and every measurable function $f\colon [0,R]^{d}\to [0,1]$ , then (3.2) will follow simply by choosing $t_k=\lambda ^{a_k}b_k$ , for $k=1,\ldots ,n$ , and taking $d=n+1$ and

. However, (3.5) is just a straightforward generalization of [Reference Durcik, Guo and Roos14, Lemma 2.1] by Durcik, Guo, and Roos, which, in turn, is an elaboration of Bourgain’s [Reference Bourgain3, Lemma 6] (stated there without proof). The paper [Reference Durcik, Guo and Roos14] established the special case $d=1$ , $n=2$ , $R=1$ of (3.5) elegantly, by dominating convolutions $f\ast \varphi _{t_k}$ pointwise from below by a dyadic martingale. We are about to reuse this idea to give a quick proof of (3.5) in general. Even if this generalization is quite clear and expected, we still choose to be sufficiently detailed, because we will have to argue similarly in an even greater generality in Section 5.1.

Let us turn $[0,R]^d$ into a probability space by taking $\mathbb{P}$ to be the Lebesgue measure normalized by the factor $R^{-d}$ . The dyadic filtration $(\mathcal {G}_m)_{m=0}^{\infty }$ of $[0,R]^d$ is obtained by defining $\mathcal {G}_m$ to be a finite $\sigma $ -algebra (i.e., a finite algebra of sets) generated with $2^{dm}$ congruent cubes of side length $2^{-m}R$ that partition $[0,R]^d$ . Let $m_k$ be the smallest nonnegative integer such that $2^{-m_k}R d^{1/2} < t_k$ . By this choice, a cube of side length $2^{-m_k}R$ has diameter smaller than $t_k$ , so, if it contains a point x, then it remains fully inside the ball $B(x,t_k)$ . Because of this and equation (2.17), we clearly have

$$ \begin{align*} f\ast\varphi_{t_k} \gtrsim \mathbb{E}(f|\mathcal{G}_{m_k}) = \mathbb{E}_{m_k}f \quad \textrm{a.e.} \end{align*} $$

Now, (3.5) becomes a consequence of the probabilistic inequality (2.24).

3.2 The error part: proof of (3.3)

We will keep writing f interchangeably with

, where B is as before. Using the product rule for differentiation and applying the generalized heat equation (2.7), we get

Thus, by the fundamental theorem of calculus, for any $0<\alpha <\beta $ , the difference $\mathcal {N}^{\alpha }_{\lambda }(f)-\mathcal {N}^{\beta }_{\lambda }(f)$ can be written as

$$ \begin{align*} \sum_{m=1}^{n}\mathcal{L}_{\lambda}^{\alpha,\beta,m}(f), \end{align*} $$

where

(3.6)

By symmetry, it is sufficient to consider the case $m=n$ , and the same reasoning as in the previous subsection gives

(3.7)

The following arguments will not depend on the dimension of the ambient space $\mathbb{R}^{n+1}$ . We will emphasize this fact by writing it as $\mathbb{R}^d$ and remembering that $d={n+1\geq 2}$ . This will also be convenient for recycling the same computation in Section 5. Here, we need to control

$$ \begin{align*} \sum_{j=1}^{J}|\mathcal{L}_{\lambda_j}^{\varepsilon,1,n}(f)|. \end{align*} $$

Set

(3.8) $$ \begin{align} \theta:=10^{-1/a_n}e^{-1}. \end{align} $$

Let us begin the estimation by multiplying the inner integral of $\mathcal {L}_{\lambda _j}^{\varepsilon ,1,n}(f)$ with

(3.9) $$ \begin{align} \int_{\theta t\lambda_j}^{e\theta t\lambda_j} \,\frac{\textrm{d}s}{s} = 1 \end{align} $$

and rewriting the whole expression as

For

(3.10) $$ \begin{align} j\in\{1,\ldots,J\},\quad \varepsilon\leq t\leq 1,\quad \theta t\lambda_j\leq s\leq e\theta t\lambda_j, \end{align} $$

denote

$$ \begin{align*} r = r(j,s,t) := \big((t\lambda_j)^{2a_n}-s^{2a_n}\big)^{1/2a_n}, \end{align*} $$

remembering that r depends on j, s, and t, and observing that $s\sim t\lambda _j \sim r$ . Convolution identity (2.4) gives

