Sharp estimates of noncommutative Bochner-Riesz means on two-dimensional quantum tori

Inspired by operator algebras, harmonic analysis, noncommutative geometry and quantum probability, noncommutative harmonic analysis has rapidly developed recently (see e.g. [4, 7, 15, 16, 22, 27, 28, 35, 36, 38, 40, 60, 61]). The purpose of this paper is to study the noncommutative Bochner-Riesz means. We start by introducing the classical Bochner-Riesz means in Euclidean spaces.

It is well-known that the boundedness and convergence of Bochner-Riesz means are among the most important problems in harmonic analysis. The study of Bochner-Riesz means can also be regarded as making precise the sense in which the Fourier inversion formula holds. Recall that the Bochner-Riesz means on the usual torus $\mathbb{T}^{d}$ are defined by

where $\lambda\geq 0$, $R>0$, $(x)_{+}=\max\{x,0\}$ and $\widehat{f}(m)=\int_{\mathbb{T}^{d}}f(x)e^{-2\pi i\langle m,x\rangle}dx$. The central topic of Bochner-Riesz means is to seek the optimal range of $\lambda$ such that ${\text{\rm B}}_{R}^{\lambda}(f)$ converges to $f$ in some sense. In particular, the problem of the $L_{p}$ convergence turns out to show $(1-\frac{|\cdot|^{2}}{R^{2}})_{+}^{\lambda}:{\mathbb{Z}}^{d}\rightarrow{\mathbb{R}}$ is a uniform $L_{p}$ Fourier multiplier in $R>0$, which can be formulated as the so called Bochner-Riesz conjecture as follows (see e.g. [47]).

One can also define the Bochner-Riesz means on ${\mathbb{R}}^{d}$ by

where $\widehat{f}$ is the Fourier transform of $f$ on ${\mathbb{R}}^{d}$. By the standard transference technique (see e.g. [17]), the uniform $L_{p}$ boundedness of ${\text{\rm B}}_{R}^{\lambda}$ in $R>0$ on $\mathbb{T}^{d}$ is equivalent to that of $B_{R}^{\lambda}$ on ${\mathbb{R}}^{d}$. Because of this equivalence, in modern literature, researchers prefer to study the Bochner-Riesz means on ${\mathbb{R}}^{d}$.

The study of Bochner-Riesz means originated from S. Bochner [1]. The necessity of the conditions of $\lambda$ and $p$ in the Bochner-Riesz conjecture was given by C. Herz [20]. In dimension two, this conjecture has been completely resolved by L. Carleson and P. Sjölin [6], independently later by C. Fefferman [14], L. Hörmander [24] and A. Córdoba [10]. When dimension $d\geq 3$, the Bochner-Riesz conjecture is still open. We refer to some substantial progress in [3, 19, 31, 52, 53] and the references therein.

Concerning the pointwise convergence of $B_{R}^{\lambda}(f)$, it is natural to investigate the maximal Bochner-Riesz means defined by $B_{*}^{\lambda}(f)(x)=\sup_{R>0}|B_{R}^{\lambda}(f)(x)|$. It is conjectured that $B_{*}^{\lambda}$ is bounded on $L_{p}({\mathbb{R}}^{d})$ for $\lambda>0$ and $\frac{2d-1}{d+2\lambda}<p<\frac{2d}{d-1-2\lambda}$ (see [51]), where the range of $p\leq 2$ is different from that of the Bochner-Riesz conjecture. It is clear that the study of $B_{*}^{\lambda}$ is hander than that of $B^{\lambda}$. Up to now, this maximal Bochner-Riesz conjecture is even open for two dimensions. For some important progress, we refer the reader to [5, 33, 51] in the two-dimensional case and [31, 32, 46] for higher dimensions.

