2.1 Basic notation

We denote $\mathbb {Z}_+:=\{1, 2, \ldots \}$ and $\mathbb {N}:=\{0,1,2,\ldots \}$ . For $d\in \mathbb {Z}_+$ , the sets $\mathbb {Z}^d$ , $\mathbb {R}^d$ , $\mathbb {C}^d$ , and $\mathbb {T}^d:=\mathbb {R}^d/\mathbb {Z}^d$ have a standard meaning. For any $x\in \mathbb {R}$ , we will use the floor function

$$ \begin{align*} \lfloor x \rfloor: = \max\{ n \in \mathbb{Z} : n \le x \}. \end{align*} $$

We denote $\mathbb {R}_+:=(0, \infty )$ and $\mathbb {X}:=[1, \infty )$ , and for every $N\in \mathbb {R}_+$ , we set

$$ \begin{align*} [N]:=(0, N]\cap\mathbb{Z}=\{1, \ldots, \lfloor N\rfloor\}, \end{align*} $$

and we will also write

$$ \begin{align*} \mathbb{N}_{\le N}:= [0, N]\cap\mathbb{N} \quad \text{and} \quad \mathbb{N}_{< N}:= [0, N)\cap\mathbb{N},\end{align*} $$

$$ \begin{align*} \mathbb{N}_{\ge N}:= [N, \infty)\cap\mathbb{N} \quad \text{and} \quad \mathbb{N}_{> N}:= (N, \infty)\cap\mathbb{N}. \end{align*} $$

For $\tau \in (0, 1)$ and $u\in \mathbb {Z}_+$ , we define sets

$$ \begin{align*} \mathbb{D}_{\tau}:=\{2^{n^\tau}:n\in\mathbb{N}\}\quad\text{and}\quad 2^{u\mathbb{Z}_+}:=\{2^{un}\colon n\in\mathbb{Z}_+\}. \end{align*} $$

We use to denote the indicator function of a set A. If S is a statement, we write to denote its indicator, equal to $1$ if S is true and $0$ if S is false. For instance, .

For two non-negative quantities $A, B$ , we write $A \lesssim B$ if there is an absolute constant $C>0$ such that $A\le CB$ . However, the constant C may change from line to line. If ${A \lesssim B\lesssim A}$ , then we write $A \simeq B$ . We will write $\lesssim _{\delta }$ or $\simeq _{\delta }$ to indicate that the implicit constant depends on $\delta $ . For two functions $f:X\to \mathbb {C}$ and $g:X\to [0, \infty )$ , we write $f = \mathcal {O}(g)$ if there exists a constant $C>0$ such that $|f(x)| \le C g(x)$ for all $x\in X$ . We will write $f = \mathcal {O}_{\delta }(g)$ if the implicit constant depends on $\delta $ .

2.2 Euclidean spaces

The standard inner product, the corresponding Euclidean norm, and the maximum norm on $\mathbb {R}^d$ are denoted respectively, for any $x=(x_1,\ldots , x_d)$ , $\xi =(\xi _1, \ldots , \xi _d)\in \mathbb {R}^d$ , by

$$ \begin{align*} x\cdot\xi:=\sum_{k=1}^dx_k\xi_k, \quad \lvert {x} \rvert := \lvert {x} \rvert _2:=\sqrt{x\cdot x},\quad \text{and} \quad |x|_{\infty}:=\max_{k\in[d]}|x_k|. \end{align*} $$

For any multi-index $\gamma =(\gamma _1,\ldots ,\gamma _k)\in \mathbb {N}^k$ , by abuse of notation, we will write $|\gamma |:=\gamma _1+\cdots +\gamma _k$ . This will never cause confusion since the multi-indices will always be denoted by Greek letters.

Throughout the paper, the d-dimensional torus $\mathbb {T}^d$ is a priori endowed with the periodic norm

(2.1) $$ \begin{align} \lVert {\xi} \rVert :=\bigg(\sum_{k=1}^d \lVert {\xi_k} \rVert ^2\bigg)^{1/2} \quad \text{for } \xi=(\xi_1,\ldots,\xi_d)\in\mathbb{T}^d, \end{align} $$

where $ \left \lVert {\xi _k} \right \rVert =\operatorname {\mathrm {dist}}(\xi _k, \mathbb {Z})$ for all $\xi _k\in \mathbb {T}$ and $k\in [d]$ . Identifying $\mathbb {T}^d$ with $[-1/2, 1/2)^d$ , we see that the norm $ \left \lVert {\:\cdot \:} \right \rVert $ coincides with the Euclidean norm $ \left \lvert {\:\cdot \:} \right \rvert $ restricted to $[-1/2, 1/2)^d$ .

2.3 Function spaces

In this paper, all vector spaces will be defined over $\mathbb {C}$ . The triple $(X, \mathcal B(X), \mu )$ denotes a measure space X with a $\sigma $ -algebra $\mathcal B(X)$ and a $\sigma $ -finite measure $\mu $ . The space of all $\mu $ -measurable functions $f:X\to \mathbb {C}$ will be denoted by $L^0(X)$ . The space of all functions in $L^0(X)$ whose modulus is integrable with pth power is denoted by $L^p(X)$ for $p\in (0, \infty )$ , whereas $L^{\infty }(X)$ denotes the space of all essentially bounded functions in $L^0(X)$ . These notions can be extended to functions taking values in a finite dimensional normed vector space $(B, \|\cdot \|_B)$ , for instance,

where $L^0(X;B)$ denotes the space of measurable functions from X to B (up to almost everywhere equivalence). If B is separable, these notions can be extended to infinite-dimensional B. However, in this paper, by appealing to standard approximation arguments, we will always be able to work in finite-dimensional settings.