so that, introducing the integration variable y,

Take $\nu $ to be an arbitrary probability measure supported on a subset of the standard unit sphere in $\mathbb{R}^{d}$ , even though the following reasoning will only be applied with $\nu =\sigma ^{y_1,\ldots ,y_{n-1}}$ in the current section. Expanding the convolution according to its definition and using (2.8), (2.9) enables us to estimate

so, dilating by $s^{a_n}b_n$ , we also get

(3.11)

for $j,t,s$ as in (3.10) and for $x\in \mathbb{R}^d$ . Consequently,

In the next step, we realize that $y_{n-1}$ only appears in one of the factors of the integrand and as one of the integration variables in the last expression. For that reason, we can rewrite the integral in $y_{n-1}$ as a convolution and get

(3.12)

Now, we observe that, by (2.8) and (2.9),

for a general $\nu $ as before and for any $k\in \{1,\ldots ,n-1\}$ . Rescaling this by $s^{a_k}b_k$ yields

(3.13)

Convolutions with dilates of

can be controlled pointwise by the Hardy–Littlewood maximal function $\textrm {M}$ (see [Reference Stein and Weiss54, Formula (3.9)]), so (3.13) implies

(3.14)

We use (3.14) with $k=n-1$ and $\nu =\sigma ^{y_1,\ldots ,y_{n-2}}_{-1}$ to further bound the expression (3.12). Repeating the previous step $n-2$ times more, i.e., for $k=n-2,\ldots ,2,1$ , we end up with

where a is the sum of the numbers $a_k$ , as in (3.1). Observing $f\leq \textrm {M}f$ and using the Cauchy–Schwarz inequality, we get

Now, we sum in j and observe that, for each fixed $t\in [\varepsilon ,1]$ , the intervals $[\theta t\lambda _j,e\theta t\lambda _j]$ , $j=1,\ldots ,J$ , cover any fixed point from $(0,\infty )$ at most two times. By one last application of the Cauchy–Schwarz inequality (this time for the sum in j) followed by boundedness of $\textrm {M}$ on $\textrm {L}^{2n}(\mathbb{R}^d)$ , we conclude

That way, identity (2.10) and trivial estimates $\|f\|_{\textrm {L}^{2n}}\leq R^{d/2n}$ , $\|f\|_{\textrm {L}^{2}}\leq R^{d/2}$ , together with $d=n+1$ , finish the proof of (3.3).

3.3 The uniform part: proof of (3.4)

Take $0<\vartheta <\varepsilon $ and a measurable function $f\colon [0,R]^{n+1}\to [0,1]$ . From the previous subsection, we know that $\mathcal {N}^{\vartheta }_{\lambda }(f)-\mathcal {N}^{\varepsilon }_{\lambda }(f)$ is the sum of $\mathcal {L}_{\lambda }^{\vartheta ,\varepsilon ,m}(f)$ over $m=1,\ldots ,n$ . Once again, by symmetry, it is sufficient to consider the case $m=n$ , when we have the representation (3.7). Using $0\leq f\leq 1$ and the Cauchy–Schwarz inequality in the variable x, we get

Another application of the Cauchy–Schwarz inequality, this time in the variables $y_1,\ldots ,y_{n-1}$ , followed by Plancherel’s identity, leads to

Observe that $\sigma ^{y_1,\ldots ,y_{n-1}}$ is just the circle measure inside a two-dimensional plane orthogonal to $\mathop {\textrm {span}}(\{y_1,\ldots ,y_{n-1}\})$ as long as $y_1,\ldots ,y_{n-1}$ are linearly independent, which happens almost surely. Using the decay estimate (2.2), we get

$$ \begin{align*} \big|\widehat{\sigma}^{y_1,\ldots,y_{n-1}}(\zeta)\big| \lesssim \mathop{\textrm{dist}}\big(\zeta,\mathop{\textrm{span}}(\{y_1,\ldots,y_{n-1}\})\big)^{-1/2}, \end{align*} $$

for every $\zeta \in \mathbb{R}^{n+1}$ . By rescaling $y_1,\ldots ,y_{n-1}$ , this, in turn, gives

(3.15)

where

$$ \begin{align*} \mathcal{I}(\zeta) := \int_{(\mathbb{R}^{n+1})^{n-1}} & \mathop{\textrm{dist}}\big(\zeta,\mathop{\textrm{span}}(\{y_1,\ldots,y_{n-1}\})\big)^{-1} \\ & \textrm{d}\sigma^{y_1,\ldots,y_{n-2}}(y_{n-1}) \cdots \textrm{d}\sigma^{y_1}(y_2) \,\textrm{d}\sigma(y_1). \end{align*} $$

