The Fourier-transform $M^{q,*}$ Carleson maximal estimate for exponents $1<q<2$

This problem is the surviving open core of the program behind "The Walsh model for $M_2^*$ Carleson" (Demeter-Lacey-Tao-Thiele, 2008). That paper proved, for $1<p<\infty$, an $L^p$ bound for a maximally modulated, maximally truncated Walsh-Fourier partial-sum operator measured in a maximal-multiplier norm $M^{2,*}$, and presented the Walsh argument as a simplifying model for the Fourier-transform proof (by the same authors, 2008) that underlies the extension of Bourgain's Return Times Theorem to $1<p\le\infty$, $q\ge2$. The Walsh-model estimate was later extended to exponents $1<q<2$ (in the range $1/p+1/q<3/2$) by Oberlin (2013), who states the Fourier-transform extension below exponent 2 as an open problem. The problem is that extension: for the truncated Carleson-type operator $S_k[f](\xi,x)=\mathrm{p.v.}\int f(x-t)e^{2\pi i\xi t}\Psi(2^{-k}t)\,dt/t$, prove $\|S_k[f](\xi,x)\|_{L^p_x(M^{q,*}_{\xi,k})}\le C_{p,q}\|f\|_{L^p}$ for $1<q<2$, at least when $1/p+1/q<3/2$. A positive answer would extend the return-times convergence for the ergodic Hilbert series $\sum_{n\ne0}f(T^nx)g(S^ny)/n$ beyond the known range $q\ge2$; the sibling question of the Return Times Theorem's range for averages beyond $1/p+1/q<3/2$ remains open as well.

Background

Carleson's theorem on the almost-everywhere convergence of Fourier series is quantitatively equivalent to the boundedness on $L^p$ ($1<p<\infty$) of the Carleson operator, the maximally modulated singular integral $Cf(x)=\sup_\xi|\mathrm{p.v.}\int f(x-t)e^{2\pi i\xi t}\,dt/t|$. The problem studied here is a refinement of such bounds in which the maximal modulation is measured not in $L^p$ of the line but in a maximal-multiplier norm, a scale of norms introduced in the study of Bourgain's Return Times Theorem in ergodic theory.

By an $L^q$-multiplier one means a bounded function $m$ of the frequency variable such that $T_mg=(m\widehat g)^{\vee}$ (multiply the Fourier transform $\widehat g$ by $m$ and invert the transform) defines a bounded operator on $L^q$; its norm is $\|m\|_{M^q}=\sup_{\|g\|_{L^q}=1}\|T_mg\|_{L^q}$. For a sequence $m=(m_k)_{k\in\mathbb Z}$ of such functions the maximal-multiplier norm is $\|m\|_{M^{q,*}}=\sup_{\|g\|_{L^q}=1}\|((m_k\widehat g)^{\vee})(u)\|_{L^q_u(\ell^\infty_k)}$, which additionally controls the supremum over the index $k$. At $q=2$ one has $M^2=L^\infty$, which is why the source paper's $M_2^*$ estimate could be approached with $L^\infty$-type multiplier control; for $q<2$ the norms $M^{q,*}$ are strictly stronger and far less understood.

A measure-preserving system is a probability space $(X,\mu,T)$ together with a transformation $T$ that preserves $\mu$; the Birkhoff ergodic average $\frac1N\sum_{n<N}F(T^nx)$ of a single function converges almost everywhere. Bourgain's Return Times Theorem (RTT), proved around 1988 (the original proof of Bourgain, Furstenberg, Katznelson and Ornstein is unpublished; detailed expositions appear in the thesis of Fritzsch and, as recorded statements, in the paper of Demeter-Lacey-Tao-Thiele), is a two-system refinement: if $f\in L^\infty(X)$, there is a set $X_0$ of full measure, independent of any second system, such that for every $x\in X_0$, every second measure-preserving system $(Y,\nu,S)$ and every $g\in L^\infty(Y)$, the bilinear averages $\frac1N\sum_{n<N}f(T^nx)g(S^ny)$ converge for $\nu$-almost every $y$; informally, the return-time sequence $n\mapsto f(T^nx)$ is a universal good weight for every other system. Holder's inequality and a density argument extend this to the duality range $1/p+1/q\le1$. The theorem fails at $p=q=1$ (Assani-Buczolich-Mauldin, 2005), and the conjectural full range $1<p,q\le\infty$ is open: the best known range for the averages is $1/p+1/q<3/2$ (Demeter, 2012). The series analogue with Hilbert-kernel weights, $\sum_{0<|n|\le N}\frac1n f(T^nx)g(S^ny)$ (the ergodic Hilbert series), is known for $1<p\le\infty$ and $q\ge2$, and is open below $q=2$.