For any $p\in [1,\infty ]$ , we define a weak- $L^p$ space of measurable functions on X by setting

$$ \begin{align*} L^{p,\infty}(X):=\{f:X\to\mathbb{C}\colon \lVert {f} \rVert _{L^{p,\infty}(X)}<\infty\}, \end{align*} $$

where for any $p\in [1,\infty )$ , we have

$$ \begin{align*} \lVert {f} \rVert _{L^{p,\infty}(X)}:=\sup_{\unicode{x3bb}>0}\unicode{x3bb}\mu(\{x\in X:|f(x)|>\unicode{x3bb}\})^{1/p} \quad\text{and}\quad \lVert {f} \rVert _{L^{\infty,\infty}(X)}:= \lVert {f} \rVert _{L^{\infty}(X)}. \end{align*} $$

In our case, we will mainly have $X=\mathbb {R}^d$ or $X=\mathbb {T}^d$ equipped with the Lebesgue measure, and $X=\mathbb {Z}^d$ endowed with the counting measure. If X is endowed with a counting measure, we will abbreviate $L^p(X)$ to $\ell ^p(X)$ , $L^p(X; B)$ to $\ell ^p(X; B)$ , and $L^{p,\infty }(X)$ to $\ell ^{p,\infty }(X)$ .

If $T : B_1 \to B_2$ is a continuous linear map between two normed vector spaces $B_1$ and $B_2$ , we use $\|T\|_{B_1 \to B_2}$ to denote its operator norm.

2.4 Fourier transform

We will use convention that $\boldsymbol {e}(z)=e^{2\pi {\boldsymbol i} z}$ for every $z\in \mathbb {C}$ , where ${\boldsymbol i}^2=-1$ . Let $\mathcal {F}_{\mathbb {R}^d}$ denote the Fourier transform on $\mathbb {R}^d$ defined for any $f \in L^1(\mathbb {R}^d)$ and for any $\xi \in \mathbb {R}^d$ as

$$ \begin{align*} \mathcal{F}_{\mathbb{R}^d} f(\xi) := \int_{\mathbb{R}^d} f(x) \boldsymbol{e}(x\cdot\xi) \,{d}x. \end{align*} $$

If $f \in \ell ^1(\mathbb {Z}^d)$ , we define the discrete Fourier transform (Fourier series) $\mathcal {F}_{\mathbb {Z}^d}$ , for any ${\xi \in \mathbb {T}^d}$ , by setting

$$ \begin{align*} \mathcal{F}_{\mathbb{Z}^d}f(\xi): = \sum_{x \in \mathbb{Z}^d} f(x) \boldsymbol{e}(x\cdot\xi). \end{align*} $$

Sometimes, we shall abbreviate $\mathcal {F}_{\mathbb {Z}^d}f$ to $\hat {f}$ .

Let $\mathbb {G}=\mathbb {R}^d$ or $\mathbb {G}=\mathbb {Z}^d$ . It is well known that their corresponding dual groups are ${\mathbb {G}^*=(\mathbb {R}^d)^*=\mathbb {R}^d}$ or $\mathbb {G}^*=(\mathbb {Z}^d)^*=\mathbb {T}^d$ , respectively. For any bounded function ${\mathfrak m: \mathbb {G}^*\to \mathbb {C}}$ and a test function $f:\mathbb {G}\to \mathbb {C}$ , we define the Fourier multiplier operator by

(2.2) $$ \begin{align} T_{\mathbb{G}}[\mathfrak m]f(x):=\int_{\mathbb{G}^*}\boldsymbol{e}(-\xi\cdot x)\mathfrak m(\xi)\mathcal{F}_{\mathbb{G}}f(\xi)\,{d}\xi \quad \text{for } x\in\mathbb{G}. \end{align} $$

One may think that $f:\mathbb {G}\to \mathbb {C}$ is a compactly supported function on $\mathbb {G}$ (and smooth if $\mathbb {G}=\mathbb {R}^d$ ) or any other function for which equation (2.2) makes sense.

2.5 Oscillation seminorms

Let $\mathbb {I}\subseteq \mathbb {R}$ be such that $\# \mathbb {I}\ge 2$ . For every $J\in \mathbb {Z}_+\cup \{\infty \}$ , define

where $\mathbb {N}_{\le \infty }:=\mathbb {N}$ . In other words, $\mathfrak S_J(\mathbb {I})$ is a family of all strictly increasing sequences of length $J+1$ taking their values in the index set $\mathbb {I}$ .

Let $(\mathfrak a_{t}(x): t\in \mathbb {I})\subseteq \mathbb {C}$ be a family of measurable functions defined on X. For any ${r\in [1, \infty )}$ , $\mathbb {J}\subseteq \mathbb {I}$ , and a sequence $I=(I_i : i\in \mathbb {N}_{\le J}) \in \mathfrak S_J(\mathbb {I})$ , the oscillation seminorm is defined by

(2.3) $$ \begin{align} O_{I, J}^r(\mathfrak a_{t}(x): t \in \mathbb{J}):= \bigg(\sum_{j=0}^{J-1}\sup_{t\in [I_j, I_{j+1})\cap\mathbb{J}} \lvert {\mathfrak a_{t}(x) - \mathfrak a_{I_j}(x)} \rvert ^r\bigg)^{1/r}. \end{align} $$

There will be no problems with measurability in equation (2.3) since we will always assume that $\mathbb {I}\ni t\mapsto \mathfrak a_{t}(x)\in \mathbb {C}$ is continuous for $\mu $ -almost every $x\in X$ , or $\mathbb {J}$ is countable. We also use the convention that the supremum taken over the empty set is zero.