Integrating over all rotations $U\in \textrm {SO}(n+1,\mathbb{R})$ and taking $(y_1,\ldots ,y_{n-1})=(U u_1,\ldots ,U u_{n-1})$ , we want to conclude

(3.16) $$ \begin{align} \mathcal{I}(\zeta) \lesssim |\zeta|^{-1}. \end{align} $$

We could proceed as in [Reference Bourgain4] or [Reference Lyall and Magyar45], but a more elementary argument is also available. Instead of integrating over U, we can rather take $y_1,\ldots ,y_{n-1}$ to be fixed unit vectors that span the coordinate plane $\mathbb{R}^{n-1}\times \{(0,0)\}$ and integrate over all possible directions of $\zeta $ determined by the standard unit sphere $\mathbb{S}^{n}$ in $\mathbb{R}^{n+1}$ . This will yield the same result for $\mathcal {I}(\zeta )$ up to a constant. Writing $\zeta $ in the $(n+1)$ -dimensional spherical coordinates,

$$ \begin{align*} \zeta = |\zeta| (\cos\phi_1, \sin\phi_1\cos\phi_2, \ldots, \sin\phi_1\cdots\sin\phi_{n-1}\cos\phi_n, \sin\phi_1\cdots\sin\phi_n), \end{align*} $$

$\phi _1,\ldots ,\phi _{n-1}\in [0,\pi ]$ , $\phi _n\in [0,2\pi )$ , and observing

$$ \begin{align*} \mathop{\textrm{dist}}\big(\zeta,\mathop{\textrm{span}}(\{y_1,\ldots,y_{n-1}\})\big) = |\zeta| \sin\phi_1 \cdots \sin\phi_{n-1}, \end{align*} $$

we can estimate $\mathcal {I}(\zeta )$ by a constant multiple of

$$ \begin{align*} |\zeta|^{-1} \int_{[0,\pi]^{n-1}\times[0,2\pi)} \frac{\sin^{n-1}\phi_1\sin^{n-2}\phi_2 \cdots \sin\phi_{n-1} \,\textrm{d}\phi_1\,\textrm{d}\phi_2\cdots\textrm{d}\phi_n}{\sin\phi_1\sin\phi_2 \cdots \sin\phi_{n-1}} \lesssim |\zeta|^{-1}. \end{align*} $$

This establishes (3.16) and thus also gives

(3.17)

for $\zeta \in \mathbb{R}^{n+1}$ and $t>0$ . Substituting $\zeta =\lambda ^{a_n}b_n\xi $ , plugging (3.17) into (3.15), and using Plancherel’s theorem again, we finally get

$$ \begin{align*} \big|\mathcal{L}_{\lambda}^{\vartheta,\varepsilon,n}(f)\big| \lesssim \|f\|_{\textrm{L}^2(\mathbb{R}^{n+1})}^2 \int_{\vartheta}^{\varepsilon} t^{a_n/2} \,\frac{\textrm{d}t}{t} \lesssim R^{n+1}\varepsilon^{a_n/2}. \end{align*} $$

This proves (3.4).

Arguments in this subsection reveal irrelevance of the nature of dilations for this part of the proof, which is the reason why we were able to proceed in a similar way as Bourgain [Reference Bourgain4] or Lyall and Magyar [Reference Lyall and Magyar45, Reference Lyall and Magyar47].

4.1 The structured part: proof of (4.1)

We partition a major part of the cube $([0,R]^2)^n$ into the collection of rectangular boxes $Q_1 \times \cdots \times Q_n$ , where each $Q_k$ is a square of the form $[l \lambda ^{a_k}b_k,{(l+1)} \lambda ^{a_k}b_k)\times [l' \lambda ^{a_k}b_k,(l'+1) \lambda ^{a_k}b_k)$ for some integers $0\leq l,l'\leq \lfloor \lambda ^{-a_k}b_k^{-1}R\rfloor -1$ . Each of these boxes has measure $(\lambda ^{a_k}b_k)^2$ and their total number is clearly comparable to $R^{2n}\prod _{k=1}^{n}\lambda ^{-2a_k}$ .