The bridge from these ergodic questions to Euclidean harmonic analysis is a transference scheme (introduced by Bourgain and developed by Demeter-Lacey-Tao-Thiele): the convergence statements reduce to maximal-multiplier inequalities for model operators on $\mathbb R$,

$$S_k[f](\xi,x)=\mathrm{p.v.}\int_{\mathbb R}\frac{f(x-t)}{t}\,e^{2\pi i\xi t}\,\Psi(2^{-k}t)\,dt,$$

the truncated, maximally modulated partial-sum (Hilbert-kernel) operator relevant to the series, where $\Psi$ is a Schwartz function equal to $1$ near the origin; its averages-type counterpart $A_k[f](\xi,x)=\int f(x-t)e^{2\pi i\xi t}2^{-k}\Psi(2^{-k}t)\,dt$ is relevant to the averages. The required estimates have the form $\|S_k[f](\xi,x)\|_{L^p_x(M^{q,*}_{\xi,k})}\le C\|f\|_{L^p}$: for almost every frozen $x$, the family of functions $\xi\mapsto S_k[f](\xi,x)$, indexed by $k$, is a sequence of multipliers whose $M^{q,*}$ norm is controlled, and this control is then averaged over $x$ in $L^p$. A key input on the Fourier side is Bourgain's maximal multiplier lemma (1989, his Lemma 4.11): for $N$ separated frequencies and suitable families of frequency windows $R_j$ around them, the maximal multiplier operator obeys $\|\sup_j|(\widehat f\,1_{R_j})^{\vee}|\|_{L^2}\le C(\log N)^{2}\|f\|_{L^2}$; a power of the logarithm is necessary (at least $\log^{1/4}N$, by Bourgain-Kostyukovsky-Olevskii), and this logarithmic dependence governs how the constants grow with the number of frequency windows.

The Walsh model is the dyadic laboratory for such theorems. The Walsh functions (products of Rademacher functions) are the characters of the group $(\mathbb R_+,\otimes)$, where $\otimes$ is addition without carry in binary expansion; the Walsh-Fourier transform diagonalizes it. Time-frequency analysis in this setting is organized by bitiles, dyadic rectangles $P=I_P\times\omega_P$ in the position-frequency plane each split into a lower and an upper frequency half, supporting Walsh wave packets, and by trees, which are collections of bitiles sharing a common top rectangle. Because Walsh wave packets are, up to constants, characteristic functions of dyadic intervals, they are exactly orthogonal, and the combinatorial core of Carleson-type theorems (organizing tiles into trees and estimating each tree against the position of its top) becomes transparent. The source paper proved the Walsh-model estimate $\|S_k[f](\theta,x)\|_{L^p_x(M^{2,*}_{\theta,k})}\le C_p\|f\|_{L^p}$ for $1<p<\infty$ (its Theorem 1.3) explicitly as a model for the harder Fourier-transform proof in its companion paper, which established the Fourier version of the same estimate and deduced from it the RTT for $1<p\le\infty$, $q\ge2$. Oberlin (2013) then extended the Walsh-model estimate to $1<q<2$ (for $1/p+1/q<3/2$) and proved stronger variation-multiplier-norm versions, but the Fourier-transform extension below $q=2$, the subject of this problem, remains open.

