Oscillation inequalities in ergodic theory and analysis: one-parameter and multi-parameter perspectives

In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of oscillation estimates in both the single and several parameter settings. We establish a number of new oscillation inequalities and give new proofs for known results with elementary arguments.

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Introduction {#section:1}
============

Pointwise convergence is the most natural as well as the most difficult type of convergence to establish. It requires sophisticated tools in analysis, ergodic theory and probability. In this survey, we will review variation and oscillation semi-norms as well as the $\lambda$-jump counting function which give us quantitative measures for pointwise convergence. However we will concentrate on the central role that oscillation inequalities play, both in the one-parameter and multi-parameter settings.

In the one-parameter setting we derive a simple abstract oscillation estimate for the so-called projective operators, which will result in oscillation estimates for martingales, smooth bump functions as well as the Carleson operator. The multi-parameter oscillation semi-norm is the only available tool that allows us to handle efficiently multi-parameter pointwise convergence problems with arithmetic features. This contrasts sharply with the one-parameter setting, where we have a variety of tools including oscillations, variations or $\lambda$-jumps to handle pointwise convergence problems. The multi-parameter oscillation estimates will be illustrated in the context of the Dunford–Zygmund ergodic theorem for commuting measure-preserving transformations as well as observations of Bourgain for certain multi-parameter polynomial ergodic averages.

We begin with describing methods that permit us to handle pointwise convergence problems in the context of various ergodic averaging operators. Before we do this we set up notation and terminology, which will allow us to discuss various concepts in a fairly unified way.

Throughout this survey the triple $(X, \mathcal B(X), \mu)$ denotes a $\sigma$-finite measure space. The space of all formal $k$-variate polynomials $P({\rm m}_1, \ldots, {\rm m}_k)$ with $k\in{\mathbb{Z}}_+$ indeterminates ${\rm m}_1, \ldots, {\rm m}_k$ and integer coefficients will be denoted by ${\mathbb{Z}}[{\rm m}_1, \ldots, {\rm m}_k]$. We will always identify each polynomial $P\in{\mathbb{Z}}[{\rm m}_1, \ldots, {\rm m}_k]$ with a function $(m_1,\ldots, m_k)\mapsto P(m_1,\ldots, m_k)$ from ${\mathbb{Z}}^k$ to ${\mathbb{Z}}$.

Let $d, k \in{\mathbb{Z}}_+$, and consider a family ${\mathcal T} = (T_1,\ldots, T_d)$ of invertible commuting measure-preserving transformations on $X$, polynomials ${\mathcal P} = (P_1,\ldots, P_d) \subset {\mathbb{Z}}[\mathrm m_1, \ldots, \rm m_k]$, an integer $k$-tuple $M = (M_1,\ldots, M_k)\in{\mathbb{Z}}_+^k$, and a measurable function $f:X\to{\mathbb{C}}$. We consider the multi-parameter polynomial ergodic average $$\frac{1}{M_1 \cdots M_k} \sum_{m_1=1}^{M_1} \cdots \sum_{m_k=1}^{M_k}
f\bigl(T_1^{P_1({ m_1},\ldots,{ m_k})}
 \cdots T_d^{P_d({ m_1},\ldots,{ m_k})} x\bigr).$$ We denote this average by $A_{{M};X,{\mathcal T}}^{\mathcal P}f(x)$ and we use the notation $$\begin{aligned}
\label{eq:31}
A_{{M}; X, {\mathcal T}}^{\mathcal P}f(x): =  {\mathbb{E}}_{m\in Q_{M}}f(T_1^{P_1(m)}\cdots T_d^{P_d(m)}x), \qquad x\in X,\end{aligned}$$ where $Q_{M}:=[M_1]\times\ldots\times[M_k]$ is a box in ${\mathbb{Z}}^k$ with $[N]:=(0, N]\cap{\mathbb{Z}}$ for any real number $N\ge1$, and ${\mathbb{E}}_{y\in Y}f(y):=\frac{1}{\#Y}\sum_{y\in Y}f(y)$ for any finite set $Y$ and any $f:Y\to{\mathbb{C}}$. We will often abbreviate $A_{M; X, {\mathcal T}}^{{\mathcal P}}$ to $A_{M; X}^{{\mathcal P}}$ when the tranformations are understood. Depending on how explicit we want to be, more precision may be necessary and we will write out the averages $$\begin{aligned}
A_{M;X}^{\mathcal P}f(x) =A_{M_1,\ldots, M_k;X}^{P_1,\ldots, P_d} f(x)
\qquad \text{ or } \qquad
A_{{M}; X, {\mathcal T}}^{\mathcal P}f(x) =
A_{M_1,\ldots, M_k;X,T_1,\ldots, T_d}^{P_1,\ldots, P_d} f(x).\end{aligned}$$

\[ex:1\] Due to the Calder<span>ó</span>n transference principle @Cald, the most important dynamical system, from the point of view of pointwise convergence problems, is the integer shift system. Namely, it is the $d$-dimensional lattice $({\mathbb{Z}}^d, \mathcal B({\mathbb{Z}}^d), \mu_{{\mathbb{Z}}^d})$ equipped with a family of shifts $S_1,\ldots, S_d:{\mathbb{Z}}^d\to{\mathbb{Z}}^d$, where $\mathcal B({\mathbb{Z}}^d)$ denotes the $\sigma$-algebra of all subsets of ${\mathbb{Z}}^d$, $\mu_{{\mathbb{Z}}^d}$ denotes counting measure on ${\mathbb{Z}}^d$, and $S_j(x):=x-e_j$ for every $x\in{\mathbb{Z}}^d$ (here $e_j$ is $j$-th basis vector from the standard basis in ${\mathbb{Z}}^d$ for each $j\in[d]$). Then the average $A_{M; X, {\mathcal T}}^{{\mathcal P}}$ from with ${\mathcal T} = (T_1,\ldots, T_d)=(S_1,\ldots, S_d)$ can be rewritten for any $x=(x_1,\ldots, x_d)\in{\mathbb{Z}}^d$ and any finitely supported function $f:{\mathbb{Z}}^d\to{\mathbb{C}}$ as $$\begin{aligned}
\label{eq:32}
A_{M; {\mathbb{Z}}^d}^{{\mathcal P}}f(x)={\mathbb{E}}_{m\in Q_{M}}f(x_1-P_1(m),\ldots, x_d-P_d(m)).\end{aligned}$$

Birkhoff's and von Neumann's ergodic theorems
---------------------------------------------

In the early 1930's Birkhoff @BI and von Neumann @vN established an almost everywhere pointwise ergodic theorem and a mean ergodic theorem, respectively, which we summarize in the following result.

\[Birkhoff's and von Neumann's ergodic theorem\] \[birk\] Let $(X,\mathcal B(X), \mu)$ be a $\sigma$-finite measure space equipped with a measure-preserving transformation $T:X\to X$. Then for every $p\in[1, \infty)$ and every $f\in L^p(X)$ the averages $$A_{M;X, T}^{\mathrm m}f(x)={\mathbb{E}}_{m\in [M]}f(T^{m}x), \qquad x\in X, \qquad M\in{\mathbb{Z}}_+$$ converge almost everywhere on $X$ and in $L^p(X)$ norm as $M\to\infty$.

Although there are many proofs of Theorem \[birk\] in the literature, (we refer for instance to the monographs @EW [@Peter] for more details and the historical background), there is a particular proof which is important in our context. This proof illustrates the classical strategy for handling pointwise convergence problems, which is based on a two-step procedure:

-   The first step establishes $L^p(X)$ boundedness (when $p\in(1, \infty)$), or a weak type $(1, 1)$ bound (when $p=1$) of the corresponding maximal function $\sup_{M\in{\mathbb{Z}}_+}|A_{M;X, T}^{\mathrm m}f(x)|$. This in turn, using the Calder<span>ó</span>n transference principle @Cald, can be derived from the corresponding maximal bounds for the Hardy–Littlewood maximal function $\sup_{M\in{\mathbb{Z}}_+}|A_{M;{\mathbb{Z}}}^{\mathrm m}f(x)|$ on the set of integers, see . Having these maximal estimates in hand one can easily prove that the set $$\begin{aligned}
    \qquad \qquad \mathbf P\mathbf C[L^{p}(X)]=\{f\in L^{p}(X): \lim_{M\to\infty}A_{M;X, T}^{\mathrm m}f \text{
    exists $\mu$-almost everywhere on $X$}\}\end{aligned}$$ is closed in $L^{p}(X)$.

-   In the second step one shows that $\mathbf P\mathbf C[L^{p}(X)]= L^{p}(X)$. In view of the first step the task is reduced to finding a dense class of functions in $L^{p}(X)$ for which we have pointwise convergence. In our problem let us first assume $p=2$. Then invoking a variant of Riesz decomposition @Riesz a good candidate is the space ${\rm I}_T\oplus {\rm J}_T\subseteq L^2(X)$, where $$\begin{aligned}
    \qquad\qquad {\rm I}_T:=\{f\in L^2(X): f\circ T=f\},
    \qquad \text{ and } \qquad
    {\rm J}_T:=\{g-g\circ T: g\in L^2(X)\cap L^{\infty}(X)\}.\end{aligned}$$ We then note that $A_{M;{\mathbb{Z}}}^{\mathrm m}f=f$ for $f\in {\rm I}_T$, and $\lim_{M\to\infty}A_{M;X, T}^{\mathrm m}h=0$ for $h\in {\rm J}_T$, since $$A_{M;X, T}^{\mathrm m}h=M^{-1}\big(g\circ T-g\circ T^{M+1}\big)$$ telescopes, whenever $h=g-g\circ T\in {\rm J}_T$. This establishes pointwise almost everywhere convergence of $A_{M;X, T}^{\mathrm m}$ on ${\rm I}_T\oplus {\rm J}_T$, which is dense in $L^2(X)$. These two steps guarantee that $\mathbf P\mathbf C[L^{2}(X)]= L^{2}(X)$. Consequently, $A_{M;X, T}^{\mathrm m}$ converges pointwise on $L^{p}(X)\cap L^{2}(X)$ for any $p\in[1, \infty)$. Since $L^{p}(X)\cap L^{2}(X)$ is dense in $L^{p}(X)$ we also conclude, in view of the first step, that $\mathbf P\mathbf C[L^{p}(X)]= L^{p}(X)$, and this completes a brief outline of the proof of Theorem \[birk\].

Dunford–Zygmund pointwise ergodic theorem
-----------------------------------------

In the early 1950's it was observed by Dunford @D and independently by Zygmund @Z that the two-step procedure can be applied in a multi-parameter setting. More precisely, the Dunford–Zygmund multi-parameter pointwise ergodic theorem, where the convergence is understood in the unrestricted sense, can be formulated as follows.

\[Dunford–Zygmund ergodic theorem\] \[dz\] Let $d\in{\mathbb{Z}}_+$ and let $(X,\mathcal B(X), \mu)$ be a $\sigma$-finite measure space equipped with a family $\mathcal T=(T_1,\ldots, T_d)$ of not necessarily commuting and measure-preserving transformations $T_1,  \ldots, T_d:X\to X$. Then for every $p\in(1, \infty)$ and every $f\in L^p(X)$ the averages $$A_{M_1,\ldots, M_d;X, \mathcal T}^{{\mathrm m}_1,\ldots, {\rm m}_d}f(x)
={\mathbb{E}}_{m\in Q_{M}}f(T_1^{m_1}\cdots T_d^{m_d}x), \qquad x\in X, \qquad M = (M_1,\ldots, M_d)\in{\mathbb{Z}}_+^d,$$ converge almost everywhere on $X$ and in $L^p(X)$ norm as $\min\{M_1,\ldots, M_d\}\to\infty$.

This theorem has a fairly simple proof, which is based on the following identity $$\begin{aligned}
A_{M_1,\ldots, M_d;X, \mathcal T}^{{\mathrm m}_1,\ldots, {\rm m}_d}f=
A_{M_1;X, T_1}^{{\mathrm m}_1}\circ \ldots \circ A_{M_d;X, T_d}^{{\mathrm m}_d}f.\end{aligned}$$ The $L^p(X)$ bounds (with $p\in(1, \infty]$) for the strong maximal function $\sup_{M\in{\mathbb{Z}}_+^d}|A_{M_1,\ldots, M_d;X, \mathcal T}^{{\mathrm m}_1,\ldots, {\rm m}_d}f|$ follow easily by applying $d$ times the corresponding $L^p(X)$ bounds for $\sup_{M\in{\mathbb{Z}}_+}|A_{M;X, T}^{\mathrm m}f|$. This establishes the first step in the two-step procedure described above. The second step is based on a suitable adaptation of the telescoping argument to the multi-parameter setting and an application of the classical Birkhoff ergodic theorem, see @Nevo for more details. These two steps establish Theorem \[dz\] and motivates our further discussion on multi-parameter convergence problems. One also knows that pointwise convergence in Theorem \[dz\] may fail if $p = 1$, and that the operator $f \mapsto \sup_{M\in{\mathbb{Z}}_+^d}|A_{M_1,\ldots, M_d;X, \mathcal T}^{{\mathrm m}_1,\ldots, {\rm m}_d}f|$ is not of weak type $(1, 1)$ in general (even if we assume that the transformations $T_j$, $1 \le j \le d$, commute). A model example is $X=\mathbb{Z}^d$ and $T_j x = x - e_j$, $1 \le j \le d$, where $e_j$ is the $j$th coordinate vector. Then the corresponding maximal operator is just the strong maximal operator for which it is well known that the weak type $(1,1)$ estimate does not hold.

Quantitative tools in the study of pointwise convergence
--------------------------------------------------------

The approach described in the context of Theorem \[birk\] and Theorem \[dz\] has a quantitative nature but it says nothing quantitatively about pointwise convergence. This approach is very effective in pointwise convergence questions arising in harmonic analysis as there are many natural dense subspaces in Euclidean settings, which can be used to establish pointwise convergence. However for ergodic theoretic questions, when one works with abstract measure spaces, the situation is dramatically different as Bourgain showed @B1 [@B2; @B3]. We shall see more examples below.

Consequently, the second step from the two-step procedure may require more quantitative tools to establish pointwise convergence. To overcome the difficulties with determining dense subspace for which pointwise convergence may be verified, Bourgain @B3 proposed three other approaches.

1.  The first approach is based on controlling the so-called oscillation semi-norms. Let ${\mathbb{J}}\subseteq {\mathbb{N}}$ be so that $\#{\mathbb{J}}\ge2$, let $I=(I_j:j\in{\mathbb{N}}_{\le J})$ be a strictly increasing sequence of length $J+1$ for some $J\in{\mathbb{Z}}_+$, which takes values in ${\mathbb{J}}$, and recall that for any sequence $(\mathfrak a_{t}: t\in{\mathbb{J}})\subseteq{\mathbb{C}}$, and any exponent $1 \leq r < \infty$, the $r$-oscillation seminorm is defined by $$\begin{aligned}
    \label{eq:38}
    O_{I, J}^r(\mathfrak a_{t}: t \in {\mathbb{J}}):=
    \Big(\sum_{j=0}^{J-1}\sup_{\substack{I_j\le t<I_{j+1}\\ t\in {\mathbb{J}}}}
    \abs{\mathfrak a_{t} - \mathfrak a_{I_j}}^r\Big)^{1/r}.\end{aligned}$$ We will give a more general definition of $r$-oscillations in the multi-parameter setting; see .

2.  The second approach is based on controlling the so-called $r$-variation seminorms. For any $\mathbb I\subseteq {\mathbb{N}}$, any sequence $(\mathfrak a_t: t\in\mathbb I)\subseteq {\mathbb{C}}$, and any exponent $1 \leq r < \infty$, the $r$-variation semi-norm is defined to be $$\begin{aligned}
    V^{r}(\mathfrak a_t: t\in\mathbb I):=
    \sup_{J\in{\mathbb{Z}}_+} \sup_{\substack{t_{0}<\dotsb<t_{J}\\ t_{j}\in\mathbb I}}
    \Big(\sum_{j=0}^{J-1}  |\mathfrak a_{t_{j+1}}-\mathfrak a_{t_{j}}|^{r} \Big)^{1/r},\end{aligned}$$ where the latter supremum is taken over all finite increasing sequences in $\mathbb I$.

3.  The third approach is based on studying the $\lambda$-jump counting function which is closely related to $r$-variations. For any ${\mathbb{I}}\subseteq {\mathbb{N}}$ and any $\lambda>0$, the $\lambda$-jump counting function of a sequence $(\mathfrak a_t: t\in\mathbb I)\subseteq {\mathbb{C}}$ is defined by $$\begin{aligned}
    N_\lambda(\mathfrak a_t:t\in{\mathbb{I}}):=\sup \{ J\in{\mathbb{N}}: \exists_{\substack{t_{0}<\ldots<t_{J}\\ t_{j}\in{\mathbb{I}}}}
    : \min_{0 \le j \le J-1} |\mathfrak a_{t_{j+1}}-\mathfrak a_{t_{j}}| \ge \lambda \}.\end{aligned}$$ We also refer to Section \[section:2\] for simple properties of $r$-oscillations, $r$-variations and $\lambda$-jumps. These will be illustrated in the context of bounded martingales, a toy model explaining their quantitative nature and their usefulness in pointwise convergence problems.