We recall some notation from [Reference Hytönen, van Neerven, Veraar and Weis6, §3, p. 165]. Let $(X,\mathcal {B}(X),\mu )$ be a $\sigma $ -finite measure space and let $\mathbb {I}$ be a totally ordered set. A sequence of sub- $\sigma $ -algebras $(\mathcal F_t: t\in \mathbb {I})$ is called filtration if it is increasing and the measure $\mu $ is $\sigma $ -finite on each $\mathcal F_t$ . Recall that a martingale adapted to a filtration $(\mathcal F_t: t\in \mathbb {I})$ is a family of functions $\mathfrak f=(\mathfrak f_t:t\in \mathbb {I})\subseteq L^1(X,\mathcal {B}(X),\mu )$ such that $\mathfrak f_{s}=\mathbb {E}[\mathfrak f_t|\mathcal F_s]$ for every $s, t\in \mathbb {I}$ so that $s\le t$ , where $\mathbb {E}[\cdot |\mathcal F]$ denotes the the conditional expectation with respect to a sub- $\sigma $ -algebra $\mathcal F\subseteq \mathcal B(X)$ . We say that a martingale $\mathfrak f=(\mathfrak f_t:t\in \mathbb {I})\subseteq L^p(X,\mathcal {B}(X),\mu )$ is bounded if

$$ \begin{align*} \sup_{t\in\mathbb{I}}\|\mathfrak f_t\|_{L^p(X)}\lesssim_p1. \end{align*} $$

We now establish oscillation inequalities for bounded martingales in $L^p(X,\mathcal {B}(X),\mu )$ .

Proposition 2.2. For every $p\in (1, \infty )$ , there exists a constant $C_p>0$ such that for every bounded martingale $\mathfrak f=(\mathfrak f_n:n\in \mathbb {Z})\subseteq L^p(X,\mathcal {B}(X),\mu )$ corresponding to a filtration $(\mathcal F_n: n\in \mathbb {Z})$ , one has

(2.7)

This proposition was established in [Reference Jones, Kaufman, Rosenblatt and Wierdl8, Theorem 6.4, p. 930]. The authors first established equation (2.7) for $p=2$ , then proved weak type $(1,1)$ as well as $L^{\infty }\to \mathrm {BMO}$ variants of equation (2.7), and consequently, by a simple interpolation, derived equation (2.7) for all $p\in (1, \infty )$ . Here, we give a slightly different and simplified proof based on a weighted Doob’s inequality, which avoids $L^{\infty }\to \mathrm {BMO}$ estimates and, in fact, is very much in the spirit of the estimates for $p=2$ from [Reference Jones, Kaufman, Rosenblatt and Wierdl8, Theorem 6.1, p. 927].

Proof of Proposition 2.2

We fix $N \in \mathbb {Z}_+$ and a sequence $I\in \mathfrak S_N(\mathbb {Z})$ . We first prove equation (2.7) for $p\ge 2$ . Since $r=p/2\ge 1$ , we take a non-negative $w\in L^{r'}(X)$ such that $ \left \lVert {w} \right \rVert _{L^{r'}(X)}\le 1$ and

To estimate the last sum, we will use a weighted version of Doob’s maximal inequality, see [Reference Hytönen, van Neerven, Veraar and Weis6, Theorem 3.2.3, p. 175], which asserts that for every $p\in (1, \infty )$ , every function $f\in L^p(X)$ , and a non-negative measurable weight w, and for any $n\in \mathbb {Z}$ , we have

(2.8) $$ \begin{align} \bigg(\int_X\sup_{m\le n} \lvert {\mathfrak f_m} \rvert ^pw\:\,{d}\mu\bigg)^{1/p} \le p'\bigg(\int_X \lvert {\mathfrak f_n} \rvert ^p\sup_{m\in\mathbb{Z}}|\mathbb{E}[w|\mathcal F_m]|\,{d}\mu\bigg)^{1/p}. \end{align} $$

We will also use an unweighted Doob’s inequality, see [Reference Hytönen, van Neerven, Veraar and Weis6, Theorem 3.2.2, p. 175], which yields that for every $p\in (1, \infty )$ , we have

(2.9) $$ \begin{align} \Big\lVert{\sup_{m\in\mathbb{Z}} \lvert {\mathfrak f_m}\rvert \Big\rVert}_{L^{p}(X)}\le p'\sup_{m\in\mathbb{Z}} \lVert {\mathfrak f_m} \rVert _{L^{p}(X)}. \end{align} $$

Finally, we will use for every $p\in (1, \infty )$ , the following bound

which, by Khintchine’s inequality, ensures

(2.10)

Then we conclude

This proves equation (2.7) for all $p\in [2, \infty )$ . To prove equation (2.7) for $p\in (1, 2)$ , it suffices to show the corresponding weak type $(1,1)$ estimate. This follows from [Reference Jones, Kaufman, Rosenblatt and Wierdl8, Theorem 6.2, p. 928], so we omit the details.

2.6 Variation seminorms

We also recall the definition of r-variations. For any $\mathbb I\subseteq \mathbb {R}$ , any family $(\mathfrak a_t: t\in \mathbb I)\subseteq \mathbb {C}$ , and any exponent $1 \leq r < \infty $ , the r-variation seminorm is defined to be

(2.11) $$ \begin{align} V^{r}(\mathfrak a_t: t\in\mathbb I):= \sup_{J\in\mathbb{Z}_+} \sup_{\substack{t_{0}<\dotsb<t_{J}\\ t_{j}\in\mathbb I}} \bigg(\sum_{j=0}^{J-1} |\mathfrak a_{t_{j+1}}-\mathfrak a_{t_{j}}|^{r} \bigg)^{1/r}, \end{align} $$

where the latter supremum is taken over all finite increasing sequences in $\mathbb I$ .