It should be pointed out that the study of Bochner-Riesz means is also quite related to several conjectures in harmonic analysis: Fourier restriction conjecture, local smoothing conjecture, maximal Kakeya function conjecture and Kakeya set conjecture (see e.g. [50]). To investigate the Bochner-Riesz conjecture, except some fundamental theories of maximal operators, Calderón-Zygmund operators, oscillatory integral operators, etc, researchers in harmonic analysis have invented many new and deep tools: bilinear or multilinear Fourier restriction (see [3, 31, 53]), incident geometry (see [2, 58]), decoupling and polynomial partitioning (see [19, 33]) in the last two decades. These new methods not only greatly improve the ranges of $\lambda$ and $p$ in the study of Bochner-Riesz means, but also open new promising research directions in harmonic analysis.

On the other hand, many useful theories in harmonic analysis, such as Littlewood-Paley-Stein square functions, Hardy-Littlewood maximal operators, duality of $H_{1}$-BMO, Calderón-Zygmund operators and multiplier operators, have been successfully transferred to the noncommutative setting (see e.g. [4, 21, 23, 36, 37, 40, 62]). Motivated by the development of this noncommutative harmonic analysis, the noncommutative Bochner-Riesz means on quantum tori have been investigated partially by Z. Chen, Q. Xu and Z. Yin [7] with limited indexes of $\lambda$ and $p$. Due to the lack of commutativity, the study of noncommutative Bochner-Riesz means seems to be more challenging. For example, Z. Chen, Q. Xu and Z. Yin [7] established the boundedness of the maximal Bochner-Riesz means on the $L_{p}$ space over quantum tori for $\lambda>(d-1)|\frac{1}{2}-\frac{1}{p}|$, which is an analogue of a classical result by E. M. Stein [46], but with a much more technical proof. Compared with the fruitful theories of commutative Bochner-Riesz means, the noncommutative Bochner-Riesz means deserve to be investigated further. This leads to a natural question that whether we can transfer the modern and powerful tools mentioned before into the noncommutative setting and apply them to studying noncommutative Bochner-Riesz means. Since the study of two-dimensional Bochner-Riesz means is relatively simple (note the Bochner-Riesz conjecture is resolved in this case), in this paper we focus on two dimensions and our main purpose is to obtain the full boundedness of noncommutative Bochner-Riesz means on quantum tori by developing a new tool—the noncommutative Kakeya maximal function.

Quantum tori are also known as noncommutative tori or rotational algebras (see [44]). One can regard quantum tori as analogues of usual tori. Quantum tori are basic examples in operator algebras (see [11]) and are interesting objects in noncommutative geometry which have been extensively studied (see e.g. [8, 44, 54]). The research of analysis on quantum tori was started in [45, 56, 57] and the first systematic work of harmonic analysis on quantum tori was given later in [7]. For recent work related to quantum tori in the direction of noncommutative analysis, we refer to see [27, 29, 34, 43, 60, 61] and the references therein.

To illustrate our main results, we should give the definition of quantum torus. Suppose that $d\geq 2$, $\theta=(\theta_{k,j})_{1\leq k,j\leq d}$ is a real skew symmetric $d\times d$ matrix. The $d$-dimensional noncommutative torus ${\mathcal{A}}_{\theta}$ is a universal $C^{*}$-algebra generated by $d$ unitary operators $U_{1},\cdots,U_{d}$ satisfying the following commutation relation:

A well-known fact is that ${\mathcal{A}}_{\theta}$ admits a faithful tracial state $\tau$. Define $\mathbb{T}_{\theta}^{d}$ the weak $*$-closure of ${\mathcal{A}}_{\theta}$ in the GNS representation of $\tau$. We call $\mathbb{T}_{\theta}^{d}$ the $d$-dimensional quantum torus. The state $\tau$ also extends to a normal faithful state on $\mathbb{T}_{\theta}^{d}$, which will be denoted again by $\tau$. Notice that when $\theta=0$, ${\mathcal{A}}_{\theta}=C(\mathbb{T}^{d})$ and $\mathbb{T}_{\theta}^{d}=L_{\infty}(\mathbb{T}^{d})$. Thus quantum torus $\mathbb{T}_{\theta}^{d}$ is a deformation of classical torus $\mathbb{T}^{d}$. Let $L_{p}(\mathbb{T}_{\theta}^{d})$ be the noncommutative space associated to pairs $(\mathbb{T}_{\theta}^{d},\tau)$ with the $L_{p}$ norm given by $\|x\|_{L_{p}(\mathbb{T}_{\theta}^{d})}=(\tau(|x|^{p}))^{1/p}$.