Bourgain's pointwise ergodic theorem
------------------------------------

In the early 1980's Bellow @Bel (being motivated by some problems from equidistribution theory) and independently Furstenberg @F (being motivated by some problems from additive combinatorics in the spirit of Szemer<span>é</span>di's theorem @Sem1 for arithmetic progressions) posed the problem of whether for any polynomial $P\in {\mathbb{Z}}[{\rm m}]$ and any measure-preserving map $T:X\to X$ on a probability space $(X,\mathcal B(X), \mu)$, the averages $$\begin{aligned}
\label{eq:8}
A_{M;X, T}^{P(\mathrm m)}f(x)={\mathbb{E}}_{m\in [M]}f(T^{P(m)}x), \qquad x\in X, \qquad M\in{\mathbb{Z}}_+\end{aligned}$$ converge almost everywhere on $X$ as $M\to\infty$, for any $f\in L^{\infty}(X)$.

An affirmative answer to this question was given by Bourgain in series of groundbreaking papers @B1 [@B2; @B3] which we summarize in the following theorem.

\[Bourgain's ergodic theorem\] \[bourgain\] Let $(X,\mathcal B(X), \mu)$ be a $\sigma$-finite measure space equipped with an invertible measure-preserving transformation $T:X\to X$. Assume that $P\in {\mathbb{Z}}[{\rm m}]$ is a polynomial such that $P(0)=0$. Then for every $p\in(1, \infty)$ and every $f\in L^p(X)$ the averages $A_{M;X, T}^{P}f$ from converge almost everywhere on $X$ and in $L^p(X)$ norm as $M\to\infty$.

Theorem \[bourgain\] is an instance where establishing pointwise convergence on a dense class is a challenging problem. The decomposition ${\rm I}_T\oplus {\rm J}_T$ of von Neumann (as for $A_{M;X, T}^{{\mathrm m}}$) is not sufficient if $\deg P\ge2$, though it still makes sense. Even for the squares $P(m)=m^2$ it is not clear whether $\lim_{M\to\infty}A_{M;X, T}^{{\mathrm m}^2}h=0$ for $h\in {\rm J}_T$. The reason is that the averages $A_{M;X, T}^{{\mathrm m}^2}h$ do not telescope for $h\in {\rm J}_T$ anymore, since the differences $(m+1)^2-m^2=2m+1$ have unbounded gaps.

Nearly two decades after Bourgain papers @B1 [@B2; @B3], it was discovered that the range of $p\in(1, \infty)$ in Bourgain's theorem is sharp. In contrast to Birkhoff's theorem, if $P\in{\mathbb{Z}}[\rm m]$ is a polynomial of degree at least two, the pointwise convergence at the endpoint for $p=1$ may fail as was shown by Buczolich and Mauldin @BM for $P(m)=m^2$ and by LaVictoire @LaV1 for $P(m)=m^k$ for any $k\ge2$. This also stands in sharp contrast to what happens for continuous analogues of ergodic averages and shows that any intuition that we build in Euclidean harmonic analysis (when sums are replaced with integrals) can fail dramatically in discrete problems.

Bourgain @B1 [@B2; @B3] also used the two-step procedure to prove Theorem \[bourgain\]. In the first step, it was proved that for all $p\in(1, \infty]$, there exists $C_{p, P}>0$ such that for every $f\in L^p(X)$ we have $$\begin{aligned}
\label{eq:33}
\norm[\big]{\sup_{M\in{\mathbb{Z}}_+}\abs{A_{M; X, T}^{P}f}}_{L^p(X)}
\le C_{p, P}\norm{f}_{L^p(X)}.\end{aligned}$$

However, in the second step of the two-step procedure a quantitative pointwise ergodic theorem was established by studying oscillation semi-norms, see . More, precisely, it was proved that for any $\tau>1$, any sequence of integers $I=(I_j:{j\in{\mathbb{N}}})\subseteq{\mathbb{L}}_{\tau}:=\{\lfloor \tau^n\rfloor: n\in{\mathbb{N}}\}$ such that $I_{j+1}>2I_j$ for all $j\in{\mathbb{N}}$, and any $f\in L^2(X)$ one has $$\begin{aligned}
\label{eq:34}
\norm[\big]{O_{I, J}^2(A_{M; X, T}^{P}f:M\in{\mathbb{L}}_\tau)}_{L^2(X)}
\le C_{I, \tau}(J)\norm{f}_{L^2(X)}, \qquad J\in{\mathbb{Z}}_+,\end{aligned}$$ where $C_{I, \tau}(J)$ is a constant depending on $I$ and $\tau$ and satisfies $$\begin{aligned}
\label{eq:41}
\lim_{J\to\infty} J^{-1/2}C_{I, \tau}(J)=0.\end{aligned}$$

Bourgain @B1 [@B2; @B3] had the ingenious insight to see that inequality with suffices to establish pointwise convergence of $A_{M; X, T}^{P}f$ for any $f\in L^2(X)$. Inequality with can be thought of as the weakest possible quantitative form for pointwise convergence. On the one hand, is very close to the maximal inequality, since by using with $p=2$ we can derive with a constant at most $J^{1/2}$. On the other hand, any improvement (better than $J^{1/2}$) for the constant in implies and so ensures pointwise convergence of $A_{M; X, T}^{P}f$ for any $f\in L^2(X)$, see Proposition \[prop:4\], where the details, even in the multi-parameter setting, are given. Therefore, from this point of view, inequality with is the minimal quantitative requirement necessary to establish pointwise convergence.

Bourgain's papers @B1 [@B2; @B3] were a significant breakthrough in ergodic theory, which used a variety of new tools (ranging from harmonic analysis and number theory through probability and the theory of Banach spaces) to study pointwise convergence problems in analysis understood in a broad sense. In @B3 a complete proof of Theorem \[bourgain\] is given using the notions of $r$-variations and $\lambda$-jumps (introduced by Pisier and Xu @PX), which are two important quantitative tools in the study of pointwise convergence problems. This initiated a systematic study of quantitative estimates in harmonic analysis and ergodic theory which resulted in a vast literature: in ergodic theory @BM [@jkrw; @JR1; @JRW; @JRW1; @LaV1; @MST2; @Kr; @KMT; @MSS; @MSZ3], in discrete harmonic analysis @IMMS [@IMSW; @IMW; @IW; @M10; @MSZ1; @MSW; @MT; @Pierce; @RW], and in classical harmonic analysis @BORSS [@BRS; @CJRW; @DDU; @GRY; @JG; @JSW; @La1; @La2; @OSTTW; @MST1; @MSZ2].

Not long after @B3, Lacey refined Bourgain's argument @RW [Theorem 4.23, p. 95], and showed that for every $\tau>1$ there is a constant $C_{\tau}>0$ such that for any $f\in L^2(X)$ one has $$\begin{aligned}
\label{eq:35}
\sup_{J\in{\mathbb{Z}}_+}\sup_{I\in \mathfrak S_J({\mathbb{L}}_{\tau})}
\norm[\big]{O_{I, J}^2(A_{M; X, T}^{P}f:M\in{\mathbb{L}}_\tau)}_{L^2(X)}
\le C_{\tau}\norm{f}_{L^2(X)},\end{aligned}$$ where $\mathfrak S_J({\mathbb{L}}_{\tau})$ denotes the set of all strictly increasing sequences $I=(I_j:j\in{\mathbb{N}}_{\le J})\subset {\mathbb{L}}_{\tau}$ of length $J+1$ for some $J\in{\mathbb{Z}}_+$. Inequality was the first uniform oscillation result in the class of $\tau$-lacunary sequences. Lacey's observation naturally motivated a question (which also motivates this survey) whether there are uniform estimates, independent of $\tau>1$, of oscillation inequalities in . For the Birkhoff averages $A_{M;X, T}^{{\mathrm m}}$, this was explicitly formulated in @RW [Problem 4.12, p. 80]. We will discuss below uniform oscillation estimates as well as other quantitative forms of pointwise convergence including $r$-variations and $\lambda$-jumps.

Martingales: a model to study pointwise convergence problems
------------------------------------------------------------

In order to understand the relationship between $r$-oscillations, $r$-variations and $\lambda$-jumps, we will use bounded martingales $\mathfrak f=(\mathfrak f_n:X\to {\mathbb{C}}:n\in{\mathbb{Z}}_+)$ as a toy model to help us understand the connections and various nuances. All properties that will be used in the discussion below are collected in Section \[section:2\]. The discussion will follow the development of the various notions in chronological order.

The $r$-variations for $\mathfrak f=(\mathfrak f_n:X\to {\mathbb{C}}:n\in{\mathbb{Z}}_+)$ were investigated by L<span>é</span>pingle @Lep who established that for all $r\in(2, \infty)$ and $p\in(1, \infty)$, there is a constant $C_{p, r}>0$ such that $$\begin{aligned}
\label{eq:42}
\norm{V^r(\mathfrak f_n: n\in{\mathbb{Z}}_+)}_{L^p(X)}
\le C_{p, r}\sup_{n\in{\mathbb{Z}}_+}\norm{\mathfrak f_n}_{L^p(X)}.\end{aligned}$$ In fact, L<span>é</span>pingle @Lep also proved a weak type $(1, 1)$ estimate. A counterexample from @JG for $r=2$ shows that holds with sharp ranges of exponents. This counterexample plays an important role showing $r$-variation estimates only hold when $r>2$. In fact this is the best we can expect in applications in analysis and ergodic theory.

Inequality can be thought of as an extension of Doob's maximal inequality for martingales, which gives a quantitative form of the martingale convergence theorem. Indeed on the one hand, inequality implies that the sequences $(\mathfrak f_n: n\in{\mathbb{Z}}_+)$ converges almost everywhere on $X$ as $n\to \infty$. One the other hand, one has $$\begin{aligned}
\norm{\sup_{n\in{\mathbb{Z}}_+}|\mathfrak f_n|}_{L^p(X)}\le \norm{V^r(\mathfrak f_n: n\in{\mathbb{Z}}_+)}_{L^p(X)}+\norm{\mathfrak f_{n_0}}_{L^p(X)}\end{aligned}$$ for any $n_0\in{\mathbb{Z}}_+$ (see below), which shows that $r$-variational estimates lie deeper than maximal function estimates. We refer to @PX [@B3; @MSZ1] for generalizations and different proofs of .

Interestingly, Bourgain @B3 gave a new proof of inequality , where it was used to address the issue of pointwise convergence of $A_{M;X, T}^{P}f$, see . This initiated a systematic study of $r$-variations and other quantitative estimates in harmonic analysis and ergodic theory, which resulted in a vast literature @jkrw [@JR1; @JSW; @JG; @OSTTW; @MST2; @MSZ1; @MSZ2; @MSZ3; @zk] and recently @IMMS [@KMT; @MSS]. Due to below one has $$\begin{aligned}
\label{eq:43}
\sup_{\lambda>0}\|\lambda N_{\lambda}(\mathfrak{f}_n: n\in{\mathbb{Z}}_+)^{1/r}\|_{L^p(X)}
\le \|V^r(\mathfrak{f}_n: n\in{\mathbb{Z}}_+)\|_{L^p(X)},\end{aligned}$$ which combined with implies $\lambda$-jump inequalities for martingales for any $r>2$. Although the right hand side of blows up when $r\to 2$, it is possible to prove that for every $p\in(1, \infty)$ there exists a constant $C_p>0$ such that $$\begin{aligned}
\label{eq:44}
\sup_{\lambda>0}\|\lambda N_{\lambda}(\mathfrak{f}_n: n\in{\mathbb{Z}}_+)^{1/2}\|_{L^p(X)}
\le C_p\sup_{n\in{\mathbb{Z}}_+}\norm{\mathfrak f_n}_{L^p(X)}.\end{aligned}$$ Inequality was first established by Pisier and Xu @PX on $L^2(X)$ and then extended by Bourgain @B3 [inequality (3.5)] on $L^p(X)$ for all $p\in(1, \infty)$. In fact, Bourgain used to prove by noting that can be reversed in the sense that for every $p\in[1,\infty]$ and $1\le \rho<r\le\infty$ one has $$\begin{aligned}
\label{eq:45}
\| V^{r}( \mathfrak{f}_n : n \in {\mathbb{Z}}_+ ) \|_{L^{p,\infty} (X)}
\lesssim_{p,\rho,r}
\sup_{\lambda>0} \| \lambda N_{\lambda}(\mathfrak{f}_n: n\in {\mathbb{Z}}_+)^{1/\rho} \|_{L^{p,\infty}(X)},\end{aligned}$$ which follows from below. One cannot replace $L^{p,\8} (X)$ with $L^{p} (X)$ in , see @MSS for more details. Combining and with $\rho=2$ and interpolating, one obtains . Therefore uniform $\lambda$-jump estimates from can be thought of as endpoint estimates for $r$-variations where we have seen that $r$-variations may be unbounded at the endpoint in question. We have already noted the failure of L<span>é</span>pingle's inequality when $r=2$.

Even though we have a fairly complete picture of the relationship between $r$-variations and $\lambda$-jumps, the relations with $r$-oscillations are less obvious. It follows from below that $$\begin{aligned}
\label{eq:46}
\sup_{J\in{\mathbb{Z}}_+}\sup_{I\in\mathfrak{S}_{J}({\mathbb{Z}}_+)}\|O^{r}_{I, J}( \mathfrak{f}_n : n \in {\mathbb{Z}}_+ ) \|_{L^{p} (X)}
\le \| V^{r}( \mathfrak{f}_n : n \in {\mathbb{Z}}_+ ) \|_{L^{p} (X)},\end{aligned}$$ where $\mathfrak{S}_{J}({\mathbb{Z}}_+)$ denotes the set of all strictly increasing sequences $I=(I_j:j\in{\mathbb{N}}_{\le J})\subset {\mathbb{Z}}_+$ of length $J+1$ for some $J\in{\mathbb{Z}}_+$. In view of , this immediately implies $r$-oscillations estimates for martingales on $L^p(X)$ for all $r\in(2, \infty)$ and $p\in(1, \infty)$.

It was shown by Jones, Kaufman, Rosenblatt and Wierdl @jkrw [Theorem 6.4, p. 930] that for every $p\in(1, \infty)$ there is a constant $C_p>0$ such that $$\begin{aligned}
\label{eq:47}
\sup_{J\in{\mathbb{Z}}_+}\sup_{I\in\mathfrak{S}_{J}({\mathbb{Z}}_+)}\|O^{2}_{I, J}( \mathfrak{f}_n : n \in {\mathbb{Z}}_+ ) \|_{L^{p} (X)}
\le C_{p}\sup_{n\in{\mathbb{Z}}_+}\norm{\mathfrak f_n}_{L^p(X)}.\end{aligned}$$ Inequality is also an extension of Doob's maximal inequality for martingales, as one has $$\begin{aligned}
\norm{\sup_{n\in{\mathbb{Z}}_+}|\mathfrak f_n|}_{L^p(X)}\le \sup_{J\in{\mathbb{Z}}_+}\sup_{I\in\mathfrak{S}_{J}({\mathbb{Z}}_+)}\|O^{2}_{I, J}( \mathfrak{f}_n : n \in {\mathbb{Z}}_+ ) \|_{L^{p} (X)}+\norm{\mathfrak f_{n_0}}_{L^p(X)}\end{aligned}$$ for any $n_0\in{\mathbb{Z}}_+$. This follows from Proposition \[prop:5\] below. Moreover, in view of Proposition \[prop:4\], inequality also gives a quantitative form of the martingale convergence theorem.

In Section \[section:3\] we give a new proof of inequality , which follows from an abstract result formulated for certain projections, see Theorem \[thm:p\] in Section \[section:3\]. This abstract theorem will also establish oscillation inequalities for smooth bump functions (see Proposition \[prop:2\] and Theorem \[thm:1\]), and establish oscillation inequalities for the Carleson operator (see Proposition \[prop:car\] as well as Proposition \[prop:6\]). It will also show that oscillation estimates are very close to maximal estimates even though it follows from Proposition \[prop:5\] that oscillations always dominate maximal functions, see the discussion below Theorem \[bourgain\].

Inequalities and are similar to inequalities and , respectively, and this raises a natural question whether $2$-oscillations can be interpreted as an endpoint for $r$-variations when $r>2$ in the sense of inequality . Recently this problem was investigated in @MSS [Theorem 1.9] and answered in the negative. Specifically, one can show if $1 \le p < \infty$ and $1 < \rho \le r < \infty$ are fixed, then it is **not true** that the estimates $$\begin{aligned}
\label{eq:49}
\begin{split}
\sup_{\lambda > 0} \| \lambda N_{\lambda}(f(\cdot,t):t\in{\mathbb{N}})^{1/r} \|_{\ell^{p,\infty}({\mathbb{Z}})} 
&\le
C_{p,\rho,r}
\sup_{I\in\mathfrak S_\infty ({\mathbb{N}})} 
\| O_{I,\infty}^{\rho}(f(\cdot,t):t\in{\mathbb{N}}) \|_{\ell^{p}({\mathbb{Z}})},\\
\| V^r(f(\cdot,t):t\in{\mathbb{N}}) \|_{\ell^{p,\infty}({\mathbb{Z}})} 
&\le
C_{p,\rho,r}
\sup_{I\in\mathfrak S_\infty ({\mathbb{N}})} 
\| O_{I,\infty}^{\rho} (f(\cdot,t):t\in{\mathbb{N}}) \|_{\ell^{p}({\mathbb{Z}})}
\end{split}\end{aligned}$$ hold uniformly for every measurable function $f \colon {\mathbb{Z}}\times {\mathbb{N}}\to {\mathbb{R}}$. The failure of the inequalities shows that the space induced by $\rho$-oscillations is different from the spaces induced by $r$-variations and $\lambda$ jumps whenever $\rho\le r$. Also the failure of the inequalities shows that $\rho$-oscillation inequalities cannot be seen (at least in a straightforward way, understood in the sense of inequality ) as endpoint estimates for $r$-variations, though it still makes sense to ask whether a priori bounds for $2$-oscillations imply bounds for $r$-variations for any $r>2$. This is an intriguing question from the point of view of quantitative pointwise convergence problems. If true, it would reduce pointwise convergence problems to the study of $2$-oscillations, which in certain cases are simpler since they are closer to square functions.