When estimating

from below, we restrict the domain of integration with additional constraints requiring that $x_k^0$ , $x_k^1$ lie in the same square $Q_k$ mentioned above, for each $k=1,\ldots ,n$ . Using the box–Gowers–Cauchy–Schwarz inequality [Reference Green and Tao30, Reference Shkredov50, Reference Tao57], or simply by several applications of the ordinary Cauchy–Schwarz inequality, we obtain

Applying this to each of the choices of $Q_1\times \cdots \times Q_n$ , using

and recalling $|Q_k|\sim \lambda ^{2a_k}$ , we get

so discrete Jensen’s inequality for the power function $t\mapsto t^{2^n}$ gives (4.1).

4.2 The error part: proof of (4.2)

Using the product rule and the generalized heat equation (2.7), we obtain

for $\lambda>0$ , and $y_1,\ldots ,y_n\in \mathbb{R}^2$ . By the fundamental theorem of calculus, the difference $\mathcal {N}^{\alpha }_{\lambda }(f)-\mathcal {N}^{\beta }_{\lambda }(f)$ is, for any $0<\alpha <\beta $ , equal to the sum of n terms given by

for $m=1,\ldots ,n$ . By symmetry, it is sufficient to fix $m=n$ and prove an upper bound for

(4.4) $$ \begin{align} \sum_{j=1}^{J}|\mathcal{L}_{\lambda_j}^{\varepsilon,1,n}(f)|. \end{align} $$

Take $\theta $ as in (3.8) and use (3.9), transforming $\mathcal {L}_{\lambda _j}^{\varepsilon ,1,n}(f)$ into

For $j,s,t$ as in (3.10), this time, we denote

$$ \begin{align*} r = r(j,s,t) := \big((t\lambda_j)^{2a_n}-2s^{2a_n}\big)^{1/2a_n} \end{align*} $$

and observe, once again, that $s\sim t\lambda _j \sim r$ . From (2.4) and (2.5), we see

so (4.4) is at most a constant times

The same computation leading to (3.13) still applies, so we can use this formula again, this time with $d=2$ and $\nu =\sigma $ :

(4.5)

for $k=1,\ldots ,n-1$ . Very similarly, imitating the proof of (3.11), we get

(4.6)

Estimates (4.5) and (4.6) bound (4.4) by a constant multiple of

Using the Cauchy–Schwarz inequality in all variables other than $x_n^0$ and $x_n^1$ , observing that the two obtained terms are equal, expanding out the square, and collapsing back the convolution using identity (2.4), we see that this expression is at most

(4.7)

For a fixed t and varying j the number of times, the intervals $[\theta t\lambda _j,e\theta t\lambda _j]$ cover any fixed point from $(0,\infty )$ is at most a constant depending only on $a_n$ . Using this observation in connection with (4.7) and recognizing the inner expression as (2.11), we finally obtain

$$ \begin{align*} \sum_{j=1}^{J}|\mathcal{L}_{\lambda_j}^{\varepsilon,1,n}(f)| \lesssim \varepsilon^{-3a} \log(1/\varepsilon) \int_{[1,\infty)^n} \Theta^{n,n}_{\gamma_1,\ldots,\gamma_{n-1},2^{1/2}}(f) \,\frac{\textrm{d}\gamma_1}{\gamma_1^2} \cdots \frac{\textrm{d}\gamma_n}{\gamma_n^2}. \end{align*} $$

Thus, the bound for $\Theta ^{n,n}_{\gamma _1,\ldots ,\gamma _{n-1},2^{1/2}}(f)$ coming from identity (2.15) completes the proof of (4.2).

4.3 The uniform part: proof of (4.3)

Take $0<\vartheta <\varepsilon \leq 1$ . In order to control $\mathcal {N}^{\vartheta }_{\lambda }(f)-\mathcal {N}^{\varepsilon }_{\lambda }(f)$ , we need to bound $|\mathcal {L}_{\lambda }^{\vartheta ,\varepsilon ,m}(f)|$ for $m=1,\ldots ,n$ .