2.7 Jumps

The r-variation is closely related to the $\unicode{x3bb} $ -jump counting function. Recall that for any $\unicode{x3bb}>0$ , the $\unicode{x3bb} $ -jump counting function of a function $f : \mathbb {I} \to \mathbb {C}$ is defined by

(2.16) $$ \begin{align} N_{\unicode{x3bb}} f:=N_\unicode{x3bb}(f(t):t\in\mathbb{I}):=\sup \Big\{ J\in\mathbb{N} : \exists_{\substack{t_{0}<\ldots<t_{J}\\ t_{j}\in\mathbb{I}}} : \min_{0 \le j \le J-1} |f(t_{j+1})-f(t_{j})| \ge \unicode{x3bb} \Big\}. \end{align} $$

We close this discussion by showing that we cannot replace $L^{p,{\infty }} (X)$ with $L^{p} (X)$ in equation (2.18).

Lemma 2.5. For a fixed $1 \le p < \infty $ , there exists a function $f \colon \mathbb {Z}_+ \times \mathbb {Z}_+ \to \mathbb {R}$ such that

$$ \begin{align*} \| V^r(f(\cdot,n):n\in\mathbb{Z}_+) \|_{\ell^p(\mathbb{Z}_+)} = {\infty}, \quad 2 \le r \le \infty \end{align*} $$

and

$$ \begin{align*} \sup_{\unicode{x3bb}> 0} \| \unicode{x3bb} N_{\unicode{x3bb}}(f(\cdot,n):n\in\mathbb{Z}_+)^{1/2} \|_{\ell^p(\mathbb{Z}_+)} < {\infty}. \end{align*} $$

Proof. Take a sequence $(a_x: x\in \mathbb {Z}_+)\subseteq \mathbb {C}$ , to be specified later, such that $a_1> a_2 > \ldots >0$ and such that $a_x \to 0$ as $x \to \infty $ (actually, $a_x = x^{-1/p}$ will work). We define a function f as

$$ \begin{align*} f(x,1) = a_x, \quad f(x,n) = 0, \quad n \in\mathbb{N}_{\ge2}, \quad x \in \mathbb{Z}_+. \end{align*} $$

Then, for every $2 \le r \le \infty $ , we have $V^r f(x) = a_x$ and, consequently, we have

$$ \begin{align*} \| V^r(f(\cdot,n):n\in\mathbb{Z}_+) \|_{\ell^p(\mathbb{Z}_+)}^p = \sum_{j\in\mathbb{Z}_+} a_j^p, \quad 2 \le r \le \infty. \end{align*} $$

We now compute $N_{\unicode{x3bb} }f$ . Observe that

$$ \begin{align*} N_{\unicode{x3bb}}f (x) = \begin{cases} 0, & \unicode{x3bb}> a_x,\\ 1, & \unicode{x3bb} \le a_x. \end{cases} \end{align*} $$

Using this, we see that $N_{\unicode{x3bb} }f (x) = 0$ for all $x \in \mathbb {Z}_+$ if $\unicode{x3bb}> a_1$ . Otherwise, if $a_j \ge \unicode{x3bb}> a_{j+1}$ for some $j \ge 1$ , then and, consequently, we get

$$ \begin{align*} \| (N_{\unicode{x3bb}}f)^{1/2} \|_{\ell^p(\mathbb{Z}_+)}^p = \begin{cases} 0, & \unicode{x3bb}> a_1,\\ j, & a_j \ge \unicode{x3bb} > a_{j+1}, \quad j \in\mathbb{Z}_+. \end{cases} \end{align*} $$

Therefore, we obtain

$$ \begin{align*} \sup_{\unicode{x3bb}> 0} \| \unicode{x3bb} (N_{\unicode{x3bb}}f)^{1/2} \|_{\ell^p(\mathbb{Z}_+)}^p & = \sup_{\unicode{x3bb} > 0} \unicode{x3bb}^p \begin{cases} 0, & \unicode{x3bb} > a_1,\\ j, & a_j \ge \unicode{x3bb} > a_{j+1}, \quad j \in\mathbb{Z}_+. \end{cases} \\ & = \sup_{j \in\mathbb{Z}_+} j a_j^p. \end{align*} $$

Taking $a_j = j^{-1/p}$ , the desired conclusion follows and the proof of Lemma 2.5 is finished.

Remark 2.6. Note that the same example can be used to show that $\unicode{x3bb} $ -jumps with the parameter $r=2$ may not imply r-oscillations with $r\ge 2$ in the $\ell ^p$ sense. To be more precise, there exists a function $f:\mathbb {Z}_+\times \mathbb {Z}_+\to \mathbb {R}$ such that

$$ \begin{align*} \sup_{N\in\mathbb{Z}_+}\sup_{I\in\mathfrak{S}_N(\mathbb{Z}_+)} \lVert {O_{I,N}^r(f(\cdot,n):n\in\mathbb{Z}_+)} \rVert _{\ell^p(\mathbb{Z}_+)} =\infty,\quad 2\le r\le\infty \end{align*} $$

and

$$ \begin{align*} \sup_{\unicode{x3bb}> 0} \| \unicode{x3bb} N_{\unicode{x3bb}}(f(\cdot,n):n\in\mathbb{Z}_+)^{1/2} \|_{\ell^p(\mathbb{Z}_+)} < \infty. \end{align*} $$

3.1 Ionescu–Wainger multiplier theorem

The key tool, which will be used in the proof of Theorem 3.2, is the Ionescu–Wainger theorem [Reference Ionescu and Wainger7]. We now recall the d-dimensional vector-valued Ionescu–Wainger multiplier theorem from [Reference Mirek, Stein and Zorin-Kranich18, §2].