Quantitative forms of Bourgain's ergodic theorem
------------------------------------------------

Quantitative bounds in the context of ergodic polynomial averaging operators have been intensively studied over the last decade. These investigations were the subject of the following papers @MST1 [@MST2; @MSZ3; @MSS], which generalized Bourgain's papers @B1 [@B2; @B3] in various ways and can be summarized as follows.

\[thm:mainet\] Let $d, k\in{\mathbb{Z}}_+$, and $\mathcal P=(P_1, \ldots, P_d)\subset{\mathbb{Z}}[{\rm m}_1, \ldots, {\rm m}_k]$ such that $P_j(0)=0$ for $j\in[d]$ be given. Let $(X, {\mathcal{B}}(X), \mu)$ be a $\sigma$-finite measure space endowed with a family ${\mathcal{T}}=(T_1,\ldots, T_d)$ of commuting invertible measure-preserving transformations on $X$. Let $f\in L^p(X)$ for some $1\le p\le \infty$, and for $M\in{\mathbb{Z}}_+$ let $A_{{M}; X, {\mathcal T}}^{\mathcal P}f = A_{M_1,\ldots, M_k;X,T_1,\ldots, T_d}^{P_1,\ldots, P_d} f$ be the polynomial ergodic average defined in with the parameters $M_1=\ldots=M_k=M$.

-   *(Mean ergodic theorem)* If $1<p<\infty$, then the averages $A_{{M}; X, {\mathcal T}}^{\mathcal P}f$ converge in $L^p(X)$ norm as $M\to \infty$.

-   *(Pointwise ergodic theorem)* If $1<p<\infty$, then the averages $A_{{M}; X, {\mathcal T}}^{\mathcal P}f$ converge pointwise almost everywhere on $X$ as $M\to\infty$.

-   *(Maximal ergodic theorem)* If $1<p\le\infty$, then one has $$\begin{aligned}
    \label{eq:5}
    \big\|\sup_{M\in{\mathbb{Z}}_+}|A_{{M}; X, {\mathcal T}}^{\mathcal P}f|\big\|_{L^p(X)}\lesssim_{d,k,p, \deg \mathcal P}\|f\|_{L^p(X)}.\end{aligned}$$

-   *(Variational ergodic theorem)* If $1<p <\infty$ and $2<r<\infty$, then one has $$\begin{aligned}
    \label{eq:7}
    \big\|V^r(A_{{M}; X, {\mathcal T}}^{\mathcal P}f: M\in{\mathbb{Z}}_+)\big\|_{L^p(X)}\lesssim_{d,k,p, r,  \deg \mathcal P}\|f\|_{L^p(X)}.\end{aligned}$$

-   *(Jump ergodic theorem)* If $1<p<\infty$, then one has $$\begin{aligned}
    \label{eq:36}
    \sup_{\lambda>0}\big\|\lambda N_{\lambda}(A_{{M}; X, {\mathcal T}}^{\mathcal P}f:M\in{\mathbb{Z}}_+)^{1/2}\big\|_{L^p(X)}\lesssim_{d,k,p, \deg \mathcal P}\|f\|_{L^p(X)}.\end{aligned}$$

-   *(Oscillation ergodic theorem)* If $1<p<\infty$, then one has $$\begin{aligned}
    \label{eq:37}
    \sup_{J\in{\mathbb{Z}}_+}\sup_{I\in\mathfrak S_J({\mathbb{Z}}_+) }\big\|O_{I, J}^2(A_{{M}; X, {\mathcal T}}^{\mathcal P}f: M\in{\mathbb{Z}}_+)\big\|_{L^p(X)}\lesssim_{d,k,p, \deg \mathcal P}\|f\|_{L^p(X)}.\end{aligned}$$

Moreover, the implicit constants in , , and can be taken to be independent of the coefficients of the polynomials from $\mathcal P$, depending only on $p$ and the degree of the family $\mathcal P$.

We now give some remarks about Theorem \[thm:mainet\].

1.  Theorem \[thm:mainet\] is a multi-dimensional, quantitative counterpart of Theorem \[bourgain\] with sharp ranges of parameters $1<p<\infty$ and $2<r<\infty$, which contributes to the Furstenberg–Bergelson–Leibman conjecture @BL [Section 5.5, p. 468] in the linear case for the class of commuting measure-preserving transformations. The Furstenberg–Bergelson–Leibman conjecture is a central open problem in pointwise ergodic theory. Moreover, inequalities and are the strongest possible quantitative forms of pointwise convergence. By taking $d=k=1$ and $P_1(m)=m$ in Theorem \[thm:mainet\], we recover Birkhoff's and von Neumann's results stated in Theorem \[birk\]. Taking $d=k=1$ and $P_1\in{\mathbb{Z}}[{\rm m}]$ in Theorem \[thm:mainet\], we also recover Bourgain's polynomial ergodic theorem from Theorem \[bourgain\] above.

2.  The mean ergodic theorem in (i) is a consequence of the dominated convergence theorem combined with (ii) and (iii). Each of the conclusions from (iv), (v) and (vi) individually implies pointwise convergence from (ii), as well as the maximal estimates from (iii). It also follows from that (v) implies (iv). Details about these implications can be easily derived from the properties of oscillations, variations and jumps collected in Section \[section:2\].

3.  Sharp $r$-variational estimates were obtained for the first time in @MST2, with a conceptually new proof which also works for other discrete operators with arithmetic features @MST1. Not long afterwards, the ideas from @MST2 were extended @MSZ3 to establish uniform $\lambda$-jump estimates . Partial result for $r$-variational estimates were obtained in @Kr [@zk; @MT].

4.  It was observed in @MST2 that and Hölder's inequality imply that for every $p\in(1, \infty)$, for any $r>2$, every $f\in L^p(X)$ and every $J\in{\mathbb{Z}}_+$ one has $$\begin{aligned}
    \qquad \qquad \sup_{I\in\mathfrak S_J({\mathbb{Z}}_+) }\big\|O_{I, J}^2(A_{{M}; X, {\mathcal T}}^{\mathcal P}f: M\in{\mathbb{Z}}_+)\big\|_{L^p(X)}
    \lesssim_{d,k,p, r, \deg \mathcal P} J^{\frac{1}{2}-\frac{1}{r}} \norm{f}_{L^p(X)},\end{aligned}$$ with the same implicit constant as in and so blows up as $r$ tends to 2. This inequality is a non-uniform version of in the spirit of Bourgain's oscillation inequality . However it was observed recently @MSS that the methods from @MST2 [@MSZ3] give the uniform oscillation inequality in . From this point of view (and from the discussion above for martingales) inequality can be thought of as an endpoint for at $r=2$, though it is not an endpoint in the sense of inequality below. It would be nice to know whether it is possible (if at all) to use to recover .

5.  Inequality is also a contribution to an interesting problem from the early 1990's of Rosenblatt and Wierdl @RW [Problem 4.12, p. 80] about uniform estimates of oscillation inequalities for ergodic averages. In @jkrw Jones, Kaufman, Rosenblatt and Wierdl proved for the classical Birkhoff averages with $d=k=1$ and $P_1(m)=m$, giving an affirmative answer to @RW [Problem 4.12, p. 80]. In @MSS it was shown that @RW [Problem 4.12, p. 80] remains true even for multidimensional polynomial ergodic averages.

6.  The proof of Theorem \[thm:mainet\] is an elaboration of methods developed in @MST2 [@MSZ3] and also recently in @MSS. The main tools are the Hardy–Littlewood circle method (major arcs estimates); Weyl's inequality (minor arcs estimates); the Ionescu–Wainger multiplier theory (see @IW [@M10; @MSZ3] and also @Pierce, @TaoIW); the Rademacher–Menshov argument (see for instance @MSZ2) and the sampling principle of Magyar–Stein–Wainger (see @MSW and also @MSZ1). The methods from @M10 [@MST2; @MSZ1; @MSZ2; @MSZ3] were further developed by the first author in collaboration with Krause and Tao @KMT, which resulted in establishing pointwise convergence for the so-called bilinear Furstenberg–Weiss ergodic averages. This was a long-standing open problem, which makes a significant contribution towards the Furstenberg–Bergelson–Leibman conjecture @BL.

A multi-parameter variant of the Bellow and Furstenberg problem
---------------------------------------------------------------

After completing @B1 [@B2; @B3], Bourgain observed that the Dunford–Zygmund theorem (see Theorem \[dz\]) can be extended to the polynomial setting at the expense of imposing that the measure-preserving transformations in Theorem \[dz\] commute. Bourgain's result can be formulated as follows.

\[Polynomial Dunford–Zygmund ergodic theorem\] \[thm:main\] Let $d\in{\mathbb{Z}}_+$, and $P_1, \ldots, P_d\in{\mathbb{Z}}[{\rm m}]$ such that $P_j(0)=0$ for $j\in[d]$ be given. Let $(X, {\mathcal{B}}(X), \mu)$ be a $\sigma$-finite measure space endowed with a family ${\mathcal{T}}=(T_1,\ldots, T_d)$ of commuting invertible measure-preserving transformations on $X$. Let $f\in L^p(X)$ for some $1\le p\le \infty$, and for $M\in{\mathbb{Z}}_+^d$ let $A_{M; X, {\mathcal{T}}}^{P_1 ({\mathrm m}_1),\ldots, P_d ({\mathrm m}_d)}f = A_{M_1,\ldots, M_d;X,T_1,\ldots, T_d}^{P_1 ({\mathrm m}_1),\ldots, P_d ({\mathrm m}_d)}f$ be the polynomial ergodic average defined in .

-   *(Mean ergodic theorem)* If $1<p<\infty$, then the averages $A_{M; X, {\mathcal{T}}}^{P_1 ({\mathrm m}_1),\ldots, P_d ({\mathrm m}_d)}f$ converge in $L^p(X)$ norm as $\min\{M_1,\ldots, M_d \}\to\infty$.

-   *(Pointwise ergodic theorem)* If $1<p<\infty$, then the averages $A_{M; X, {\mathcal{T}}}^{P_1 ({\mathrm m}_1),\ldots, P_d ({\mathrm m}_d)}f$ converge pointwise almost everywhere on $X$ as $\min\{M_1,\ldots, M_d\}\to\infty$.

-   *(Maximal ergodic theorem)* If $1<p\le\infty$, then one has $$\begin{aligned}
    \label{eq:51}
    \big\|\sup_{M\in{\mathbb{Z}}_+^d}|A_{M; X, {\mathcal{T}}}^{P_1 ({\mathrm m}_1),\ldots, P_d ({\mathrm m}_d)}f|\big\|_{L^p(X)}\lesssim_{d,p, \deg P_1,\ldots, \deg P_d}\|f\|_{L^p(X)}.\end{aligned}$$

-   *(Oscillation ergodic theorem)* If $1<p<\infty$, then one has $$\begin{aligned}
    \label{eq:52}
    \sup_{J\in{\mathbb{Z}}_+}\sup_{I\in\mathfrak S_J({\mathbb{Z}}_+^d) }\big\|O_{I, J}^2(A_{M; X, {\mathcal{T}}}^{P_1 ({\mathrm m}_1),\ldots, P_d ({\mathrm m}_d)}f: M\in{\mathbb{Z}}_+^d)\big\|_{L^p(X)}\lesssim_{d,p, \deg P_1,\ldots, \deg P_d}\|f\|_{L^p(X)}.\end{aligned}$$

We refer to Section \[section:2\] for the definitions of the sets $\mathfrak S_J({\mathbb{Z}}_+^d)$, see , and the multi-parameter oscillations, see . Moreover, the implicit constants in and can be taken to be independent of the coefficients of the polynomials $P_1,\ldots, P_d$, depending only on $p$ and $\deg P_1,\ldots, \deg P_d$.

We now give some remarks about Theorem \[thm:main\].

1.  Theorem \[thm:main\](i)-(iii) is attributed to Bourgain, though it has never been published. The first and third authors learned about this result from Bourgain in October 2016, when they started to work with Bourgain and Stein on some aspects of multi-parameter ergodic theory @BMSWer.

2.  In this paper we prove Theorem \[thm:main\] using a general abstract principle, see Proposition \[prop:8\] in Section \[section:4\]. In contrast to Bourgain's original observation our proof of Theorem \[thm:main\] relies on uniform bounds for multi-parameter oscillation inequalities.

3.  Theorem \[thm:main\](iv) with linear polynomials $P_1(m)=\ldots=P_d(m)=m$ was established in @JRW, where it was essential that ${\mathcal{T}}=(T_1,\ldots, T_d)$ is a commuting family of measure-preserving transformations on $X$. It is straightforward to see that (iv) implies (iii) by , as well as (ii) by appealing to Proposition \[prop:4\]. Using the dominated convergence theorem with (ii) and (iii) we also obtain (i). So it suffices to prove , which we do in Section \[section:4\].

4.  To prove Theorem \[thm:main\] it is essential to note that $$\begin{aligned}
    \label{eq:50}
    A_{M; X, {\mathcal{T}}}^{P_1 ({\mathrm m}_1),\ldots, P_d ({\mathrm m}_d)}f=A_{M_1,\ldots, M_d; X, T_1,\ldots, T_d}^{P_1 ({\mathrm m}_1),\ldots, P_d ({\mathrm m}_d)}f=A_{M_1; X, T_1}^{P_1({\mathrm m}_1)}\circ\ldots\circ A_{M_d; X, T_d}^{P_d({\mathrm m}_d)}f,\end{aligned}$$ where the latter averages (defined as in ) commute as long as the family ${\mathcal{T}}=(T_1,\ldots, T_d)$ is commuting. Using identity and iterating appropriately with $k=d=1$ we will be able to derive . We refer to Section \[section:4\] for details.

Theorem \[thm:main\] can be thought of as a simple case of a multi-parameter variant of the Bellow and Furstenberg problem, which is a central open problem in modern ergodic theory and can be subsumed under the following conjecture:

\[con:1\] Let $d, k\in{\mathbb{Z}}_+$ be given and let $(X, \mathcal B(X), \mu)$ be a probability measure space endowed with a family ${\mathcal{T}}=(T_1,\ldots, T_d)$ of invertible commuting measure-preserving transformations on $X$. Assume that ${\mathcal{P}}=(P_1,\ldots, P_d) \subset{\mathbb{Z}}[{\rm m}_1, \ldots, {\rm m}_k]$ such that $P_j(0)=0$ for $j\in[d]$ are given. Then for any $f\in L^{\infty}(X)$ the multi-parameter polynomial averages $A_{M; X, {\mathcal{T}}}^{{\mathcal{P}}}f(x)=A_{M_1,\ldots, M_k;X,T_1,\ldots, T_d}^{P_1,\ldots, P_d} f(x)$ defined in converge for $\mu$-almost every $x\in X$, as $\min\{M_1,\ldots, M_k\}\to\infty$.

A few remarks about this conjecture, its history, and the current state of the art are in order.

1.  As we have seen above the case $d=k=1$ of Conjecture \[con:1\] with $P_1(m)=m$ follows from Birkhoff's ergodic theorem, see Theorem \[birk\]. The case $d=k=1$ of Conjecture \[con:1\] with arbitrary polynomials $P_1\in{\mathbb{Z}}[\rm n]$ was the famous open problem of Bellow @Bel and Furstenberg @F and solved by Bourgain @B1 [@B2; @B3] in the mid 1980's, see Theorem \[bourgain\]. The general case $d,k\in{\mathbb{Z}}_+$ of Conjecture \[con:1\] with arbitrary polynomials $P_1,\ldots, P_d\in{\mathbb{Z}}[{\rm m}_1, \ldots, {\rm m}_k]$ in the diagonal setting $M_1=\ldots=M_k$, that is, the multi-dimensional one-parameter setting, follows from Theorem \[thm:mainet\].

2.  A genuinely multi-parameter case $d=k\ge2$ of Conjecture \[con:1\] for averages with $P_{j}(m_1, \ldots, m_{d})=P_j(m_j)$, where $P_j\in{\mathbb{Z}}[{\rm m}_j]$ for $j\in[d]$ follows from Theorem \[thm:main\], which extends the case of linear polynomials $P_1(m)=\ldots=P_d(m)=m$ established independently by Dunford @D and Zygmund @Z in the early 1950's, see Theorem \[dz\].