Once we fix m and a number t such that $\vartheta \leq t\leq \varepsilon $ , Plancherel’s theorem yields

(4.8)

We use the decay estimate (2.1) for $\widehat {\sigma }$ to conclude

(4.9)

for each $\zeta \in \mathbb{R}^2$ . Then, we simply take $\zeta =\lambda ^{a_m}b_m\xi $ in (4.9) and combine it with (4.8). By another application of Plancherel’s theorem, we see that (4.8) is at most a constant times

$$ \begin{align*} t^{a_m/2} \|\mathcal{F}^{(m)}\|_{\textrm{L}^2(\mathbb{R}^2)}^2 \leq t^{a_m/2} R^2. \end{align*} $$

Recall that

. Integrating in all of the remaining variables, we get

and thus also

It remains to send $\vartheta \to 0^+$ .

5.1 The structured part: proof of (5.1)

Denote $t_k:=\lambda ^{a_k}b_k$ for each $k\in E$ . Just as in Section 3.1, turn $[0,R]^2$ into a probability space and define the dyadic filtration $(\mathcal {G}_m)_{m=0}^{\infty }$ on it.

Let us declare arbitrary particular vertex $v_{\mathcal {T}}\in V$ as the tree root or the tree top. We can imagine that the tree $\mathcal {T}$ is “hanged” upside down by holding it by the vertex $v_{\mathcal {T}}$ . This naturally yields to relations is a child of and is a parent of on the set of vertices V. For any $v\in V$ , let $\mathcal {T}_v=(V_v,E_v)$ be the subtree of $\mathcal {T}$ consisting of v as its root and of all descendants of v with respect to $\mathcal {T}$ .

For any tree $\mathcal {T}=(V,E)$ with root $v_{\mathcal {T}}$ , we define the following function of only one variable $x_{v_{\mathcal {T}}}$ :

By induction on the tree structure, we are going to show that for any such tree $\mathcal {T}$ , there exist nonnegative integers $m_1,m_2,\ldots ,m_{|E|}$ (depending also on the numbers $t_k$ ) such that the following pointwise inequality holds:

(5.4) $$ \begin{align} \mathcal{A}_{\mathcal{T}}f \geq f (\mathbb{E}_{m_1}f) (\mathbb{E}_{m_2}f) \cdots (\mathbb{E}_{m_{|E|}}f) \quad \textrm{a.e.} \end{align} $$

for any nonnegative bounded measurable function f. Indeed, the induction basis is the case when the tree $\mathcal {T}$ has only one vertex and no edges, and then (5.4) reduces to a trivial inequality $f\geq f$ . For the induction step, let $k_1,\ldots ,k_s$ be all edges incident with the root $v_{\mathcal {T}}$ and let $v_1,\ldots ,v_s$ , respectively, be the corresponding children of $v_{\mathcal {T}}$ . Clearly,

i.e.,

For each $i\in \{1,\ldots ,s\}$ , let $l_i$ be the smallest nonnegative integer such that $2^{-{l_i}}R<t_{k_i}$ . That way, we get

(5.5) $$ \begin{align} \mathcal{A}_{\mathcal{T}}f \gtrsim f \ \prod_{i=1}^{s} \mathbb{E}_{l_i} \mathcal{A}_{\mathcal{T}_{v_i}}f \quad \textrm{a.e.} \end{align} $$

Let us apply the induction hypothesis of (5.4) to each of the subtrees $\mathcal {T}_{v_i}$ and then resolve the nested conditional expectations using inequality (2.25). Plugging these into (5.5) and multiplying them for all i, we conclude that $\mathcal {A}_{\mathcal {T}}f$ also satisfies a lower bound of the form (5.4) for some positive integers $m_1,\ldots ,m_{|E|}$ , which we do not need to make explicit. This finalizes the induction step and establishes the claim (5.4).

Combining (5.4) with (2.24) and applying it to

, we finally conclude

which is precisely (5.1).

5.2 The error part: proof of (5.2)

Generalized heat equation (2.7) and the fundamental theorem of calculus allow us to rewrite the difference $\mathcal {N}^{\varepsilon }_{\lambda }(f)-\mathcal {N}^{1}_{\lambda }(f)$ as

$$ \begin{align*} \sum_{m\in E}\mathcal{L}_{\lambda}^{\varepsilon,1,m}(f), \end{align*} $$

where, this time,

The proof of (3.3) presented in Section 3.2 does not recognize any graph structure at all. Thus, the same proof carries over here, replacing measures $\sigma ^{y_1,\ldots ,y_{k-1}}$ (associated with varying spheres of different dimensions) always with the same circle measure $\sigma $ .