Theorem 3.3. For every $\varrho>0$ , there exists a family $(P_{\leq N})_{N\in \mathbb {Z}_+}$ of subsets of $\mathbb {Z}_+$ satisfying the following properties.

1. (i) One has $[N]\subseteq P_{\leq N}\subseteq [\max \{N, e^{N^{\varrho }}\}]$ .

2. (ii) If $N_1\le N_2$ , then $P_{\leq N_1}\subseteq P_{\leq N_2}$ .

3. (iii) If $q \in P_{\leq N}$ , then all factors of q also lie in $P_{\leq N}$ .

4. (iv) One has $\operatorname {\mathrm {lcm}}{(P_N)}\le 3^N$ .

Furthermore, for every $p \in (1,\infty )$ , there exists $0<C_{p, \varrho , d}<\infty $ such that, for every $N\in \mathbb {Z}_+$ , the following holds.

Let $0<\varepsilon _N \le e^{-N^{2\varrho }}$ , and let $\Theta : \mathbb {R}^{d} \to L(H_0,H_1)$ be a measurable function supported on $\varepsilon _{N}\mathbf {Q}$ , where $\mathbf {Q}:=[-1/2, 1/2)^d$ is a unit cube, with values in the space $L(H_{0},H_{1})$ of bounded linear operators between separable Hilbert spaces $H_{0}$ and $H_{1}$ . Let $0 \leq \mathbf A_{p} \leq \infty $ denote the smallest constant such that, for every function $f\in L^2(\mathbb {R}^d;H_0)\cap L^{p}(\mathbb {R}^d;H_0)$ , we have

(3.6)

Then the multiplier

(3.7) $$ \begin{align} \Delta_N(\xi) :=\sum_{b \in\Sigma_{\leq N}} \Theta(\xi - b), \end{align} $$

where $\Sigma _{\leq N}$ is $1$ -periodic subsets of $\mathbb {T}^d$ defined by

(3.8) $$ \begin{align} \Sigma_{\leq N} := \bigg\{ \frac{a}{q}\in\mathbb{Q}^d\cap\mathbb{T}^d: q \in P_{\leq N} \text{ and } \mathrm{gcd}(a, q)=1\bigg\}, \end{align} $$

satisfies, for every $f\in L^p(\mathbb {Z}^d;H_0)$ , the following inequality:

(3.9)

An important feature of Theorem 3.3 is that one can directly transfer square function estimates from the continuous to the discrete setting. The constant $\mathbf A_{p}$ in equation (3.6) remains unchanged when $\Theta $ is replaced by $\Theta (A\cdot )$ for any invertible linear transformation $A:\mathbb {R}^d\to \mathbb {R}^d$ . By property i, we see

(3.10) $$ \begin{align} |\Sigma_{\leq N}|\lesssim e^{(d+1)N^\varrho}. \end{align} $$

A scalar-valued version of Theorem 3.3 was originally established by Ionescu and Wainger [Reference Ionescu and Wainger7] with the factor $(\log N)^D$ in place of $\log N$ in equation (3.9). Their proof was based on a delicate inductive argument that exploited super-orthogonality phenomena arising from disjoint supports in the definition of equation (3.7). A somewhat different proof with factor $\log N$ in equation (3.9) was given in [Reference Mirek14]. The latter proof, instead of induction as in [Reference Ionescu and Wainger7], used certain recursive arguments, which clarified the role of the underlying square functions and strong-orthogonalities that lie behind of the proof of the Ionescu–Wainger multiplier theorem. A much broader context of the super-orthogonality phenomena was recently discussed in the survey article of Pierce [Reference Pierce19]. A detailed proof of Theorem 3.3 (in the spirit of [Reference Mirek14]) can be found in [Reference Mirek, Stein and Zorin-Kranich18, §2]. We also refer to the recent remarkable paper of Tao [Reference Tao23], where the factor $\log N$ was removed from equation (3.9).

An important ingredient in the proof of Theorem 3.3 is the sampling principle of Magyar–Stein–Wainger from [Reference Magyar, Stein and Wainger13]. We recall this principle as it will play an essential role in our further discussion.

Proposition 3.4. Let $d\in \mathbb {Z}_+$ be fixed. There exists an absolute constant $C>0$ such that the following holds. Let $p \in [1,\infty ]$ and $q\in \mathbb {Z}_+$ , and let $B_1, B_2$ be finite-dimensional Banach spaces. Let $\mathfrak m : \mathbb {R}^d \to L(B_1, B_2)$ be a bounded operator-valued function supported on $q^{-1}{\textbf{Q}}$ and denote the associated Fourier multiplier operator over $\mathbb {R}^d$ by $T_{\mathbb {R}^d}[\mathfrak m]$ . Let $\mathfrak m^{q}_{\mathrm {per}}$ be the periodic multiplier

$$ \begin{align*} \mathfrak m^{q}_{\mathrm{per}}(\xi) : = \sum_{n\in\mathbb{Z}^d} \mathfrak m(\xi-n/q),\quad \xi\in\mathbb{T}^d. \end{align*} $$

Then,

$$ \begin{align*} \|T_{\mathbb{Z}^d}[\mathfrak m^{q}_{\mathrm{per}}]\|_{\ell^{p}(\mathbb{Z}^d;B_1)\to \ell^{p}(\mathbb{Z}^d;B_2)}\le C\|T_{\mathbb{R}^d}[\mathfrak m]\|_{L^{p}(\mathbb{R}^d;B_1)\to L^{p}(\mathbb{R}^d;B_2)}. \end{align*} $$

The proof can be found in [Reference Magyar, Stein and Wainger13, Corollary 2.1, p. 196]. We also refer to [Reference Mirek, Stein and Zorin-Kranich16] for generalization of Proposition 3.4 to real interpolation spaces, which in particular covers the case of jump inequalities.