3.  Thanks to the product structure of Theorem \[dz\], as well as Theorem \[thm:main\], have relatively simple one-parameter proofs, which are based on iterative applications of Theorem \[birk\] and Theorem \[thm:mainet\], respectively. This is explained in Proposition \[prop:8\] below. However, the situation is dramatically different when orbits in are defined along genuinely $k$-variate polynomials $P_1,\ldots, P_d\in{\mathbb{Z}}[{\rm m}_1, \ldots, {\rm m}_k]$ since then we lose the product structure . This can be illustrated by considering averages for $d=1$, $k=2$ with, let us say, $P_1(m_1, m_2)=m_1^2m_2^3$. Then Conjecture \[con:1\] becomes challenging. Surprisingly, even in this simple case, it seems that there is no simple way (like changing variables or interpreting the average from as a composition of simpler one-parameter averages as in ) that would help us reduce the matter to the setup where pointwise convergence is known. This was one of the motivations leading to Conjecture \[con:1\].

4.  The Dunford–Zygmund theorem (see Theorem \[dz\] above) was originally proved for not necessarily commuting, measure-preserving transformations ${\mathcal{T}}=(T_1,\ldots, T_d)$ on $X$. However, it is well known for instance from the Bergelson–Leibman paper @BL that the commutation assumption imposed on the family $T_1,\ldots, T_d:X\to X$ in is essential in order to have an ergodic theorem if $\deg P_j\ge2$ for at least one $j\in[d]$ and $d\ge2$. Even in the one-parameter case (assuming $k=1$) in an ergodic theorem may fail. The question to what extent one can relax commutation relations among $T_1,\ldots, T_d$ in , even in the one-parameter case, is very intriguing. This also motivates the desire to understand Conjecture \[con:1\] in the commutative setting first, as it is unclear whether Conjecture \[con:1\] is true for all polynomials $P_1,\ldots, P_d\in{\mathbb{Z}}[{\rm m}_1, \ldots, {\rm m}_k]$.

5.  With respect to the noncommutative setting, we mention that recently the first and second authors with Ionescu and Magyar @IMMS established Conjecture \[con:1\] with $k=1$, $d\in{\mathbb{Z}}_+$ and arbitrary polynomials $P_1,\ldots, P_d\in{\mathbb{Z}}[\rm m]$ in the diagonal nilpotent setting, i.e. one-parameter and multi-dimensional, when ${\mathcal{T}}=(T_1,\ldots, T_d)$ is a family of invertible measure-preserving transformations of a $\sigma$-finite measure space $(X, \mathcal B(X), \mu)$ that generates a nilpotent group of step two. In view of the Bergelson–Leibman paper @BL, the nilpotent setting is probably the most general setting where Conjecture \[con:1\] might be true, at least in the one-parameter case.

6.  We finally mention that progress towards establishing Conjecture \[con:1\] was recently made by the first and third authors in collaboration with Bourgain and Stein @BMSWer. This conjecture was verified for any integer $d\ge2$ with $k=d-1$ for averages with polynomials $$\begin{aligned}
    \label{eq:53}
    \begin{split}
    P_{j}({\rm m}_1, \ldots, {\rm m}_{d-1})&={\rm m}_j \quad \text{for}\quad  j\in[d-1]; \quad \text{and} \\
    P_{d}({\rm m}_1, \ldots, {\rm m}_{d-1})&=P({\rm m}_1, \ldots, {\rm m}_{d-1}),
    \end{split}\end{aligned}$$ whenever $P\in{\mathbb{Z}}[{\rm m}_1,\ldots,  {\rm m}_{d-1}]$ is a polynomial such that $$\begin{aligned}
    \qquad P(0,\ldots, 0)=\partial_1P(0,\ldots, 0)=\ldots=\partial_{d-1}P(0,\ldots, 0)=0,\end{aligned}$$ which has partial degrees (as a polynomial of the variable ${\rm m}_i$ for any $i\in[d-1]$) at least two. Furthermore, it follows from @BMSWer that for any $P\in{\mathbb{Z}}[{\rm m}_1,\ldots, {\rm m}_{d}]$ the following averages $$\begin{aligned}
    \label{eq:3}
    \qquad \qquad A_{{M}; X, {\mathcal T}}^{P}f(x): =  {\mathbb{E}}_{(m_1,\ldots, m_{d})\in Q_{M}}f(T^{P(m_1,\ldots, m_{d})}x), \qquad x\in X,\end{aligned}$$ where $M=(M_1,\ldots, M_{d})\in{\mathbb{Z}}_+^{d},$ converge almost everywhere on $X$ as $\min\{M_1,\ldots, M_{d}\}\to\infty$. In fact, Conjecture \[con:1\] was originally formulated with averages , the authors learned about this from Jean Bourgain in a private communication in October 2016. The proof from @BMSWer developed new methods from Fourier analysis and number theory. Even though the averages with polynomials from share a lot of difficulties that arise in the general case, there are some cases that are not covered by the methods developed in @BMSWer. At this moment it is not clear whether Conjecture \[con:1\] is true in full generality. The work in @BMSWer is a significant step towards understanding Conjecture \[con:1\] that sheds new light on the general case and will either lead to its full resolution or to a counterexample. The authors plan to investigate this question in the near future.

Overview of the paper
---------------------

In this paper we prove an abstract principle for the so-called projective operators, see Theorem \[thm:p\] in Section \[section:3\], which allows us to deal with one-parameter oscillation inequalities in a fairly unified way. As a consequence of Theorem \[thm:p\] we give a simple proof of Jones, Kaufman, Rosenblatt and Wierdl @jkrw [Theorem 6.4, p. 930] oscillation inequality for martingales, see Proposition \[prop:1\], then we prove oscillation inequalities for smooth bumps, see Proposition \[prop:2\] and Theorem \[thm:1\]. Further, we discuss oscillation estimates for projection operators corresponding to orthonormal systems in Hilbert spaces, see Proposition \[prop:6\], and finally we obtain new oscillations inequalities for the Carleson operator, see Proposition \[prop:car\]. In Section \[section:4\] we build a multi-parameter theory of oscillation estimates, see Proposition \[prop:8\] and Corollary \[cor:2\]. As an application of our method, we give a simple proof of Theorem \[thm:main\].

This paper can be viewed as a fairly systematic treatment of oscillation estimates in the one-parameter as well as multi-parameter settings in ergodic theory and analysis. In the multi-parameter setting, oscillation semi-norms seem to be the only viable tool that allows us to handle efficiently multi-parameter pointwise convergence problems. This is especially the case in @BMSWer where operators with arithmetic features were studied. It also contrasts sharply with the one-parameter setting, where we have a variety of available tools to handle pointwise convergence problems: including oscillations, variations or jumps, see @JSW [@MST2] and the references given there.

Notation and useful tools {#section:2}
=========================

We now set some notation that will be used throughout the paper. Basic properties of one-parameter as well as multi-parameter $r$-oscillation semi-norms, $r$-variation semi-norms and $\lambda$-jump counting functions will be also gathered here. We borrow notation from @BMSWer [Section 2] and @MSS [Section 2].

Basic notation
--------------

Let ${\mathbb{Z}}_+:=\{1, 2, \ldots\}$, ${\mathbb{N}}:=\{0,1,2,\ldots\}$ and ${\mathbb{R}}_+:=(0, \infty)$. 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 standard meaning. We will also consider the set of dyadic numbers ${\mathbb{D}}:=\{2^n: n\in{\mathbb{Z}}\}$. For any $x\in{\mathbb{R}}$ we define the floor function $$\lfloor x \rfloor: = \max\{ n \in {\mathbb{Z}}: n \le x \}.$$ For $x, y\in{\mathbb{R}}$ let $x \wedge y := \min\{x,y\}$ and $x \vee y := \max\{x,y\}$. For every $N\in{\mathbb{R}}_+$ and ${\mathbb{A}}\subseteq{\mathbb{R}}$ define $$[N]:=(0, N]\cap{\mathbb{Z}}=\{1, \ldots, \lfloor N\rfloor\},$$ as well as $$\begin{aligned}
{\mathbb{A}}_{\le N}:= [0, N]\cap{\mathbb{A}},\ \: \quad &\text{ and } \quad
{\mathbb{A}}_{< N}:= [0, N)\cap{\mathbb{A}},\\
{\mathbb{A}}_{\ge N}:= [N, \infty)\cap{\mathbb{A}}, \quad &\text{ and } \quad
{\mathbb{A}}_{> N}:= (N, \infty)\cap{\mathbb{A}}.\end{aligned}$$

We use ${\mathds{1}_{{A}}}$ to denote the indicator function of a set $A$. If $S$ is a statement we write ${\mathds{1}_{{S}}}$ to denote its indicator, equal to $1$ if $S$ is true and $0$ if $S$ is false. For instance ${\mathds{1}_{{A}}}(x)={\mathds{1}_{{x\in A}}}$.

For two nonnegative quantities $A, B$ we write $A \lesssim B$ if there is an absolute constant $C>0$ such that $A\le CB$, however $C>0$ may change from occurrence to occurrence. We will write $A \simeq B$ when $A \lesssim B\lesssim A$. We will write $\lesssim_{\delta}$ or $\simeq_{\delta}$ to emphasize that the implicit constant depends on $\delta$. For two functions $f:X\to {\mathbb{C}}$ and $g:X\to [0, \infty)$, write $f = O(g)$ if there exists $C>0$ such that $|f(x)| \le C g(x)$ for all $x\in X$. We will also write $f = O_{\delta}(g)$ if the implicit constant depends on $\delta$.

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 $$\begin{aligned}
x\cdot\xi:=\sum_{k=1}^dx_k\xi_k, \qquad \text{ and } \qquad
\abs{x}:=\abs{x}_2:=\sqrt{{x\cdotx}}, \qquad \text{ and } \qquad |x|_{\infty}:=\max_{k\in[d]}|x_k|.\end{aligned}$$

Function spaces
---------------

Throughout this paper all vector spaces will be defined over ${\mathbb{C}}$. For a continuous linear map $T : B_1 \to B_2$ between two normed vector spaces $B_1$ and $B_2$, its operator norm will be denoted by $\|T\|_{B_1 \to B_2}$.

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 $p$-th 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 separable normed vector space $(B, \|\cdot\|_B)$, for instance $$\begin{aligned}
L^{p}(X;B)
:=\big\{F\in L^0(X;B):\|F\|_{L^{p}(X;B)} \coloneqq \left\|\|F\|_B\right\|_{L^{p}(X)}<\infty\big\},\end{aligned}$$ where $L^0(X;B)$ denotes[^2] the space of measurable functions from $X$ to $B$ (up to almost everywhere equivalence). For any $p\in[1,\infty]$ we define a weak-$L^p$ space of measurable functions on $X$ by setting $$L^{p,\infty}(X):=\{f:X\to{\mathbb{C}}\colon\norm{f}_{L^{p,\infty}(X)}<\infty\},$$ where for any $p\in[1,\infty)$ we have $$\begin{aligned}
\norm{f}_{L^{p,\infty}(X)}:=\sup_{\lambda>0}\lambda\mu(\{x\in X:|f(x)|>\lambda\})^{1/p},
\qquad\text{\and}\qquad
\norm{f}_{L^{\infty,\infty}(X)}:=\norm{f}_{L^{\infty}(X)}.\end{aligned}$$

In our case we will mainly take $X={\mathbb{R}}^d$ 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 $\ell^p(X)$ and $L^p(X; B)$ to $\ell^p(X; B)$ and $L^{p,\infty}(X)$ to $\ell^{p,\infty}(X)$.

Fourier transform
-----------------

We will use the convention that ${\bm{e}}(z)=e^{2\pi {\bm i} z}$ for every $z\in{\mathbb{C}}$, where ${\bm i}^2=-1$. 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 $$\begin{aligned}
{\mathcal{F}}_{{\mathbb{R}}^d} f(\xi) := \int_{{\mathbb{R}}^d} f(x) {\bm{e}}(x\cdot\xi) {\rm d}x.\end{aligned}$$ We can also consider the Fourier transform for finite Borel measures $\sigma$ on ${\mathbb{R}}^d$. 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 $$\begin{aligned}
{\mathcal{F}}_{{\mathbb{Z}}^d}f(\xi): = \sum_{x \in {\mathbb{Z}}^d} f(x) {\bm{e}}(x\cdot\xi).\end{aligned}$$ Sometimes we shall abbreviate ${\mathcal{F}}_{{\mathbb{Z}}^d}f$ or ${\mathcal{F}}_{{\mathbb{R}}^d}f$ to $\hat{f}$, if the context will be clear.

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 $$\begin{aligned}
\label{eq:1}
T_{{\mathbb{G}}}[\mathfrak m]f(x):=\int_{{\mathbb{G}}^*}{\bm{e}}(-\xi\cdot x)\mathfrak m(\xi){\mathcal{F}}_{{\mathbb{G}}}f(\xi){\rm d}\xi, \quad \text{ for } \quad x\in{\mathbb{G}}.\end{aligned}$$ 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 makes sense.

Littlewood–Paley theory {#lp}
-----------------------

Often we will control oscillation and variation semi-norms by certain square functions of the form $$S(f)(x) := \Bigl(\sum_{k\in {\mathbb Z}} |\sigma_k*f(x)|^2 \Bigr)^{1/2},$$ where $(\sigma_k)_{k \in {\mathbb{Z}}}$ is a sequence of Borel measures on ${\mathbb R}^d$ with bounded total variation satisfying $|{\widehat{\sigma_k}}(\xi)| \le C \min\{|a_{k+1} \xi|^{\alpha}, |a_k \xi|^{-\alpha}\}$ for some $\alpha>0$ and all $k\in {\mathbb{Z}}$. Here $\inf_{k\in{\mathbb{Z}}} a_{k+1}/a_k > 1$. What we call <span>*standard Littlewood–Paley arguments*</span> sometimes refer to the arguments developed in the seminal paper @DR. In particular, Theorem B in @DR implies that the square function $S$ satisfies $L^p$ bounds $\|S(f)\|_{L^p} \le C_p \|f\|_{L^p}$ for all $p\in(1, \infty)$ whenever the corresponding maximal function $\sigma^{*}$ associated to the measures $(\sigma_k)_{k \in {\mathbb{Z}}}$ satisfies the same $L^p$ bounds.

At one point we will use a powerful square function bound of Rubio de Francia associated to any pairwise disjoint collection of intervals $(I_j : j \in {\mathbb{Z}})$ on ${\mathbb{R}}$. It states $$\Big\|\Bigl(\sum_{j\in{\mathbb{Z}}} |T_{{\mathbb{R}}}[{\mathds{1}_{{I_j}}}] f|^2 \Bigr)^{1/2} \Big\|_{L^p({{\mathbb{R}}})} \ \lesssim \ \|f\|_{L^p({\mathbb{R}})}$$ whenever $p\in [2, \infty)$. See @RdF [Theorem 1.2].

Coordinatewise order $\preceq$
------------------------------

For any $x=(x_1,\ldots, x_k)\in{\mathbb{R}}^k$ and $y=(y_1,\ldots, y_k)\in{\mathbb{R}}^k$ we say $x\preceq y$ if an only if $x_i\le y_i$ for each $i\in[k]$. We also write $x\prec y$ if an only if $x\preceq y$ and $x\neq y$, and $x\prec_{\rm s} y$ if an only if $x_i< y_i$ for each $i\in[k]$. Let ${\mathbb{I}}\subseteq {\mathbb{R}}^k$ be an index set such that $\# {\mathbb{I}}\ge2$ and for every $J\in{\mathbb{Z}}_+\cup\{\infty\}$ define the set $$\begin{aligned}
\label{eq:2}
\mathfrak S_J({\mathbb{I}}):
=
\Set[\big]{(t_i:i\in{\mathbb{N}}_{\le J})\subseteq {\mathbb{I}}}{t_{0}\prec_{\rm s}
t_{1}\prec_{\rm s}\ldots \prec_{\rm s}t_{J}},\end{aligned}$$ where ${\mathbb{N}}_{\le \infty}:={\mathbb{N}}$. In other words, $\mathfrak S_J({\mathbb{I}})$ is the family of all strictly increasing sequences (with respect to the coordinatewise order) of length $J+1$ taking their values in the set ${\mathbb{I}}$.

Oscillation semi-norms
----------------------

Let ${\mathbb{I}}\subseteq {\mathbb{R}}^k$ be an index set such that $\#{{\mathbb{I}}}\ge2$. Let $(\mathfrak a_{t}(x): t\in{\mathbb{I}})$ be a $k$-parameter family of complex-valued measurable functions defined on $X$. For any ${\mathbb{J}}\subseteq {\mathbb{I}}$, any $1\le r<\infty$ and a sequence $I=(I_i : i\in{\mathbb{N}}_{\le J}) \in \mathfrak S_J({\mathbb{I}})$ the multi-parameter $r$-oscillation seminorm is defined by $$\begin{aligned}
\label{eq:102}
O_{I, J}^r(\mathfrak a_{t}(x): t \in {\mathbb{J}}):=
\Big(\sum_{j=0}^{J-1}\sup_{t\in {\mathbb{B}}[I_j]\cap{\mathbb{J}}}
\abs{\mathfrak a_{t}(x) - \mathfrak a_{I_j}(x)}^r\Big)^{1/r},\end{aligned}$$ where ${\mathbb{B}}[I_i]:=[I_{i1}, I_{(i+1)1})\times\ldots\times[I_{ik}, I_{(i+1)k})$ is a box determined by the element $I_i=(I_{i1}, \ldots, I_{ik})$ of the sequence $I\in \mathfrak S_J({\mathbb{I}})$. In order to avoid problems with measurability we 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.

\[rem:1\] Let $1\le r<\infty$. Some remarks are in order.