5.3 The uniform part: proof of (5.3)

The proof of (3.4) given in Section 3.3 does not see any graph structure either. It keeps cancellation at only one crucial place, i.e., associated with only one chosen edge. For this reason, we can proceed with only very minor modifications of the arguments from either Section 3.3 or 4.3.

6.1 Multilinear anisotropic singular integrals

As we have already said, a part of the motivation behind this work lies in encouraging connections between the techniques from multilinear harmonic analysis and the problems on combinatorics of the Euclidean space. In this subsection, we want to single out singular integrals that can naturally be associated with problems studied in Theorems 1–3. This correspondence should not be understood literally, but rather on the level of heuristics and methodology. After all, the proofs of the above theorems did not use any estimates for the integral operators that will be mentioned here. In Sections 3–5, we preferred to use ad hoc shortcuts in the form of the Gaussian domination estimate (2.9) and identities (2.10) and (2.15). However, it would be a pity not to mention analytical counterparts of these problems, if for nothing else, then to point out where to look for the techniques for handling similar or more general combinatorial questions.

First, for a tuple $\textbf {f}=(f_0,f_1,\ldots ,f_n)$ of measurable functions on $\mathbb{R}^d$ and for a certain singular kernel K, we define the translation-invariant multilinear form

(6.1) $$ \begin{align} \Lambda_{K}(\textbf{f}) := \textrm{p.v.} \int_{(\mathbb{R}^{d})^{n+1}} K(x_1-x_0,\ldots,x_n-x_0) \,\left(\prod_{k=0}^{n} f_k(x_k)\,\textrm{d}x_k \right). \end{align} $$

From either (3.6) or (3.7), we are naturally lead to the study of multilinear singular integral forms (6.1). For instance, if we were allowed to replace the measures $\mu $ or $\sigma ^{y_1,\ldots ,y_{k-1}}$ by the Dirac delta measure at the origin, we would obtain the kernel K given by

(6.2)

The study of multilinear singular integrals (6.1) with Calderón–Zygmund kernels K was initiated by Coifman and Meyer in the 1970s (for instance, see [Reference Coifman and Meyer7–Reference Coifman and Meyer9]), while a more systematic treatment was first given by Grafakos and Torres [Reference Grafakos and Torres29]. However, (6.2) is not the most usual singular kernel. It satisfies the Calderón–Zygmund estimates with respect to quasinorms associated with anisotropic power-type dilations (1.1). Such more general dilation structures have been studied by Stein and Wainger [Reference Stein and Wainger53]. Strictly speaking, classical results, such as those from [Reference Grafakos and Torres29], do not apply to kernels (6.2), but the same techniques do, and the proofs can be repeated mutatis mutandis (also see [Reference Ghosh, Bhojak, Mohanty and Shrivastava28]). Unsurprisingly, Section 3.2 already comes quite close to proving some $\textrm {L}^p$ bounds for (6.1), with its use of maximal and square function estimates, the trivial case of the latter being identity (2.10). However, it is conceivable that, in the future, one reduces a certain combinatorial problem to the study of (6.1) for quite different kernels K.

Next, for a tuple of measurable functions $\textbf {f}=(f_{r_1,\ldots ,r_n})_{(r_1,\ldots ,r_n)\in \{0,1\}^n}$ on $(\mathbb{R}^d)^{n}$ indexed by points from $\{0,1\}^n$ and for a singular kernel K, we define

(6.3) $$ \begin{align} \Theta_{K}(\textbf{f}) := \textrm{p.v.} \int_{(\mathbb{R}^d)^{2n}} & \left( \prod_{(r_1,\ldots,r_n)\in\{0,1\}^n} f_{r_1,\ldots,r_n}(x_1 + r_1 y_1, \ldots, x_n + r_n y_n) \right) \nonumber \\ & \times \,K(y_1,\ldots,y_n) \left( \prod_{k=1}^{n}\textrm{d}x_k\,\textrm{d}y_k \right) . \end{align} $$