3.2 Proof of inequality (3.5) from Theorem 3.2

Fix $p\in (1,\infty )$ and let $f\in \ell ^p(\mathbb {Z}^\Gamma )$ be a finitely supported function. We fix $p_0>1$ close to $1$ such that $p\in (p_0,p_0')$ . We take $\tau \in (0,1)$ such that

(3.11) $$ \begin{align} \tau<\tfrac{1}{2}\min\{p_0-1,1\}. \end{align} $$

Now by equation (2.15), one can split equation (3.5) into long oscillations and short variations respectively by

(3.12)

To handle the short variations, we may proceed as in [Reference Mirek, Stein and Zorin-Kranich18] (see also [Reference Zorin-Kranich24]) and conclude that

(3.13)

since $q(1-\tau )>1$ by equation (3.11); here, $q=\min \{p,2\}$ . The first inequality in equation (3.13) follows from a simple observation that for any finite sequence ${t_0<t_1<\dotsc <t_J}$ contained in $[n^\tau ,(n+1)^\tau )$ , one has

The latter quantity is controlled by

which immediately follows by invoking the following proposition.

Proposition 3.5. ([Reference Mirek, Stein and Zorin-Kranich18, Proposition 4.16, p. 45], see also [Reference Mirek, Stein and Zorin-Kranich17, Lemma A.1, p. 554])

Let $\Omega \subset \mathbb {R}^{k}$ be a bounded and convex set and let $1\leq s\leq \operatorname {\mathrm {diam}}(\Omega )$ . Then,

(3.14) $$ \begin{align} \#\{x \in \mathbb{Z}^{k} : \operatorname{\mathrm{dist}}(x,\partial\Omega)<s\}\lesssim_k s \operatorname{\mathrm{diam}}(\Omega)^{k-1}. \end{align} $$

The implicit constant depends only on the dimension k, but not on the convex set $\Omega $ .

Now we have to bound long oscillations from equation (3.12). By Davenport’s result [Reference Davenport5], we know that

$$ \begin{align*} \#(\Omega_{2^{n^\tau}}\cap\mathbb{Z}^k)=|\Omega_{2^{n^\tau}}|+\mathcal{O}(2^{n^\tau(k-1)}). \end{align*} $$

Consequently, one has the following estimate:

where

$$ \begin{align*} m_{t}(\xi):=\frac{1}{|\Omega_{t}|}\sum_{y \in\Omega_{t}\cap\mathbb{Z}^{k}}\boldsymbol{e}(\xi\cdot (y)^\Gamma),\quad\xi\in\mathbb{T}^\Gamma,\quad t\in \mathbb{R}_+. \end{align*} $$

Thus, we are reduced to prove

(3.15)

3.3 Proof of inequality (3.15)

Now, let $\chi \in (0,1/10)$ . The proof of equation (3.15) will require several appropriately chosen parameters. We choose $\alpha>0$ such that

$$ \begin{align*} \alpha>\bigg(\frac{1}{p_0}-\frac{1}{2}\bigg)\bigg(\frac{1}{p_0}-\frac{1}{\min\{p,p'\}}\bigg)^{-1}. \end{align*} $$

Let $u\in \mathbb {Z}_+$ be a large natural number to be specified later. We set

(3.16) $$ \begin{align} \varrho:=\min\bigg\{\frac{1}{10u},\frac{\delta}{8\alpha}\bigg\}, \end{align} $$

where $\delta>0$ is the exponent from the Gauss sum estimates in equation (3.28). Let ${S_0:=\max (2^{u\mathbb {Z}_+}\cap [1,n^{\tau u}])}$ . We shall, by convenient abuse of notation, write

$$ \begin{align*} \Sigma_{\le n^{\tau u}}:=\Sigma_{\leq S_0}. \end{align*} $$

Next, for dyadic integers $S\in 2^{u\mathbb {Z}_+}$ , we define

Then, it is easy to see that

(3.17) $$ \begin{align} \Sigma_{\le n^{\tau u}}=\bigcup_{\substack{S\le S_0,\\S\in 2^{u\mathbb{Z}_+}}}\Sigma_S. \end{align} $$

Now we define the Ionescu–Wainger projection multipliers. For this purpose, we introduce a diagonal matrix A of size $|\Gamma | \times |\Gamma |$ such that $(A v)_\gamma := \left \lvert {\gamma } \right \rvert v_\gamma $ for any $\gamma \in \Gamma $ and $v\in \mathbb {R}^\Gamma $ , and for any $t> 0$ , we also define corresponding dilations by setting $t^A x=(t^{|\gamma |}x_{\gamma }: \gamma \,{\in}\, \Gamma )$ for every $x\in \mathbb {R}^\Gamma $ . Let $\eta :\mathbb {R}^\Gamma \to [0,1]$ be a smooth function such that

$$ \begin{align*} \eta(x)=\begin{cases} 1, & |x|\le1/(32|\Gamma|), \\ 0, & |x|\ge1/(16|\Gamma|).\end{cases} \end{align*} $$