1.  Clearly $O_{I, J}^r(\mathfrak a_{t}: t \in {\mathbb{J}})$ defines a semi-norm.

2.  Let ${\mathbb{I}}\subseteq {\mathbb{R}}^k$ be an index set such that $\#{{\mathbb{I}}}\ge2$, and let ${\mathbb{J}}_1, {\mathbb{J}}_2\subseteq {\mathbb{I}}$ be disjoint. Then for any family $(\mathfrak a_t:t\in{\mathbb{I}})\subseteq {\mathbb{C}}$, any $J\in{\mathbb{Z}}_+$ and any $I\in\mathfrak S_J({\mathbb{I}})$ one has $$\begin{aligned}
    O_{I, J}^r(\mathfrak a_{t}: t\in{\mathbb{J}}_1\cup{\mathbb{J}}_2)
    \le O_{I, J}^r(\mathfrak a_{t}: t\in{\mathbb{J}}_1)
    +O_{I, J}^r(\mathfrak a_{t}: t\in{\mathbb{J}}_2).\end{aligned}$$

3.  Let ${\mathbb{I}}\subseteq {\mathbb{R}}^k$ be a countable index set such that $\#{{\mathbb{I}}}\ge2$ and ${\mathbb{J}}\subseteq {\mathbb{I}}$. Then for any family $(\mathfrak a_t:t\in{\mathbb{I}})\subseteq {\mathbb{C}}$, any $J\in{\mathbb{Z}}_+$, any $I\in\mathfrak S_J({\mathbb{I}})$ one has $$\begin{aligned}
    \label{eq:4}
     O_{I, J}^r(\mathfrak a_{t}: t \in {\mathbb{J}})\lesssim \Big(\sum_{t\in{\mathbb{I}}}|\mathfrak a_{t}|^r\Big)^{1/r}.
     \end{aligned}$$

4.  Let $(\mathfrak a_t:t\in{\mathbb{I}}^k)$ be a $k$-parameter family of measurable functions on $X$. For any ${\mathbb{I}}\subseteq{\mathbb{R}}$ with $\#{{\mathbb{I}}}\ge2$ and any sequence $I=(I_i : i\in{\mathbb{N}}_{\le J}) \in \mathfrak S_J({\mathbb{I}})$ of length $J\in{\mathbb{Z}}_+\cup\{\infty\}$ we define the diagonal sequence $\bar{I}=(\bar{I}_i : i\in{\mathbb{N}}_{\le J}) \in \mathfrak S_J({\mathbb{I}}^k)$ by setting $\bar{I}_i=(I_i,\ldots,I_i)\in{\mathbb{I}}^k$ for each $i\in{\mathbb{N}}_{\le J}$. Then for any $p\in[1, \infty]$ and for any ${\mathbb{J}}\subseteq {\mathbb{I}}^k$ one has $$\begin{aligned}
    \sup_{I\in \mathfrak S_J({\mathbb{I}})}\norm[\big]{O_{\bar{I}, J}^r(\mathfrak a_{t}: t \in {\mathbb{J}})}_{L^p(X)}
    \le
    \sup_{I\in \mathfrak S_J({\mathbb{I}}^k)}\norm[\big]{O_{I, J}^r(\mathfrak a_{t}: t \in {\mathbb{J}})}_{L^p(X)}.\end{aligned}$$

We now show that oscillation semi-norms always dominate maximal functions.

\[prop:5\] Assume that $k\in{\mathbb{Z}}_+$, ${\mathbb{I}}\subseteq {\mathbb{R}}$ be such that $\#{{\mathbb{I}}}\ge2$ and let $(\mathfrak a_{t}: t\in{\mathbb{I}}^k)$ be a $k$-parameter family of measurable functions on $X$. Then for every $p\in[1, \infty]$ and $r\in[1, \infty)$ we have $$\begin{aligned}
\label{eq:9}
\norm[\big]{\sup_{t \in ({\mathbb{I}}\setminus\{\sup{\mathbb{I}}\})^k}\abs{\mathfrak a_{t}}}_{L^p(X)}
\le \sup_{t \in {\mathbb{I}}^k}\norm{\mathfrak a_{t}}_{L^p(X)}
+\sup_{J\in{\mathbb{Z}}_+}\sup_{I\in \mathfrak S_J({\mathbb{I}})}\norm[\big]{O_{\bar{I}, J}^r(\mathfrak a_{t}: t \in {\mathbb{I}}^k)}_{L^p(X)},\end{aligned}$$ where $\bar{I}\in\mathfrak S_J({\mathbb{I}}^k)$ is the diagonal sequence corresponding to a sequence $I\in \mathfrak S_J({\mathbb{I}})$ as in Remark \[rem:1\].

Let $a=\inf{\mathbb{I}}$ and $b=\sup{\mathbb{I}}$. We see that $a<b$, since $\#{{\mathbb{I}}}\ge2$. We choose a decreasing sequence $(a_n:n\in{\mathbb{N}})\subseteq{\mathbb{I}}$ and an increasing sequence $(b_n:n\in{\mathbb{N}})\subseteq{\mathbb{I}}$ such that $a\le a_n\le b_n\le b$ for every $n\in{\mathbb{N}}$ satisfying $$\lim_{n\to\infty}a_n=a,
\qquad \text{and}\qquad
\lim_{n\to\infty}b_n=b$$ and such that $a_n = a$ for all $n \in {\mathbb{N}}$ if $a \in {\mathbb{I}}$. By the monotone convergence theorem we get $$\begin{aligned}
\norm[\big]{\sup_{t \in ({\mathbb{I}}\setminus\{\sup{\mathbb{I}}\})^k}\abs{\mathfrak a_{t}}}_{L^p(X)}
&=\lim_{n\to\infty}\norm[\big]{\sup_{t \in [a_n, b_n)^k\cap{\mathbb{I}}^k}\abs{\mathfrak a_{t}}}_{L^p(X)}\\
&\le\sup_{n} \ \norm{\mathfrak a_{\bar{a}_n}}_{L^p(X)}+
\sup_{n} \ \norm[\big]{\sup_{t \in [a_n, b_n)^k\cap{\mathbb{I}}^k}\abs{\mathfrak a_{t}-\mathfrak a_{\bar{a}_n}}}_{L^p(X)},\end{aligned}$$ where $\bar{a}_n=(a_n,\ldots, a_n)\in [a_n, b_n)^k \cap {\mathbb{I}}^k$, and consequently we obtain .

A remarkable feature of the oscillation seminorms is that they imply pointwise convergence. This property is formulated precisely in the following proposition.

\[prop:4\] Let $(X, {\mathcal{B}}(X), \mu)$ be a $\sigma$-finite measure space. For $k\in{\mathbb{Z}}_+$ let $(\mathfrak a_{t}: t\in{\mathbb{R}}_+^k)$ be a $k$-parameter family of measurable functions on $X$. Suppose that there are $p, r\in[1, \infty)$ such that for any $J\in{\mathbb{Z}}_+$ one has $$\begin{aligned}
\sup_{I\in \mathfrak S_J({\mathbb{R}}_+)}
\norm[\big]{O_{\bar{I}, J}^r(\mathfrak a_{t}: t \in {\mathbb{R}}_+^k)}_{L^p(X)}\le C_{p, r}(J),\end{aligned}$$ where $$\begin{aligned}
\lim_{J\to \infty}J^{-\frac{1}{p\vee r}}C_{p, r}(J)=0,\end{aligned}$$ and $\bar{I}\in\mathfrak S_J({\mathbb{R}}_+^k)$ is the diagonal sequence corresponding to a sequence $I\in \mathfrak S_J({\mathbb{R}}_+)$ as in Remark \[rem:1\]. Then the limits $$\begin{aligned}
\label{eq:161}
\lim_{\min\{t_1,\ldots, t_k\}\to\infty}\mathfrak a_{(t_1,\ldots, t_k)},
\qquad \text{ and } \qquad
\lim_{\max\{t_1,\ldots, t_k\}\to 0}\mathfrak a_{(t_1,\ldots, t_k)},\end{aligned}$$ exist $\mu$-almost everywhere on $X$.

We only prove the first conclusion of as the second one can be proved in much the same way. Suppose by contradiction that the first limit in does not exist $\mu$ almost everywhere on $X$. Since $\mu$ is a $\sigma$-finite measure then there exists $X_0\subseteq X$ such that $\mu(X_0)<\infty$, and also there is a small $\delta>0$ such that $$\begin{aligned}
\mu\big(\Set{x\in X_0}{\lim_{N\to\infty}\sup_{s, t\succeq \bar{N}}
\abs{\mathfrak a_{s}(x)-\mathfrak a_{t}(x)}>2\delta}\big)>2\delta,\end{aligned}$$ where $\bar{N}=(N,\ldots, N)\in{\mathbb{Z}}_+^k$. For $N\in{\mathbb{Z}}_+$ define $$\begin{aligned}
A_N:=\Set{x\in X_0}{\sup_{s, t\succeq \bar{N}}\abs{\mathfrak a_{s}(x)-\mathfrak a_{t}(x)}>2\delta}.\end{aligned}$$ Note that $A_{N+1}\subseteq A_N$ for every $N\in{\mathbb{Z}}_+$, and consequently from the continuity of measure one has $$\begin{aligned}
\lim_{N\to\infty}\mu\big(\Set{x\in X_0}{\sup_{s, t\succeq \bar{N}}\abs{\mathfrak a_{s}(x)-\mathfrak a_{t}(x)}>2\delta}\big)>2\delta.\end{aligned}$$ Hence there is an $N_0\in{\mathbb{Z}}_+$ such that for every $N\ge N_0$, we have $$\begin{aligned}
\mu\big(\Set{x\in X_0}{\sup_{t\succeq \bar{N}}\abs{\mathfrak a_{t}(x)-\mathfrak a_{\bar{N}}(x)}>\delta}\big)>\delta.\end{aligned}$$ For $M, N\in{\mathbb{Z}}_+$ we now define $$\begin{aligned}
B_M^N:=\Set{x\in X_0}{\sup_{\bar{N}\preceq t\prec_{\rm s}\bar{M}}\abs{\mathfrak a_{t}(x)-\mathfrak a_{\bar{N}}(x)}>\delta}.\end{aligned}$$ We observe that $B_M^N\subseteq B_{M+1}^N$ for every $M, N\in{\mathbb{Z}}_+$ and using once again continuity of measure, we obtain for every $N\ge N_0$, $$\begin{aligned}
\label{eq:162}
\lim_{M\to\infty}\mu(B_M^N)=
\mu\big(\Set{x\in X_0}{\sup_{t\succeq \bar{N}}\abs{\mathfrak a_{t}(x)-\mathfrak a_{\bar{N}}(x)}>\delta}\big)>\delta.\end{aligned}$$ Using recursively we can construct a strictly increasing sequence $(I_i:i\in{\mathbb{N}})\subset {\mathbb{R}}_+$ with $I_0=N_0$ such that for every $i\in{\mathbb{N}}$ we have $$\begin{aligned}
\label{eq:163}
\mu\big(\Set{x\in X_0}{\sup_{t\in{\mathbb{B}}[\bar{I}_i]}
\abs{\mathfrak a_{t}(x)-\mathfrak a_{\bar{I}_i}(x)}>\delta}\big)>\delta,\end{aligned}$$ where $\bar{I}_i=(I_i,\ldots,I_i)\in{\mathbb{R}}_+^k$. Then by we obtain for every $J\in{\mathbb{Z}}_+$ that $$\begin{aligned}
J\delta^{p+1}=\sum_{j=0}^{J-1}\delta^{p+1}&\le
\int_{X}\sum_{j=0}^{J-1}\sup_{t\in{\mathbb{B}}[\bar{I}_j]}
\abs{\mathfrak a_{t}(x)-\mathfrak a_{\bar{I}_j}(x)}^p{\mathrm{d}}\mu(x)\\
&\le J^{1-q/r}\sup_{I\in \mathfrak S_J({\mathbb{Z}}_+)}
\norm[\big]{O_{\bar{I}, J}^r(\mathfrak a_{t}: t \in {\mathbb{R}}_+^k)}_{L^p(X)}^p,\end{aligned}$$ where $q:=p\wedge r$. Thus $$\begin{aligned}
J^{q/r}\delta^{p+1}
\le \sup_{I\in \mathfrak S_J({\mathbb{R}}_+)}
\norm[\big]{O_{\bar{I}, J}^r(\mathfrak a_{t}: t \in {\mathbb{R}}_+^k)}_{L^p(X)}^p
\le C_{p, r}(J)^p.\end{aligned}$$ Letting $J\to \infty$ we get a contradiction. This completes the proof of Proposition \[prop:4\].

Variation semi-norms
--------------------

We 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 semi-norm is defined to be $$\begin{aligned}
\label{eq:67}
V^{r}(\mathfrak a_t: t\in\mathbb I):=
\sup_{J\in{\mathbb{Z}}_+} \sup_{\substack{t_{0}<\dotsb<t_{J}\\ t_{j}\in\mathbb I}}
\Big(\sum_{j=0}^{J-1}  |\mathfrak a_{t_{j+1}}-\mathfrak a_{t_{j}}|^{r} \Big)^{1/r},\end{aligned}$$ where the latter supremum is taken over all finite increasing sequences in $\mathbb I$.

\[rem:2\] Some remarks about definition are in order.

1.  Clearly $V^{r}(\mathfrak a_t: t\in\mathbb I)$ defines a semi-norm.

2.  The function $r\mapsto V^r(\mathfrak{a}_t: t\in {\mathbb{I}})$ is non-increasing. Moreover, if ${\mathbb{I}}_1\subseteq {\mathbb{I}}_2$, then $$V^r(\mathfrak{a}_t: t\in {\mathbb{I}}_1)\le V^r(\mathfrak{a}_t: t\in {\mathbb{I}}_2).$$

3.  Let ${\mathbb{I}}\subseteq {\mathbb{R}}$ be such that $\#{{\mathbb{I}}}\ge2$. Let $(\mathfrak a_t:t\in{\mathbb{R}})\subseteq {\mathbb{C}}$ be given, and let $r\in[1, \infty)$. If $V^r(\mathfrak{a}_t: t\in {\mathbb{R}})<\infty$ then $\lim_{t\to\infty}\mathfrak{a}_t$ exists. Moreover, for any $t_0\in{\mathbb{I}}$ one has $$\begin{aligned}
    \label{eq:68}
    \sup_{t\in{\mathbb{I}}}|\mathfrak{a}_t|\le |\mathfrak{a}_{t_0}|+V^r(\mathfrak{a}_t: t\in {\mathbb{I}}).\end{aligned}$$

4.  Let ${\mathbb{I}}\subseteq {\mathbb{R}}$ be such that $\#{{\mathbb{I}}}\ge2$. Then for any $r\ge1$, and any family $(\mathfrak a_t:t\in{\mathbb{I}})\subseteq {\mathbb{C}}$, any $J\in{\mathbb{Z}}_+\cup\{\infty\}$, any $I\in\mathfrak S_J({\mathbb{I}})$ one has $$\begin{aligned}
    \label{eq:62}
     O_{I, J}^r(\mathfrak a_{t}: t \in {\mathbb{I}})\le V^r(\mathfrak{a}_t: t\in {\mathbb{I}})\le 2\Big(\sum_{t\in{\mathbb{I}}}|\mathfrak a_{t}|^r\Big)^{1/r}.
     \end{aligned}$$

5.  Let $(\mathfrak{a}_t(x): t\in{\mathbb{R}}_+)$ be a family of complex-valued measurable functions on a $\sigma$-finite measure space $(X,{\mathcal{B}}(X),\mu)$. Then for any $p\ge1$ and $r \ge 2$ we have $$\begin{aligned}
    \label{eq:70}
    \begin{gathered}
    \qquad \sup_{N\in{\mathbb{Z}}_+}\sup_{I\in\mathfrak{S}_N({\mathbb{R}}_+)}\norm{O_{I,N}^r(\mathfrak{a}_t:t\in{\mathbb{R}}_+)}_{L^p(X)}\lesssim \sup_{N\in{\mathbb{Z}}_+}\sup_{I\in\mathfrak{S}_N({\mathbb{D}})}\norm{O_{I,N}^r(\mathfrak{a}_{t}:t\in {\mathbb{D}})}_{L^p(X)}\\
    \qquad +\norm[\Big]{\Big(\sum_{n\in{\mathbb{Z}}} V^r(\mathfrak{a}_{t}:t\in[2^{n},2^{n+1}])^2\Big)^{1/2}}_{L^p(X)}.
    \end{gathered}\end{aligned}$$ The inequality is an analogue of @JSW [Lemma 1.3, p. 6716] for oscillation semi-norms.

Jumps
-----

The $r$-variation is closely related to the $\lambda$-jump counting function. Recall that for any $\lambda>0$ the $\lambda$-jump counting function of a function $f : {\mathbb{I}}\to {\mathbb{C}}$ is defined by $$\begin{aligned}
\label{eq:71}
N_{\lambda} f:=N_\lambda(f(t):t\in{\mathbb{I}}):=\sup \{ J\in{\mathbb{N}}: \exists_{\substack{t_{0}<\ldots<t_{J}\\ t_{j}\in{\mathbb{I}}}}  : \min_{0 \le j \le J-1} |f(t_{j+1})-f(t_{j})| \ge \lambda \}.\end{aligned}$$

\[rem:3\] Some remarks about definition are in order.