These forms have also been studied extensively when K is the usual Calderón–Zygmund kernel, i.e., satisfying Calderón–Zygmund estimates with respect to the Euclidean metric. The first $\textrm {L}^p$ bounds for the forms (6.3) were established in the case $d=1$ and $n=2$ by Durcik [Reference Durcik12, Reference Durcik13]. Prior to that, their dyadic model has been investigated by the present author [Reference Kovač40, Reference Kovač41] and by Thiele and the present author [Reference Kovač and Thiele43]. Estimates for rather general “entangled” multilinear singular integrals of the above type (i.e., with cubical structure) have been shown recently by Durcik and Thiele [Reference Durcik and Thiele21]. It is also interesting to mention that the study of multiparameter variants of these objects has only recently been initiated by Bernicot and Durcik [Reference Bernicot and Durcik2]. It could be interesting to study (6.3) for more general dilation structures, i.e., when K is a generalized Calderón–Zygmund kernel, such as in the case (6.2), which is relevant here. In this context, a similar but still different object has been studied by Škreb and the present author [Reference Kovač and Škreb42]. Some generalizations of the result by Durcik [Reference Durcik12] are straightforward: the single $\textrm {L}^{2^n}\times \cdots \times \textrm {L}^{2^n}$ bound for (6.3) can be extracted easily from the presented proof of Theorem 2 combined with a cone decomposition of the kernel. It is quite likely that $\Theta _K$ still satisfies the same $\textrm {L}^p$ estimates from [Reference Durcik13], but we do not attempt such generalizations here.

Finally, after Theorem 3, a closely related topic of further investigation could be establishing estimates for entangled multilinear singular integrals associated with bipartite graphs or r-partite r-regular hypergraphs. Dyadic models of these problems are significantly easier: they have already been handled quite generally by the present author [Reference Kovač40] and Stipčić [Reference Stipčić55], respectively. Otherwise, the only cases and variants of entangled singular integral forms studied so far are the so-called “twisted paraproduct operator” [Reference Durcik and Roos20, Reference Kovač41], the operators with cubical structure [Reference Durcik12, Reference Durcik13, Reference Durcik and Thiele21], and the operators that resemble multilinear Hilbert transforms [Reference Durcik and Kovač16, Reference Durcik, Kovač and Thiele19, Reference Tao58, Reference Zorin-Kranich60].

6.2 Comments on the anisotropic setting

Configurations generated by anisotropic power-type dilations (1.1) have not been studied prior to this work. We found this setting sufficiently interesting, because it fits nicely to the general method explained in Section 1.3. Still, the results formulated in Section 1.2 are far from being definite.

In relation to simplices, the fact that we are dilating by power functions $\lambda \mapsto \lambda ^a b$ plays little role in the proof of Theorem 1. Structured and uniform parts are handled much more generally, while the control of the error part essentially requires control of the multiscale objects of the form (6.1). Perhaps by studying multilinear singular integral forms (6.1) more closely, one can hope for further generalizations of Theorem 1. In the present paper, the setting (1.1) was just very convenient.

The same comment does not apply to rectangular boxes. So far, the only known way of handing entangled singular forms (6.3) is quite rigid and uses the same steps as those employed in the proof of Theorem 2. All of the papers [Reference Durcik12, Reference Durcik13, Reference Durcik, Kovač, Škreb and Thiele18, Reference Durcik, Kovač and Thiele19, Reference Durcik and Thiele21] used domination by Gaussians (2.9) and some form of the product rule (2.16), or equivalently, integration by parts. This leaves less flexibility, so algebraic properties, such as (2.3)–(2.7), are now crucial in the proof.

Finally, one might wonder why we do not study general distance graphs in Theorem 3. When vertices of the graph (or points of the configuration) are added one by one, it is desirable that the location of the new vertex depends on the locations of previous vertices and $\lambda $ in a reasonably simple way. In general, dilations (1.1) with different exponents $a_k$ can make this dependence “very nonlinear.” This complication arises already when the distance graph in question is a cycle of length $3$ . Suppose that, for all sufficiently large $\lambda $ , we want to find points $x_0,x_1,x_2\in A\subseteq \mathbb{R}^d$ such that $|x_0-x_1|=\lambda $ , $|x_0-x_2|=|x_1-x_2|=\lambda ^2$ . Once $x_0$ and $x_1$ are located, the third point $x_2$ has to lie on the hyperplane H orthogonal to the segment joining $x_0$ and $x_1$ and passing through its midpoint $(x_0+x_1)/2$ . However, the distance from $x_3$ to $(x_0+x_1)/2$ needs to be $\lambda \sqrt {4\lambda ^2-1}/2$ , and we would need to stretch the corresponding spherical measure by this radical factor. For a similar reason in Theorem 1, we do not determine a simplex solely by lengths of its edges, but rather fix the angles between all pairs of its edges meeting at a single vertex.