For any $n\in \mathbb {Z}_+$ , we define

(3.18) $$ \begin{align} \Pi_{\le n^\tau,\,n^\tau(A-\chi \mathrm{Id})}(\xi):=\sum_{a/q\in\Sigma_{\le n^{\tau u}}}\eta(2^{n^\tau(A-\chi \mathrm{Id})}(\xi-a/q)),\quad \xi\in\mathbb{T}^\Gamma, \end{align} $$

as well as

(3.19) $$ \begin{align} \Pi_{S,\, n^\tau(A-\chi \mathrm{Id})}(\xi):=\sum_{a/q\in\Sigma_{S}}\eta(2^{n^\tau(A-\chi \mathrm{Id})}(\xi-a/q)),\quad \xi\in\mathbb{T}^\Gamma,\quad S\in 2^{u\mathbb{Z}_+}. \end{align} $$

Note that Theorem 3.3 applies for equations (3.18) and (3.19) since $2^{-n^\tau (|\gamma |-\chi )}\le e^{-n^{\tau /10}}\le e^{-S_0^{2\varrho }}$ for sufficiently large $n\in \mathbb {Z}_+$ . Using equation (3.18), one sees that

(3.20)

(3.21)

The first and second terms in the above inequality correspond to minor and major arcs, respectively.

3.4 Minor arcs: estimates for equation (3.20)

A key ingredient in estimating equation (3.20) will be a multidimensional Weyl’s inequality.

Theorem 3.6. [Reference Mirek, Stein and Zorin-Kranich18, Theorem A.1, p. 49]

For every $k\in \mathbb {Z}_+$ and a finite non-empty set

, there exists $\epsilon>0$ such that, for every polynomial

$$ \begin{align*} P(n) = \sum_{\gamma\in\Gamma} \xi_{\gamma} n^{\gamma}, \end{align*} $$

every $N>1$ , convex set $\Omega \subseteq B(0,N) \subset \mathbb {R}^{k}$ , function $\phi : \Omega \cap \mathbb {Z}^{k}\to \mathbb {C}$ , multi-index ${\gamma _{0}\in \Gamma }$ , and integers $0\leq a<q$ with $\mathrm {gcd}(a, q) = 1$ and

we have

(3.22)

where

$$ \begin{align*} \kappa := \min \{q,N^{ \lvert {\gamma_{0}} \rvert }/q\}. \end{align*} $$

The implicit constant in equation (3.22) is independent on the coefficients of P and the numbers a, q, and N.

To estimate equation (3.20), we first observe, using equation (2.13), that

(3.23)

Using Theorem 3.6 and proceeding as in [Reference Mirek, Stein and Zorin-Kranich18, Lemma 3.29, p. 34], we may conclude that

(3.24) $$ \begin{align} \lVert {T_{\mathbb{Z}^\Gamma}[m_{2^{n^\tau}}(1-\Pi_{\le n^\tau,\,n^\tau(A-\chi \mathrm{Id})})]f} \rVert _{\ell^p(\mathbb{Z}^\Gamma)} \lesssim(n+1)^{-2}\|f\|_{\ell^p(\mathbb{Z}^\Gamma)}, \end{align} $$

provided that u is large. Combining equation (3.23) with equation (3.24), we obtain the desired bound for equation (3.20).

3.5 Major arcs: estimates for equation (3.21)

Our aim is to prove

(3.25)

For $n\in \mathbb {N}$ and $\xi \in \mathbb {T}^{\Gamma }$ , define a new multiplier

(3.26) $$ \begin{align} {\boldsymbol m}_n(\xi):=\sum_{a/q\in\Sigma_{\le n^{\tau u}}}G(a/q)\Phi_{2^{n^\tau}}(\xi-a/q)\eta(2^{n^\tau(A-\chi \mathrm{Id})}(\xi-a/q)), \end{align} $$

where $\Phi _{N}$ is a continuous version of the multiplier $m_{N}$ given by

$$ \begin{align*} \Phi_N(\xi):=\frac{1}{|\Omega_{N}|}\int_{\Omega_{N}}{\boldsymbol{e}(\xi\cdot(t)^\Gamma)\,{d}t}, \end{align*} $$

and $G(a/q)$ is the Gauss sum defined by

$$ \begin{align*} G(a/q):=\frac{1}{q^k}\sum_{r\in[q]^k}{\boldsymbol{e}((a/q)\cdot({r})^\Gamma)}. \end{align*} $$

It is well known from the multidimensional Van der Corput lemma [Reference Stein22, Proposition 5, p. 342] that

(3.27) $$ \begin{align} |\Phi_N(\xi)|\lesssim |N^{A}\xi|_\infty^{-1/|\Gamma|}\quad\text{and}\quad|\Phi_N(\xi)-1|\lesssim |N^{A}\xi|_\infty. \end{align} $$

By Theorem 3.6 (see also [Reference Mirek, Stein and Zorin-Kranich18, Lemma 4.14, p. 44]), one can easily find $\delta \in (0, 1)$ such that

(3.28) $$ \begin{align} |G(a/q)|\lesssim_k q^{-\delta} \end{align} $$

for every $q\in \mathbb {Z}_+$ and $a=(a_{\gamma }:\gamma \in \Gamma )\in [q]^{\Gamma }$ such that $\mathrm {gcd}(a, q)=1$ .

We claim that

(3.29)

To establish equation (3.29), one can proceed as in [Reference Mirek, Stein and Zorin-Kranich18, Lemma 3.38, p. 36] by appealing to the proposition stated below and the estimate in equation (3.10), Theorem 3.3, and Proposition 3.4.