1.  For any $\lambda>0$ and a function $f : {\mathbb{I}}\to {\mathbb{C}}$ let us also define the following quantity $$\begin{aligned}
    \qquad \mathcal{N}_{\lambda} f:={\mathcal{N}}_\lambda(f(t):t\in{\mathbb{I}}):=\sup \{ J\in{\mathbb{N}}: \exists_{\substack{s_1 < t_{1}  \le \ldots \le s_J < t_{J} \\ s_j , t_{j}\in{\mathbb{I}}}}  : \min_{1 \le j \le J} |f(t_{j})-f(s_{j})| \ge \lambda \}.  \end{aligned}$$ Then on has $N_{\lambda} f \le \mathcal{N}_{\lambda} f \le N_{\lambda/2} f$.

2.  It is clear from these definitions that $f\mapsto \sup_{\lambda>0} \big\| \lambda N_{\lambda}(f(\cdot, t): t\in {\mathbb{I}})^{1/\rho} \big\|_{L^{p}(X)}$ satisfies a quasi-triangle inequality. However it is not obvious whether a genuine triangle inequality is available for $\lambda$-jumps. In many applications, the problem can be overcome since there is always a comparable semi-norm in the following sense. Namely, for every $p \in (1,\infty)$, and $\rho \in (1,\infty)$ there exists a constant $0<C<\infty$ such that for every measure space $(X,{\mathcal{B}}(X),\mu)$, and ${\mathbb{I}}\subseteq {\mathbb{R}}$, there exists a (subadditive) seminorm ${{\left\lvert\kern-0.25ex\left\lvert\kern-0.25ex\left\lvert \cdot 
        \right\rvert\kern-0.25ex\right\rvert\kern-0.25ex\right\rvert}}$ such that the following two-sided inequality $$\begin{aligned}
    C^{-1} {{\left\lvert\kern-0.25ex\left\lvert\kern-0.25ex\left\lvert f 
        \right\rvert\kern-0.25ex\right\rvert\kern-0.25ex\right\rvert}} \leq \sup_{\lambda>0} \big\| \lambda N_{\lambda}(f(\cdot, t): t\in {\mathbb{I}})^{1/\rho} \big\|_{L^{p}(X)} \leq C {{\left\lvert\kern-0.25ex\left\lvert\kern-0.25ex\left\lvert f 
        \right\rvert\kern-0.25ex\right\rvert\kern-0.25ex\right\rvert}}\end{aligned}$$ holds for all measurable functions $f : X \times {\mathbb{I}}\to {\mathbb{C}}$. This was established in @MSZ1 [Corollary 2.2, p. 805].

3.  Let $(\mathfrak{a}_t(x): t\in{\mathbb{R}})$ be a family of measurable functions on a $\sigma$-finite measure space $(X,{\mathcal{B}}(X),\mu)$. Let ${\mathbb{I}}\subseteq{\mathbb{R}}$ and $\#{\mathbb{I}}\geq2$, then for every $p\in[1,\infty]$ and $r\in[1,\infty)$ we have $$\begin{aligned}
    \label{eq:72}
    \sup_{\lambda>0}\|\lambda N_{\lambda}(\mathfrak{a}_t: t\in{\mathbb{I}})^{1/r}\|_{L^p(X)}\le \|V^r(\mathfrak{a}_t: t\in{\mathbb{I}})\|_{L^p(X)},\end{aligned}$$ since for all $\lambda>0$ we have the following pointwise estimate $$\lambda N_{\lambda}(\mathfrak{a}_t(x): t\in{\mathbb{I}})^{1/r}\le V^r(\mathfrak{a}_t(x): t\in{\mathbb{I}}).$$

4.  Let $(X,\mathcal{B}(X),\mu)$ be a $\sigma$-finite measure space and ${\mathbb{I}}\subseteq{\mathbb{R}}$. Fix $p \in [1,\8]$, and $1 \le \rho < r \le \infty$. Then for every measurable function $f : X \times {\mathbb{I}}\to {\mathbb{C}}$ we have the estimate $$\begin{aligned}
     \label{estt1}
    \big\| V^{r}\big( f(\cdot, t) : t \in {\mathbb{I}}\big) \big\|_{L^{p,\8} (X)}
    \lesssim_{p,\rho,r}
    \sup_{\lambda>0} \big\| \lambda N_{\lambda}(f(\cdot, t): t\in {\mathbb{I}})^{1/\rho} \big\|_{L^{p,\8}(X)}.\end{aligned}$$ The inequality can be thought of as an inverse to inequality . A proof of can be found in @MSZ1 [Lemma 2.3, p. 805]. Moreover, one cannot replace $L^{p,\8} (X)$ with $L^{p} (X)$ in , see @MSS [Lemma 2.24]. One can also show that there is $f:{\mathbb{Z}}_+\times{\mathbb{Z}}_+\to{\mathbb{R}}$ such that $$\sup_{N\in{\mathbb{Z}}_+}\sup_{I\in\mathfrak{S}_N({\mathbb{Z}}_+)}\norm{O_{I,N}^r(f(\cdot,n):n\in{\mathbb{Z}}_+)}_{\ell^p({\mathbb{Z}}_+)}
    =\infty,\quad 2\le r\le\infty,$$ but $$\sup_{\lambda > 0} \| \lambda N_{\lambda}(f(\cdot,n):n\in{\mathbb{Z}}_+)^{1/2} \|_{\ell^p({\mathbb{Z}}_+)} < \infty.$$

One-parameter oscillation estimates {#section:3}
===================================

We state a simple one-parameter oscillation estimate for projections, which has many interesting implications. Here we are inspired by observations of M. Lacey who highlighted and pointed out the importance of projections in pointwise ergodic theory; see @RW.

\[thm:p\] Let $(X,\mathcal B(X), \mu)$ be a $\sigma$-finite measure space and let ${\mathbb{I}}\subseteq {\mathbb{R}}$ be such that $\#{\mathbb{I}}\ge2$. Let $(P_t)_{t \in {\mathbb{I}}}$ be a family of projections; that is, the linear operators $P_t:L^0(X)\to L^0(X)$ satisfying $$\begin{aligned}
\label{eq:10}
P_s P_t = P_{s \wedge t}, \qquad \text{ for} \qquad  s \not=  t. \end{aligned}$$ If the set ${\mathbb{I}}$ is uncountable then we assume in addition that ${\mathbb{I}}\ni t\mapsto P_tf$ is continuous $\mu$-almost everywhere on $X$ for every $f\in L^0(X)$. Let $p, r \in(1, \infty)$ be fixed. Suppose that $P_t$ are bounded on $L^p(X)$, and suppose that the following two estimates hold $$\begin{aligned}
 \label{1.15}
\sup_{J\in{\mathbb{Z}}_+}\sup_{I\in \mathfrak{S}_J({\mathbb{I}})} 
\norm[\Big]{ \Big( \sum_{j=0}^{J-1} 
\abs{ (P_{I_{j+1}}- P_{I_{j}}) f}^r \Big)^{1/r} }_{L^p(X)}
\lesssim_{p, r} 
\norm{f}_{L^p(X)}, \qquad f \in L^p(X),\end{aligned}$$ and the vector-valued estimate $$\begin{aligned}
 \label{1.2}
\norm[\Big]{ \Big( \sum_{j \in {\mathbb{Z}}} \sup_{t \in {\mathbb{I}}} \abs{P_{t} f_j}^r \Big)^{1/r} }_{L^p(X)} 
\lesssim_{p, r} 
\norm[\Big]{ \Big( \sum_{j \in {\mathbb{Z}}} \abs{f_j}^r \Big)^{1/r} }_{L^p(X)}, \qquad (f_j)_{j\in{\mathbb{Z}}}\in L^p(X; \ell^r({\mathbb{Z}})).\end{aligned}$$ Then the following one-parameter oscillation estimate holds: $$\begin{aligned}
 \label{1.3}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{I}}) }
\norm{ O^r_{I, J} ( P_{t} f : t \in {\mathbb{I}}) }_{L^p(X)} 
\lesssim_{p, r}
\norm{ f }_{L^p(X)}, 
\qquad f \in L^p(X).\end{aligned}$$

Fix $J \in {\mathbb{Z}}_+$ and $I \in \mathfrak S_{J}({\mathbb{I}})$ and observe, using , that $$\begin{aligned}
(P_t - P_{I_{j}} )f = P_t (P_{I_{j+1}} - P_{I_{j}}) f, 
\qquad \text{ whenever } \qquad I_j < t < I_{j+1}.\end{aligned}$$ Using this identity and then we see that $$\begin{aligned}
\norm[\Big]{ \Big( \sum_{j=0}^{J-1} \sup_{\substack{I_j < t<I_{j+1}\\ t\in {\mathbb{I}}}} \abs{P_t f - P_{I_{j}} f}^r \Big)^{1/r} }_{L^p(X)}
& \le 
\norm[\Big]{ \Big( \sum_{j=0}^{J-1} \sup_{t \in {\mathbb{I}}} 
\abs{P_t (P_{I_{j+1}} - P_{I_{j}}) f}^r \Big)^{1/r} }_{L^p(X)} \\
& \lesssim_{p, r}
\norm[\Big]{ \Big( \sum_{j=0}^{J-1} 
\abs{ (P_{I_{j+1}} - P_{I_{j}}) f}^r \Big)^{1/r} }_{L^p(X)}.\end{aligned}$$ Now applying we arrive at . The proof of Theorem \[thm:p\] is complete.

\[rem:33\] A few remarks are in order.

1.  Theorem \[thm:p\] will be applied mainly when $r=2$. Then the estimate in is a square function estimate, which can be deduced from the estimate $$\begin{aligned}
    \label{1.1}
    \sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{I}}) } \sup_{\substack{\abs{\varepsilon_j} \le 1\\ 0 \le j\le J}}
    \norm[\Big]{\sum_{j = 0}^{J-1} \varepsilon_j (P_{I_{j+1}} f - P_{I_{j}} f) }_{L^p(X)} 
    \lesssim_p 
    \norm{f}_{L^p(X)}, \qquad f \in L^p(X).\end{aligned}$$ In fact, the implication from to is a simple consequence of Khintchine's inequality.

2.  Let $(X,\mathcal B(X), \mu)$ be a $\sigma$-finite measure space, ${\mathbb{I}}\subseteq {\mathbb{R}}$ be countable and let $(T_t)_{t \in {\mathbb{I}}}$ be a family of bounded operators on $L^p(X)$ for $p\in(1, \infty)$ satisfying $$\begin{aligned}
    \label{eq:12}
    \norm[\Big]{ \Big(\sum_{t \in {\mathbb{I}}} \abs{(T_t-P_{t})f}^2 \Big)^{1/2} }_{L^p(X)} 
    \lesssim_p 
    \norm{f}_{L^p(X)}, \qquad f \in L^p(X),\end{aligned}$$ where $(P_t)_{t \in {\mathbb{I}}}$ is a family of projections as in Theorem \[thm:p\] satisfying and with $r=2$. Then one has $$\begin{aligned}
    \label{eq:13}
    \sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{I}}) }
    \norm{ O^2_{I, J} ( T_{t} f :t \in {\mathbb{I}}) }_{L^p(X)} 
    \lesssim_{p}
    \norm{ f }_{L^p(X)}, 
    \qquad f \in L^p(X).\end{aligned}$$ In fact, in view of the inequality easily reduces the 2-oscillation estimate for $(T_t)_{t \in {\mathbb{I}}}$ to a 2-oscillation estimate for $(P_t)_{t \in {\mathbb{I}}}$. This observation will be very useful in many applications. We will see how it works in the case of smooth bump functions, see Theorem \[thm:1\].

3.  As we know oscillation inequalities are important in pointwise convergence problems, and in the vast majority of applications it suffices to understand for $p=2$. This can be nicely illustrated as follows: suppose for $p\in(1, \infty)$ one has an a priori maximal bound $$\begin{aligned}
    \label{eq:14}
    \norm[big]{\sup_{t \in {\mathbb{I}}} \abs{P_{t} f}}_{L^p(X)}\lesssim_p\norm{f}_{L^p(X)}, \qquad f \in L^p(X).\end{aligned}$$ Then with $p=2$ can be used to verify with $p=r=2$. Finally, it remains to verify with $p=r=2$, which in many cases can be deduced by using Fourier techniques or exploiting almost-orthogonality phenomena invoking $TT^*$ arguments, see Proposition \[prop:6\].

We now derive some consequences of Theorem \[thm:p\].

Oscillation inequalities for martingales
----------------------------------------

We recall some basic facts about martingales. We will follow notation from @HNVW [Section 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 a *filtration* if it is increasing and the measure $\mu$ is $\sigma$-finite on each $\mathcal F_t$. 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 operator 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 $$\sup_{t\in{\mathbb{I}}}\|\mathfrak f_t\|_{L^p(X)}\lesssim_p1.$$

Applying Theorem \[thm:p\] we immediately recover the oscillation inequality of Jones–Kaufman–Rosenblatt–Wierdl @jkrw, which in fact is an oscillation inequality for bounded martingales.

\[prop:1\] 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 $$\begin{aligned}
\label{eq:11}
\sup_{J\in{\mathbb{Z}}_+}\sup_{I\in\mathfrak S_J({\mathbb{Z}})}
\norm[\big]{O^2_{I,J}(\mathfrak f_n:n\in{\mathbb{Z}})}_{L^p(X)}
&\leq C_p
\sup_{n\in{\mathbb{Z}}}\|\mathfrak f_n\|_{L^p(X)}.\end{aligned}$$

Inequality was established in @jkrw [Theorem 6.4, p. 930]. The authors first established for $p=2$, then proved weak type $(1,1)$ as well as $L^{\infty}\to {\rm BMO}$ variants of , and consequently derived for all $p\in(1, \infty)$ by interpolation. Our approach is direct and will avoid using any interpolation arguments in the proof.

\[Proof of Proposition \[prop:1\]\] Fix $p\in(1, \infty)$. Define projections by $P_n(f):={\mathbb{E}}[f|{\mathcal{F}}_n]$ for any $n\in{\mathbb{Z}}$ and $f\in L^p(X)$. Since $\mathfrak f=(\mathfrak f_n:n\in{\mathbb{Z}})$ is a martingale then $\mathfrak f_n=P_n(\mathfrak f_n)$ for any $n\in{\mathbb{Z}}$ and consequently holds. Moreover, by Burkholder @Bu1, see also @Bu2, it is very well known that holds, which in view of Remark \[rem:33\], implies $$\begin{aligned}
\sup_{J\in{\mathbb{Z}}_+}\sup_{I\in \mathfrak{S}_J({\mathbb{Z}}_+)} 
\norm[\Big]{ \Big( \sum_{j=0}^{J-1} 
\abs{ P_{I_{j+1}}(\mathfrak f_{I_J})- P_{I_{j}}(\mathfrak f_{I_J})}^2 \Big)^{1/2} }_{L^p(X)}
\lesssim_{p} 
\sup_{n\in{\mathbb{Z}}}\|\mathfrak f_n\|_{L^p(X)}.\end{aligned}$$ This consequently verifies inequality . Invoking the Fefferman–Stein inequality for non-negative submartingales @HNVW [Theorem 3.2.7, p. 178] we obtain $$\begin{aligned}
\norm[\Big]{ \Big( \sum_{j \in {\mathbb{Z}}} \sup_{n \in {\mathbb{Z}}} \abs{{\mathbb{E}}[|f_j||{\mathcal{F}}_n]}^2 \Big)^{1/2} }_{L^p(X)} 
\lesssim_{p} 
\norm[\Big]{ \Big( \sum_{j \in {\mathbb{Z}}} \abs{f_j}^2 \Big)^{1/2} }_{L^p(X)}, \qquad (f_j)_{j\in{\mathbb{Z}}}\in L^p(X; \ell^2({\mathbb{Z}})),\end{aligned}$$ which in turn verifies the vector-valued estimate from . Appealing to Theorem \[thm:p\], the oscillation inequality follows and the proof of Proposition \[prop:1\] is complete.

Oscillation inequalities for smooth bump functions
--------------------------------------------------

Our aim will be to show that oscillation inequalities hold for $L^1$-dilated smooth bump functions. We begin with the main estimate.

\[prop:2\] For $d\in{\mathbb{Z}}_+$ let $\chi:{\mathbb{R}}^d\to[0, 1]$ be a smooth function satisfying $$\begin{aligned}
\label{eq:15}
{\mathds{1}_{{[-1,1]^d}}} \le \chi \le {\mathds{1}_{{[-2,2]^d}}}
\quad \text{ for } \quad
\xi\in{\mathbb{R}}^d.\end{aligned}$$ For every $n\in {\mathbb{Z}}$ and $\xi\in{\mathbb{R}}^d$ define $\chi_{2^n}(\xi):=\chi(2^{-n}\xi)$. Then for every $p\in(1, \infty)$ one has $$\begin{aligned}
\label{eq:16}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{Z}}) }
\norm{ O^2_{I, J} ( T_{{\mathbb{R}}^d}[\chi_{2^n}]f : n \in {\mathbb{Z}}) }_{L^p({\mathbb{R}}^d)} 
\lesssim_{p}
\norm{ f }_{L^p({\mathbb{R}}^d)}, 
\qquad f \in L^p({\mathbb{R}}^d).\end{aligned}$$

Setting $P_nf:= T_{{\mathbb{R}}^d}[\chi_{2^{n}}]f$ for every $n\in{\mathbb{Z}}$, and using one sees that $P_n$ is a projection in the sense of . Standard arguments based on the Littlewood–Paley theory (see Section \[lp\]) show that with $r=2$ holds. By the Fefferman–Stein inequality @bigs we also obtain . An application of Theorem \[thm:p\] now gives as desired.