Problem Statement

Let $\Psi$ be a Schwartz function on $\mathbb R$ with $\Psi\equiv1$ on a neighborhood of the origin, and for a Schwartz function $f$ on $\mathbb R$, indices $\xi,x\in\mathbb R$ and $k\in\mathbb Z$ consider the truncated, maximally modulated singular-integral (partial Fourier-sum) operator

$$S_k[f](\xi,x)=\mathrm{p.v.}\int_{\mathbb R}\frac{f(x-t)}{t}\,e^{2\pi i\xi t}\,\Psi(2^{-k}t)\,dt.$$

For a sequence $m=(m_k)_{k\in\mathbb Z}$ of bounded functions define the maximal-multiplier norm

$$\|m\|_{M^{q,*}}=\sup_{\|g\|_{L^q(\mathbb R)}=1}\Big\|\big((m_k\,\widehat g\,)^{\vee}\big)(u)\Big\|_{L^q_u(\ell^\infty_k)},$$

where $\widehat{\cdot}$ and $(\cdot)^{\vee}$ denote the Fourier and inverse Fourier transforms, and define the mixed norm

$$\|S_k[f](\xi,x)\|_{L^p_x(M^{q,*}_{\xi,k})}=\Big(\int_{\mathbb R}\Big(\sup_{\|g\|_{L^q}=1}\ \int_{\mathbb R}\sup_{k\in\mathbb Z}\big|\big(S_k[f](\cdot,x)\,\widehat g\,\big)^{\vee}(u)\big|^{q}\,du\Big)^{p/q}dx\Big)^{1/p};$$

that is, for each frozen $x$ the family $\{\xi\mapsto S_k[f](\xi,x)\}_{k\in\mathbb Z}$ is regarded as a sequence of frequency-side multipliers, measured in $M^{q,*}$, and the result is integrated over $x$ in $L^p$.

What is known: (i) for every $1<p<\infty$, $\|S_k[f](\xi,x)\|_{L^p_x(M^{2,*}_{\xi,k})}\le C_p\|f\|_{L^p(\mathbb R)}$ (Demeter-Lacey-Tao-Thiele, 2008, in the proof of their return-times theorem); (ii) the identical estimate, with the Walsh-Fourier operator in place of $S_k$, holds for all $1<p<\infty$ and $1<q<2$ with $1/p+1/q<3/2$ (Oberlin, 2013).

Open problem. Prove that, for $1<p<\infty$ and $1<q<2$, at least in the range $1/p+1/q<3/2$,

$$\|S_k[f](\xi,x)\|_{L^p_x(M^{q,*}_{\xi,k})}\le C_{p,q}\|f\|_{L^p(\mathbb R)}.$$

The estimate is motivated in part by the associated ergodic statement: as Oberlin (2013) notes, an extension of the range of exponents in the estimate could be used to extend the range of exponents for the return-times convergence of the truncated ergodic Hilbert series

$$\sum_{0<|n|\le N}\frac{1}{n}\,f(T^nx)\,g(S^ny),$$

that is, the statement that for every $f\in L^p(X)$, $1<p\le\infty$, on a probability-preserving system $(X,\mu,T)$ there exists a universal set $X_f\subset X$ of full measure such that for every $x\in X_f$, every second system $(Y,\nu,S)$ and every $g\in L^q(Y)$, the series converges as $N\to\infty$ for $\nu$-almost every $y$, currently known only for $q\ge2$. The transference scheme of Demeter-Lacey-Tao-Thiele (2008) that establishes the $q\ge2$ case is $L^2$-based; carrying it out below $q=2$ is itself open work and is not part of the problem posed here. (For the averages-type operator $A_k[f](\xi,x)=\int f(x-t)e^{2\pi i\xi t}2^{-k}\Psi(2^{-k}t)\,dt$ the corresponding $M^{q,*}$ bound is known for $1/p+1/q<3/2$, and the Return Times Theorem for averages is known in the same range; extending that range toward the conjectural $1<p,q\le\infty$ is the sibling open problem.)

References

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