Proposition 3.7. [Reference Mirek, Stein and Zorin-Kranich18, Proposition 4.18, p. 47]

Let $\Omega \subseteq B(0,N)\subset \mathbb {R}^k$ be a convex set and $\mathcal {K}\colon \Omega \to \mathbb {C}$ be a continuous function. Then for any $q\in \mathbb {Z}_+$ , $a=(a_{\gamma }:\gamma \in \Gamma )\in [q]^{\Gamma }$ such that $\mathrm {gcd}(a, q)=1$ and $\xi = a/q + \theta \in \mathbb {R}^{\Gamma }$ , we have

$$ \begin{align*} &\bigg|\sum_{y\in\Omega\cap\mathbb{Z}^k}{\boldsymbol{e}(\xi\cdot(y)^\Gamma)\mathcal{K}(y)}-G(a/q)\int_{\Omega}{\boldsymbol{e}(\theta\cdot(t)^\Gamma)\mathcal{K}(t)\,{d}t}\bigg|\\ &\quad\lesssim_k\frac{q}{N}N^k\|\mathcal{K}\|_{L^\infty(\Omega)}+N^k\|\mathcal{K}\|_{L^\infty(\Omega)}\sum_{\gamma\in\Gamma}{(q|\theta_\gamma|N^{|\gamma|-1})^{\varepsilon_\gamma}}\\&\qquad+N^k\sup_{x,y\in\Omega\colon|x-y|\le q}|\mathcal{K}(x)-\mathcal{K}(y)|, \end{align*} $$

for any sequence $(\varepsilon _\gamma \colon \gamma \in \Gamma )\subseteq [0,1]$ . The implicit constant is independent of $a,q,N,\theta $ and the kernel $\mathcal {K}$ .

Using equations (3.29) and (2.13), the proof of the inequality (3.25) is reduced to showing

(3.30)

3.6 Major arcs: estimates for equation (3.30)

For $n\in \mathbb {Z}_+$ , $S\in 2^{u\mathbb {Z}_+}$ , and $\xi \in \mathbb {T}^{\Gamma }$ , we define a new multiplier

(3.31) $$ \begin{align}{\boldsymbol m}_n^S(\xi):=\sum_{a/q\in\Sigma_S}G(a/q)\Phi_{2^{n^\tau}}(\xi-a/q)\eta(2^{n^\tau(A-\chi \mathrm{Id})}(\xi-a/q)). \end{align} $$

Then using equations (3.17) and (3.31), we see that to prove equation (3.30), it suffices to show that

(3.32)

since $S^{-4\varrho }$ is summable in $S\in 2^{u\mathbb {Z}_+}$ .

To establish equation (3.32), we define $\tilde {\eta }(x):=\eta (x/2)$ and two new multipliers

$$ \begin{align*} \nu_n^S(\xi)&:=\sum_{a/q\in\Sigma_S}\Phi_{2^{n^\tau}}(\xi-a/q)\eta(2^{n^\tau(A-\chi \mathrm{Id})}(\xi-a/q)),\\ \mu_S(\xi)&:=\sum_{a/q\in\Sigma_S}G(a/q)\tilde{\eta}(2^{S^{1/u}(A-\chi \mathrm{Id})}(\xi-a/q)). \end{align*} $$

Obviously, we have ${\boldsymbol m}_n^S= \nu _n^S\mu _S$ and we see that the estimate in equation (3.32) will follow if we show that

(3.33)

(3.34)

Using Proposition 3.7, Theorem 3.3, and the Gauss sum estimates in equation (3.28), we can easily establish equation (3.33), and we refer to [Reference Mirek, Stein and Zorin-Kranich18, Lemma 3.47 and estimate (3.49) pp. 38–39] for more details. Now we return to equation (3.34). We define $\kappa _S:= \left \lceil {S^{2\varrho }} \right \rceil $ and split equation (3.34) into small and large scales respectively by

where $\mathbb {D}^\tau _{\le S}:=\{n\in \mathbb {Z}_+:n\in [S^{1/(\tau u)},2^{\kappa _S+1}]\}$ and $\mathbb {D}^\tau _{\ge S}:=\{n\in \mathbb {Z}_+:n \ge 2^{\kappa _S}\}$ .

3.7 Major arcs: small scales estimates

Our aim is to prove

(3.35)

Using the Rademacher–Menschov inequality from equation (2.14), one has

where $j\in \mathbb {Z}_+$ runs over the set of integers such that $U_j^i=[j2^i,(j+1)2^i)\subseteq [S^{1/(\tau u)},2^{\kappa _S+1}]$ . By Theorem 3.3, it suffices to justify that uniformly in $0\le i \le \kappa _S + 1$ , we have

(3.36)

This, in turn, is a fairly straightforward matter by appealing to equation (3.27) and standard arguments from the Littlewood–Paley theory. We refer to [Reference Mirek, Stein and Zorin-Kranich17] for more details, see also discussion below [Reference Mirek, Stein and Zorin-Kranich18, Theorem 4.3, p. 42].

3.8 Major arcs: large scales estimates

Finally, our aim is to prove

(3.37)

Proceeding as in [Reference Mirek, Stein and Zorin-Kranich18, §3.6, pp. 40–41] and invoking Proposition 3.4, the estimate in equation (3.37) is reduced to showing for every $p\in (1, \infty )$ and $f\in L^p(\mathbb {R}^{\Gamma })$ , the following uniform oscillation inequality:

(3.38)

The inequality (3.38) can be reduced to the martingale setting from Proposition 2.2 by invoking square function arguments [Reference Jones, Seeger and Wright10, Lemma 3.2, p. 6722] and the standard Littlewood–Paley theory. The details may be found in [Reference Mirek, Stein and Zorin-Kranich17]. This completes the proof of Theorem 3.2.