Now our aim will be to extend inequality to continuous times and general smooth bump functions.

\[rem:100\] A few remarks concerning Proposition \[prop:2\] are in order.

1.  An important feature of our approach in Proposition \[prop:2\] is that we do not need to invoke the corresponding inequality for martingales in the proof. This stands in sharp contrast to variants of inequality involving $r$-variations, where all arguments to the best of our knowledge use the corresponding $r$-variational inequalities for martingales.

2.  Of course, inequality can be reduced to the martingale setting from Proposition \[prop:1\] by invoking square function arguments @JSW [Lemma 3.2, p. 6722] and standard Littlewood–Paley theory. The details may be found in @MSZ2.

3.  With respect to the previous two remarks, it would be interesting to know whether the $r$-variational counterpart of Proposition \[prop:2\] can be proved without appealing to $r$-variational inequalities for martingales, see L<span>é</span>pingle's inequality .

\[thm:1\] For $d\in{\mathbb{Z}}_+$ let $\phi : {\mathbb{R}}^d\to {\mathbb{C}}$ be a Schwartz function. For $t\in{\mathbb{R}}_+$ and $x\in{\mathbb{R}}^d$ define $\phi_t (x) := t^{-d} \phi(t^{-1}x)$. Then for every $p\in (1, \infty)$ one has $$\begin{aligned}
\label{id:1}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{R}}_+) }
\norm{ O^2_{I, J} ( \phi_t * f : t \in{\mathbb{R}}_+) }_{L^p({\mathbb{R}}^d)} 
\lesssim_p 
\norm{ f }_{L^p({\mathbb{R}}^d)}, 
\qquad f \in L^p({\mathbb{R}}^d).\end{aligned}$$

\[rem:partial\] Theorem \[thm:1\] immediately extends to families of partial convolution operators. If ${\mathbb{R}}^d = {\mathbb{R}}^n \times {\mathbb{R}}^m$, we write elements $x\in {\mathbb{R}}^d$ as $x = (x',x'')$ where $x'\in {\mathbb{R}}^n$ and $x''\in {\mathbb{R}}^m$. Let $\phi$ be a Schwartz function on ${\mathbb{R}}^n$ and define $$T_t f(x) \ = \ \int_{{\mathbb{R}}^n} f(x' - y,x'') \phi_t(y) \, dy.$$ The oscillation inequality implies the corresponding oscillation inequality for the family of partial convolution operators $(T_t)_{t \in {\mathbb{R}}_+}$.

\[Proof of Theorem \[thm:1\]\] To prove , in view of , it suffices to show $$\begin{aligned}
\label{4.2}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{D}}) }
\norm{ O^2_{I, J} ( \phi_t * f : t \in {\mathbb{D}}) }_{L^p({\mathbb{R}}^d)} 
\lesssim_p 
\norm{ f }_{L^p({\mathbb{R}}^d)}, 
\qquad f \in L^p({\mathbb{R}}^d),\end{aligned}$$ and $$\begin{aligned}
 \label{7.1}
\norm[\Big]{ \Big( \sum_{k \in {\mathbb{Z}}} V^2 \big(\phi_t * f : t \in [2^k,2^{k+1}] \big)^2 \Big)^{1/2} }_{L^{p}({\mathbb{R}}^d)}
\lesssim
\norm{f }_{L^{p}({\mathbb{R}}^d)}, \qquad f \in L^{p}({\mathbb{R}}^d).\end{aligned}$$

Short 2-variational estimates were treated in @JSW and in particular, the estimate follows directly from @JSW [Lemma 6.1]. To establish we first observe that we may assume that $\int_{{\mathbb{R}}^d} \phi(x)dx = 0$. Indeed, if $\int_{{\mathbb{R}}^d} \phi(x)dx \ne 0$, then by scaling we may assume that $\int_{{\mathbb{R}}^d} \phi(x)dx = \chi(0) = 1$ where $\chi$ appears in Proposition \[prop:2\]. By standard Littlewood–Paley arguments (see Section \[lp\]), we note that holds with $T_t f = \phi_t * f$ and $P_t f = T_{{\mathbb{R}}^d}[\chi_t] f$. Therefore by Remark \[rem:33\], we see that follows from the oscillation inequality and so we may assume $\phi$ has mean zero. Using we see that $$\begin{aligned}
\text{ LHS of } \eqref{4.2} 
& \lesssim
\norm[\Big]{ \Big( \sum_{k \in {\mathbb{Z}}} \abs{\phi_{2^k} * f}^2 \Big)^{1/2} }_{L^p({\mathbb{R}}^d)} 
\lesssim \|f\|_{L^p({\mathbb{R}}^d)},\end{aligned}$$ the last inequality following directly from @DR [Theorem B]; see Section \[lp\]. This completes the proof of Theorem \[thm:1\].

Oscillation inequalities for orthonormal systems
------------------------------------------------

We now state a result which justifies in a strong sense the importance of oscillation inequalities.

\[prop:6\] Let $(X,{\mathcal{B}}(X),\mu)$ be a $\sigma$-finite measure space such that the corresponding Hilbert space $L^2(X)$ is endowed with an orthonormal basis $(\Phi_n)_{n\in{\mathbb{N}}}$. Then the projection operators $$\begin{aligned}
\label{eq:17}
P_nf:=\sum_{k=0}^n\langle f, \Phi_k\rangle \Phi_k, \qquad f\in L^2(X),\end{aligned}$$ satisfy the oscillation estimate $$\begin{aligned}
\label{eq:20-}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{N}}_{\le N}) }
\norm{ O^2_{I, J} ( P_{n} f : n \in {\mathbb{N}}_{\le N}) }_{L^2(X)} 
\lesssim \ \log(N+1) \,
\norm{ f }_{L^2(X)}.\end{aligned}$$ Furthermore if the projection operators $P_n$ satisfy the following maximal estimate $$\begin{aligned}
\label{eq:18}
\norm[\big]{\sup_{n\in{\mathbb{N}}}|P_nf|}_{L^2(X)}\lesssim \norm{f}_{L^2(X)} , \qquad f\in L^2(X),\end{aligned}$$ then one has the uniform bound $$\begin{aligned}
\label{eq:20}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{N}}) }
\norm{ O^2_{I, J} ( P_{n} f : n \in {\mathbb{N}}) }_{L^2(X)} 
\lesssim
\norm{ f }_{L^2(X)}, 
\qquad f \in L^2(X).\end{aligned}$$

It is easy to see that $P_n$ from satisfies . To verify we fix $J \in {\mathbb{Z}}_+$ and $I \in \mathfrak S_{J}({\mathbb{N}})$ and note that by orthognality we have $$\begin{aligned}
\norm[\Big]{ \Big( \sum_{j=0}^{J-1} 
\abs{ (P_{I_{j+1}}- P_{I_{j}}) f}^2 \Big)^{1/2} }_{L^2(X)}^2&=
\sum_{j=0}^{J-1} \sum_{k_1=I_j+1}^{I_{j+1}}\sum_{k_2=I_j+1}^{I_{j+1}}
\langle f, \Phi_{k_1}\rangle\overline{\langle f, \Phi_{k_2}\rangle}
\langle\Phi_{k_1}, \Phi_{k_2}\rangle\\
&\le\sum_{k\in{\mathbb{N}}}
|\langle f, \Phi_{k}\rangle|^2\\
&=\|f\|_{L^2(X)}^2,\end{aligned}$$ where in the last line we have used Parseval's identity for orthonormal bases. This proves with $p=r=2$. A famous result of Rademacher @Rad and Menshov @Men asserts that there is a constant $C>0$ such that for any $N\in{\mathbb{Z}}_+$ the projection operator $P_n$ from satisfies $$\begin{aligned}
\label{eq:19}
\norm[\big]{\sup_{n\in[N]}|P_nf|}_{L^2(X)}\le C\log(N+1)\Big(\sum_{n\in[N]}|\langle f, \Phi_n\rangle|^2\Big)^{1/2} \
\lesssim \ \log(N+1) \, \|f\|_{L^2(X)}.\end{aligned}$$ Using we see that holds with $p=r=2$ with constant $\log(N+1)$. Now applying Theorem \[thm:p\] we obtain .

Under condition , we see that holds with a uniform constant for $p=r=2$ and so applying Theorem \[thm:p\] again, we obtain .

Proposition \[prop:6\] is a key example in the study of oscillation semi-norms from the point of view their importance and usefulness in pointwise convergence problems. It exhibits, in view of inequality , that oscillation estimates and maximal estimates are equivalent in the class of orthonormal systems.

However, we have to emphasize that the maximal estimate from is a very strong condition. On the one hand, we have Menshov's construction @Men of an orthonormal basis $(\Psi_n)_{n\in{\mathbb{N}}}\subseteq L^2([0, 1])$ and a function $f_0\in L^2([0, 1])$ with almost everywhere diverging partial sums $\sum_{k=0}^n\langle f, \Psi_k\rangle \Psi_k$. Therefore maximal estimate for Menshov's system cannot hold. In fact, the best what we can expect in the general case is the Rademacher–Menshov bound . The above-mentioned Menshov's @Men construction also shows that is sharp and the logarithm in cannot be removed.

On the other hand, there is the famous result of Carleson @Carleson which led to establishing for the canonical trigonometric system $({\bm{e}}(n\xi))_{n\in{\mathbb{Z}}}$ on $L^2([0, 1])$ (see also @Hunt [@F1; @LT1]).

Oscillation inequalities for the Carleson operator
--------------------------------------------------

In this subsection we obtain certain $r$-oscillation estimates for partial Fourier integrals on the real line ${\mathbb{R}}$. The Carleson operator ${\mathcal{C}}_t$ is defined by $$\begin{aligned}
\label{eq:23}
{\mathcal{C}}_t f(x) := T_{{\mathbb{R}}}[{\mathds{1}_{{[-t, t]}}}]f(x)= \int_{-t}^t {\mathcal{F}}_{{\mathbb{R}}}f(\xi) {\bm{e}}(-x \xi) \, d\xi, 
\qquad f \in {\mathcal{S}}({\mathbb{R}}), \quad x\in{\mathbb{R}},\quad  t \in {\mathbb{R}}_+.\end{aligned}$$ The celebrated Carleson–Hunt theorem (see the papers of Carleson @Carleson and Hunt @Hunt) asserts that for every $p\in(1, \infty)$ there is a constant $C_p>0$ such that $$\begin{aligned}
\label{eq:21}
\norm[\big]{\sup_{t>0}|{\mathcal{C}}_t f|}_{L^p({\mathbb{R}})}\le C_p\norm{f}_{L^p({\mathbb{R}})}, \qquad f\in L^p({\mathbb{R}}).\end{aligned}$$

\[rem:4\] A few remarks about the Carleson–Hunt theorem are in order.

1.  Carleson @Carleson originally proved that the maximal partial sum operator of Fourier series corresponding to square-integrable functions on the circle is weak type $(2, 2)$. Not long afterwards this result was extended by Hunt @Hunt who proved that the maximal partial sum operator of Fourier series is bounded on $L^p({\mathbb{T}})$ for any $p\in(1, \infty)$.

2.  Kenig and Tomas @KT used a transplantation arugment to show the latter result is equivalent to inequality . This equivalence was extended to variation and oscillation inequalities in @OSTTW. The foundational work of Kolmogorov @Kol1 [@Kol2] shows that the range of $p\in(1, \infty)$ in inequality is sharp.

3.  An alternative proof of Carleson's theorem was provided by Fefferman @F1, who pioneered the ideas of the so called time–frequency analysis.

4.  Lacey and Thiele @LT1 established an independent proof on the real line of the weak type $(2, 2)$ boundedness of the maximal Fourier integral operator . The latter bound was extended by Grafakos, Tao, and Terwilleger @GTT to for all $p\in(1, \infty)$, see also @PT.

5.  Inequality was extended to vector-valued setting by Grafakos, Martell and Soria @GMS, who proved that that for every $p, r\in(1, \infty)$ there is a constant $C_{p, r}>0$ such that $$\begin{aligned}
    \label{eq:26}
    \qquad \quad \norm[\Big]{ \Big( \sum_{j \in {\mathbb{Z}}} \sup_{t>0}|{\mathcal{C}}_t f_j|^r \Big)^{1/r} }_{L^p({\mathbb{R}})} 
    \le C_{p, r}
    \norm[\Big]{ \Big( \sum_{j \in {\mathbb{Z}}} \abs{f_j}^r \Big)^{1/r} }_{L^p({\mathbb{R}})}, \qquad (f_j)_{j\in{\mathbb{Z}}}\in L^p(X; \ell^r({\mathbb{Z}})).\end{aligned}$$

6.  We finally refer to the survey of Lacey @La1, where details (including comprehensive historical background) and an extensive literature are given about this fascinating subject of pointwise convergence of Fourier series and related topics.

A far-reaching quantitative extension of was obtained by the third author in collaboration with Oberlin, Seeger, Tao and Thiele @OSTTW, which asserts that for every $p\in(1, \infty)$ and for every $r>\max\big\{2, \frac{p}{p-1}\big\}$ there is a constant $C_{p, r}>0$ such that $$\begin{aligned}
\label{eq:24}
\norm[\big]{V^r({\mathcal{C}}_t f: t\in{\mathbb{R}}_+)}_{L^p({\mathbb{R}})}\le C_{p, r}\norm{f}_{L^p({\mathbb{R}})}, \qquad f\in L^p({\mathbb{R}}).\end{aligned}$$ See also in @U for a different proof using outer measures. Furthermore a restricted weak-type bound is established at the endpoint $p = r'$ when $p\in (1,2)$ (here $r' = r/(r-1)$) and it is open whether weak type $(p,p)$ holds true. It also follows from @OSTTW that the ranges of parameter $p\in(1, \infty)$ and $r>\max\big\{2,p'\big\}$ in are sharp. In the endpoint case $p=r'$, the Lorentz space $L^{r',\infty}$ cannot be replaced by a smaller Lorentz space. For weighted variational estimates for the Carleson operator, see @DL, and @DDU and the references given there.

Inequality , in view of inequality , immediately implies that for every $p\in(1, \infty)$ and for every $r>\max\{2, p'\}$ there is a constant $C_{p, r}>0$ (actually the same as in ) such that $$\begin{aligned}
\label{eq:25}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{R}}_+) }
\norm[\big]{O^r_{I, J}({\mathcal{C}}_t f: t\in{\mathbb{R}}_+)}_{L^p({\mathbb{R}})}\le C_{p, r}\norm{f}_{L^p({\mathbb{R}})}, \qquad f\in L^p({\mathbb{R}}).\end{aligned}$$ For applications of and to the Wiener-Wintner theorem in ergodic theory, see @LT and @OSTTW.

It has been observed by M. Lacey @lacey (see @RW for the case $p=2$) that remains true for $r=2$ whenever $p\in[2, \infty)$. Furthermore, this can be extend to all $p>1$ when we restrict the $t$ parameter in ${\mathcal{C}}_t$ to dyadic numbers $t\in {\mathbb D}$. Our aim here is to show how these results follow as an immediate consequence of Theorem \[thm:p\].

\[prop:car\] Let $({\mathcal{C}}_t)_{t \in {\mathbb{R}}_+}$ be as in . Then for every $p\in[2, \infty)$, there exists a constant $C_p>0$ such that $$\begin{aligned}
\label{eq:27}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{R}}_+) }
\norm[\big]{O^2_{I, J}({\mathcal{C}}_t f: t\in{\mathbb{R}}_+)}_{L^p({\mathbb{R}})}
\le C_p 
\norm{ f }_{L^p({\mathbb{R}})}, 
\qquad f \in L^p({\mathbb{R}}).\end{aligned}$$ Furthermore for $({\mathcal{C}}_t)_{t\in {\mathbb D}}$, we have $$\begin{aligned}
\label{eq:27+}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb D}) }
\norm[\big]{O^2_{I, J}({\mathcal{C}}_t f: t\in{\mathbb D})}_{L^p({\mathbb{R}})}
\le C_p 
\norm{ f }_{L^p({\mathbb{R}})}, 
\qquad {\rm for \ all} \ p \in(1, \infty).\end{aligned}$$

Observe the operators $({\mathcal{C}}_t)_{t \in {\mathbb{R}}_+}$ are projections in the sense of . Moreover when the sequence $(I_j)_{j\in{\mathbb{N}}} \subset {\mathbb D}$ lies among the dyadic numbers, the bound $$\begin{aligned}
\sup_{J\in{\mathbb{Z}}_+}\sup_{I\in \mathfrak{S}_J({\mathbb D})} 
\norm[\Big]{ \Big( \sum_{j=0}^{J-1} 
\abs{ ({\mathcal{C}}_{I_{j+1}}- {\mathcal{C}}_{I_{j}}) f}^2 \Big)^{1/2} }_{L^p({\mathbb{R}})}
\lesssim_{p} 
\norm{f}_{L^p({\mathbb{R}})}, \qquad p \in(1, \infty),\end{aligned}$$ follows from the classical Littlewood–Paley inequality associated to dyadic intervales (no need to refer to the refinements of the theory from Section \[lp\]). This verifies with $r=2$ and $p\in(1, \infty)$ in the dyadic case. Furthermore by Rubio de Francia's square function theorem for intervals (see Section \[lp\]), one has for every $p\in[2, \infty)$ that $$\begin{aligned}
\sup_{J\in{\mathbb{Z}}_+}\sup_{I\in \mathfrak{S}_J({\mathbb{R}}_+)} 
\norm[\Big]{ \Big( \sum_{j=0}^{J-1} 
\abs{ ({\mathcal{C}}_{I_{j+1}}- {\mathcal{C}}_{I_{j}}) f}^2 \Big)^{1/2} }_{L^p({\mathbb{R}})}
\lesssim_{p} 
\norm{f}_{L^p({\mathbb{R}})}, \qquad f \in L^p({\mathbb{R}}),\end{aligned}$$ which verifies with $r=2$ and $p\in[2, \infty)$. Using with $r=2$ we also see that is verified with $r=2$ and $p\in (1, \infty)$. Thus invoking Theorem \[thm:p\] inequalities and follow.

Proposition \[prop:car\] for $p=2$ was established by Rosenblatt and Wierdl @RW [inequality (4.12), p. 82]. In @LT, Lacey and Terwilleger established for $p\in (1,\infty)$. Proposition \[prop:car\] gives a simple proof of these results.

In view of inequality it is not difficult to see that the maximal estimates and the oscillation estimates for the Carleson operator are equivalent for all $p\in[2, \infty)$.

We also remark that the proof above also gives a proof of which does not appeal to the variational inequality . Indeed, Rubio de Francia's result in @RdF [inequality (7.1), p. 10] states that for every $p\in(1, 2)$ and $r>p'$ one has $$\begin{aligned}
\label{eq:54}
\sup_{J\in{\mathbb{Z}}_+}\sup_{I\in \mathfrak{S}_J({\mathbb{R}}_+)} 
\norm[\Big]{ \Big( \sum_{j=0}^{J-1} 
\abs{ ({\mathcal{C}}_{I_{j+1}}- {\mathcal{C}}_{I_{j}}) f}^r \Big)^{1/r} }_{L^p({\mathbb{R}})}
\lesssim_{p,r} 
\norm{f}_{L^p({\mathbb{R}})}, \qquad f \in L^p({\mathbb{R}}).\end{aligned}$$ Hence using and and invoking Theorem \[thm:p\] we obtain the desired claim in .

A counterexample of Cowling and Tao @CT to Rubio de Francia's conjecture in @RdF [Conjecture 7.2] shows that for all $p\in(1, 2)$, one has $$\begin{aligned}
\sup_{\|f\|_{L^p({\mathbb{R}})}\le1}\sup_{I\in \mathfrak{S}_\infty({\mathbb{R}}_+)} 
\norm[\Big]{ \Big( \sum_{j=0}^{\infty} 
\abs{ ({\mathcal{C}}_{I_{j+1}}- {\mathcal{C}}_{I_{j}}) f}^{r} \Big)^{1/{r}} }_{L^p({\mathbb{R}})}=\infty, \qquad \text{ where } \qquad r=\frac{p}{p-1}.\end{aligned}$$ Therefore for $r=p'$ with $p\in(1, 2)$ cannot hold. This shows that the range of $p$ and $r$ in and is sharp.

Multi-parameter oscillation estimates {#section:4}
=====================================

In this section we establish Theorem \[thm:main\]. We begin with proving an abstract multi-parameter oscillation result, which may be of independent interest. Before we do this we need more notation. For linear operator $T: L^0(X)\to L^0(X)$ we shall denote by $\abs{T}$ the sublinear maximal operator taken in the lattice sense defined by $$\abs{T} f (x) = \sup_{\abs{g} \le \abs{f}} \abs{Tg (x)}, \qquad x \in X, \text{ and }  f\in L^p(X).$$ For two linear operators $S,T: L^0(X)\to L^0(X)$ we have $\abs{ST}f \le \abs{S} \abs{T}f$ whenever $f\in L^0(X)$.

\[prop:8\] Let $(X,\mathcal B(X), \mu)$ be a $\sigma$-finite measure space and let ${\mathbb{I}}\subseteq {\mathbb{R}}$ be such that $\#{\mathbb{I}}\ge2$. Let $k \in {\mathbb{N}}_{\ge2}$ and $p, r\in(1, \infty)$ be fixed. Let $(T_t)_{t\in{\mathbb{I}}^k}$ be a family of linear operators of the form $$\begin{aligned}
T_t :=  T_{t_1}^{1} \cdots T_{t_k}^{k},
\qquad t=(t_1,\ldots, t_k)\in{\mathbb{I}}^k,\end{aligned}$$ where $\{ T_{t_i}^i : i \in [k], \, t_i \in {\mathbb{I}}\}$ is a family of commuting linear operators, which are bounded on $L^p(X)$. If the set ${\mathbb{I}}$ is uncountable then we also assume that ${\mathbb{I}}\ni t\mapsto T_t^if$ is continuous $\mu$-almost everywhere on $X$ for every $f\in L^0(X)$ and $i\in[k]$. Further assume that for every $i\in[k]$, we have $$\begin{aligned}
\label{eq:28}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{I}})}
\norm{ O^r_{I, J} ( T_{t}^{i} f : t \in {\mathbb{I}}) }_{L^p(X)} 
& \lesssim_{p, r} 
\norm{ f }_{L^p(X)}, 
\qquad f \in L^p(X), \end{aligned}$$ and $$\begin{aligned}
\label{eq:29}
\norm[\Big]{ \Big( \sum_{j \in {\mathbb{Z}}} \big( \sup_{t \in {\mathbb{I}}} \abs{T_t^i} \abs{f_j} \big)^r \Big)^{1/r} }_{L^{p}(X)}
& \lesssim_{p, r}
\norm[\Big]{ \Big( \sum_{j \in {\mathbb{Z}}} \abs{f_j}^r  \Big)^{1/r} }_{L^{p}(X)}, \qquad (f_j)_{j \in {\mathbb{Z}}} \in L^{p}(X;\ell^r({\mathbb{Z}})).\end{aligned}$$ Then we have the following multi-parameter $r$-oscillation estimate: $$\begin{aligned}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{I}}^k) }
\norm{ O^r_{I, J} ( T_{t} f : t \in {\mathbb{I}}^k) }_{L^p(X)} 
\lesssim
\norm{ f }_{L^p(X)}, 
\qquad f \in L^p(X).\end{aligned}$$

For $i \in [k]$ and $n = (n_1, \ldots, n_{i-1}, n_{i+1}, \ldots, n_{k}) \in {\mathbb{I}}^{k-1}$, let us denote $$T_{n}^{(i)}:  = T_{n_1}^{1} \cdots T_{n_{i-1}}^{i-1} T_{n_{i+1}}^{i+1} 
\cdots T_{n_k}^{k}.$$ Using this definition, the bound and proceeding inductively we easily see that $$\begin{aligned}
 \label{3.3.2}
\norm[\Big]{ \Big( \sum_{j \in {\mathbb{Z}}} \big( \sup_{n \in {\mathbb{I}}^{k-1}} \abs{T_{n}^{(i)}} \abs{f_j} \big)^r \Big)^{1/r} }_{L^{p}(X)}
& \lesssim_p
\norm[\Big]{ \Big( \sum_{j \in {\mathbb{Z}}} \abs{f_j}^r  \Big)^{1/r} }_{L^{p}(X)},\end{aligned}$$ uniformly in $i \in [k]$ and $(f_j)_{j \in {\mathbb{Z}}} \in L^{p}(X;\ell^r({\mathbb{Z}}))$. Furthermore for $n \in {\mathbb{I}}^k$ and $I_j=(I_{j1},\ldots, I_{jk})\in{\mathbb{I}}^k$, we have the identity $$\begin{aligned}
\label{eq:30}
T_n f - T_{I_{j}} f 
=
\sum_{m=1}^k T_{n(m,n,I_{j})}^{(m)} (T_{n_{m}}^{m} - T_{I_{jm}}^{m}) f,\end{aligned}$$ where $n(m,n,I_{j}) := (n_1, \ldots, n_{m-1}, I_{j(m+1)}, \ldots, I_{jk}) \in {\mathbb{I}}^{k-1}$.

We now fix $J\in{\mathbb{Z}}_+$ and a sequence $I \in \mathfrak S_{J}({\mathbb{I}}^k)$. Applying the identity , the triangle inequality, the bound applied to $f_j^m = \sup_{\substack{ I_{jm} \le n_m < I_{(j+1)m} \\ n_m \in {\mathbb{I}}}}
\abs{T_{n_m}^{m} f - T_{I_{jm}}^{m} f}$ and , we obtain $$\begin{aligned}
& \norm[\Big]{ \Big( \sum_{j =0}^{J-1}   \sup_{n \in {\mathbb{B}}[I_{j}]\cap{\mathbb{I}}^k} \abs{T_{n}f - T_{I_{j}}f}^r \Big)^{1/r} }_{L^{p}(X)} \\
& \qquad \le 
\sum_{m=1}^k \norm[\Big]{ \Big( \sum_{j =0}^{J-1} 
    \big(  \sup_{n \in {\mathbb{B}}[I_{j}]\cap{\mathbb{I}}^k}\abs{T_{n(m,n,I_{j})}^{(m)}} \abs{T_{n_m}^{m} f - T_{I_{jm}}^{m} f} \big)^r \Big)^{1/r} }_{L^{p}(X)} \\
& \qquad \le 
\sum_{m=1}^k \norm[\Big]{ \Big( \sum_{j =0}^{J-1}  
 \Big( \sup_{n \in {\mathbb{I}}^{k-1}}  \abs{T_{n}^{(m)}} 
\big( \sup_{\substack{ I_{jm} \le n_m < I_{(j+1)m} \\ n_m \in {\mathbb{I}}}}
\abs{T_{n_m}^{m} f - T_{I_{jm}}^{m} f} \big) \Big)^r \Big)^{1/r} }_{L^{p}(X)} \\
& \qquad \lesssim
\sum_{m=1}^k \norm[\Big]{ \Big( \sum_{j =0}^{J-1}  
\sup_{\substack{ I_{jm} \le n_m < I_{(j+1)m} \\ n_m \in {\mathbb{I}}}}
 \abs{T_{n_m}^{m} f - T_{I_{jm}}^{m} f}^r \Big)^{1/r} }_{L^{p}(X)} \ \lesssim \ \norm{f}_{L^{p}(X)}.\end{aligned}$$ This completes the proof of Proposition \[prop:8\].

We have a simple consequence of the above result.

\[cor:2\] Let $k \in {\mathbb{N}}_{\ge2}$ and fix parameters $n_1, \ldots, n_k \in {\mathbb{Z}}_+$, and $p\in(1, \infty)$. For every $i\in[k]$ let $\phi^{i}:{\mathbb{R}}^{n_i}\to {\mathbb{C}}$ be a Schwartz function, and define $\phi_{t_i}^{i}(x):= t_{i}^{-n_i} \phi^{i} (t_i^{-1}x)$ for every $t_i \in{\mathbb{R}}_+$ and $x \in {\mathbb{R}}^{n_i}$. Set $N := n_1 + \ldots + n_k$ and for $t = (t_1, \ldots, t_k)\in {\mathbb{R}}_+^k$ and $x = (x_1, \ldots, x_k) \in {\mathbb{R}}^N:={\mathbb{R}}^{n_1}\times\ldots\times{\mathbb{R}}^{n_k}$, consider the operator $T_{t}: L^p({\mathbb{R}}^{N})\to L^p({\mathbb{R}}^{N})$ defined by $$\begin{aligned}
T_{t} f (x) 
:= \int_{{\mathbb{R}}^{n_1}}\ldots\int_{{\mathbb{R}}^{n_k}} \Big( \prod_{i=1}^k \phi_{t_i}^{i}(z_i) \Big) f(x-z) \, dz_1\ldots dz_k, \qquad z = (z_1, \ldots, z_k).\end{aligned}$$ Then we have the following multi-parameter oscillation estimate $$\begin{aligned}
 \label{id:3}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{R}}_+^k) }
\norm{ O^2_{I, J} ( T_{t} f : t \in {\mathbb{R}}_+^k) }_{L^p({\mathbb{R}}^N)} 
\lesssim_p 
\norm{ f }_{L^p({\mathbb{R}}^N)}, 
\qquad f \in L^p({\mathbb{R}}^N).\end{aligned}$$

For $i \in [k]$ and $z_i \in {\mathbb{R}}^{n_i}$ we denote by $z_i^{(i)}=(z^{(i)}_1,\ldots, z^{(i)}_k)$ the point in ${\mathbb{R}}^N={\mathbb{R}}^{n_1}\times\ldots\times{\mathbb{R}}^{n_k}$ such that $z^{(i)}_j = {\mathds{1}_{{\{j\}}}}(i) z_i\in {\mathbb{R}}^{n_i}$ for any $j \in [k]$. We define the operators $T_{t_i}^i: L^p({\mathbb{R}}^{N})\to L^p({\mathbb{R}}^{N})$ by $$\begin{aligned}
T_{t_i}^i f (x) :
= \int_{{\mathbb{R}}^{n_i}} \phi_{t_i}^{i}(z_i) f(x-z_i^{(i)}) \, dz_i,
\qquad x = (x_1, \ldots, x_k) \in {\mathbb{R}}^N, \quad t_i \in{\mathbb{R}}_+.\end{aligned}$$ These operators commute and we have $T_{t} = T_{t_1}^1 \circ\ldots\circ T_{t_k}^k$. Furthermore, these are partial convolution operators with Schwartz functions and so Theorem \[thm:1\] (see Remark \[rem:partial\]) implies that the oscillation estimate holds for the family $(T^i_t)_{t \in {\mathbb{R}}_+}$, for each $i \in [k]$. Finally, the Fefferman–Stein vector-valued maximal inequality shows that holds and so Proposition \[prop:8\] gives us the desired conclusion . This completes the proof of Corollary \[cor:2\].

We close this section by establishing the main ergodic result of this survey.

\[Proof of Theorem \[thm:main\]\] We will invoke Proposition \[prop:8\] with $k=d$ and $r=2$. As in note that $$\begin{aligned}
A_{M; X, {\mathcal{T}}}^{P_1 ({\mathrm m}_1),\ldots, P_d ({\mathrm m}_d)}f=A_{M_1,\ldots, M_d; X, T_1,\ldots, T_d}^{P_1 ({\mathrm m}_1),\ldots, P_d ({\mathrm m}_d)}f=A_{M_1; X, T_1}^{P_1({\mathrm m}_1)}\circ\ldots\circ A_{M_d; X, T_d}^{P_d({\mathrm m}_d)}f,\end{aligned}$$ where the averages $A_{M_1; X, T_1}^{P_1({\mathrm m}_1)},\ldots, A_{M_d; X, T_d}^{P_d({\mathrm m}_d)}$ commute. Thus it remains to verify and . We fix $j\in[d]$. For we refer to @MSS [Theorem 1.4], which ensures that for every $p\in(1,\infty)$ one has $$\begin{aligned}
\sup_{J \in {\mathbb{Z}}_+} \sup_{ I \in \mathfrak S_{J}({\mathbb{Z}}_+)}
\norm{ O^2_{I, J} ( A_{M_j; X, T_j}^{P_j({\mathrm m}_j)} : M_j \in {\mathbb{Z}}_+) }_{L^p(X)} 
& \lesssim_{p} 
\norm{ f }_{L^p(X)}, 
\qquad f \in L^p(X).\end{aligned}$$ For we refer to @MST1 [Theorem C], which guarantees that for every $p\in(1,\infty)$ one has $$\begin{aligned}
\norm[\Big]{ \Big( \sum_{\iota \in {\mathbb{Z}}} \big( \sup_{M_j \in {\mathbb{Z}}_+} \abs{A_{M_j; X, T_j}^{P_j({\mathrm m}_j)}}|f_{\iota}| \big)^2 \Big)^{1/2} }_{L^{p}(X)}
& \lesssim_{p}
\norm[\Big]{ \Big( \sum_{\iota \in {\mathbb{Z}}} \abs{f_\iota}^2  \Big)^{1/2} }_{L^{p}(X)}, \qquad (f_j)_{j \in {\mathbb{Z}}} \in L^{p}(X;\ell^2({\mathbb{Z}})).\end{aligned}$$ This completes the proof of the multi-parameter oscillation inequality in Theorem \[thm:main\].

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[^1]: Mariusz Mirek was partially supported by NSF grant DMS-2154712, and by the National Science Centre in Poland, grant Opus 2018/31/B/ST1/00204. Tomasz Z. Szarek was partially supported by the National Science Centre of Poland, grant Opus 2017/27/B/ST1/01623, by Juan de la Cierva Incorporaci<span>ó</span>n 2019 grant number IJC2019-039661-I funded by Agencia Estatal de Investigaci<span>ó</span>n, grant PID2020-113156GB-I00/AEI/10.13039/501100011033 and also by the Basque Government through the BERC 2018-2021 program and by Spanish Ministry of Sciences, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718.

[^2]: Note that there are various definitions of $L^0(X;B)$ in the literature.

{'source': 'arXiv_out/2209'}