In the following, we consider the Bochner-Riesz means on quantum tori which are defined by

where $U=(U_{1},\cdots,U_{d})$, $U^{m}=U_{1}^{m_{1}}\cdots U_{d}^{m_{d}}$ and $\widehat{f}(m)=\tau((U^{m})^{*}f)$. A fundamental problem raised in [7, Page 762] is that in which sense the Bochner-Riesz means converge back to $f$. In this paper we consider this problem in two-dimensional case and state our main results as follows.

This theorem is in fact a noncommutative version of the two-dimensional Bochner-Riesz conjecture. Thus we completely resolve an open problem raised in [7] in the sense of the $L_{p}$ convergence for two dimensions. In the following, we briefly introduce the strategy used in the proof of Theorem 1.1.

Notice that ${\text{\rm B}}_{R}^{\lambda}(f)$ is fully noncommutative and its analysis seems to be rather difficult. Nevertheless there is a clever trick that transfers the problem of multiplier operator on quantum tori to the operator-valued setting on usual tori (see [7]). Hence using this method, the $L_{p}$ boundedness of Bochner-Riesz means on quantum tori can be reduced to that of the operator-valued Bochner-Riesz means on usual tori $\mathbb{T}^{d}$ (see Theorem 5.1). Next we use the noncommutative transference of multiplier (see Theorem 5.4), we can transfer the study of the operator-valued Bochner-Riesz means on $\mathbb{T}^{d}$ to that on ${\mathbb{R}}^{d}$, in which case we can do analysis based on some known noncommutative theories of harmonic analysis. However to establish the full boundedness of two-dimensional operator-valued Bochner-Riesz means on ${\mathbb{R}}^{2}$, the previous noncommutative theories may not be sufficient and some new tools in harmonic analysis related to the geometry of Euclidean spaces should be brought in.

Our main new tool is the noncommutative Kakeya maximal function. Define the Kakeya average operator by

where $R$ is a rectangle centered at the origin with arbitrary orientation and eccentricity $N$ (see Section 3 for its definition). As aforementioned, the study of the Kakeya maximal function $\sup_{R}|\mathcal{K}_{R}f(x)|$ is another important problem in harmonic analysis related to Bochner-Riesz means (see e.g. [59]). Notice that the study of noncommutative Kakeya maximal functions is more difficult since it can not be defined directly. It is easy to see that $\mathcal{K}_{R}$ could be dominated by the Hardy-Littlewood average operator with bound $N$. This implies that the noncommutative Kakeya maximal operator is $L_{2}$ bounded with norm $N^{\frac{1}{2}}$. In this paper, we shall establish its sharp $L_{2}$ norm—$\log N$ (see Theorem 3.1), which is crucial to our study of noncommutative Bochner-Riesz means. To the best knowledge of the author, it is the first time that a sharp estimate of the noncommutative Kakeya maximal function is obtained in noncommutative analysis. The proof here is quite technical and our strategy is the microlocal decomposition, together with theories of the Fourier transform and noncommutative square/maximal functions.

Below we sketch out the proof of the $L_{p}$ boundedness of the operator-valued Bochner-Riesz means on ${\mathbb{R}}^{2}$ (i.e. Theorem 4.1). To get the full boundedness of Bochner-Riesz means, by the duality and the noncommutative analytic interpolation theorem (see the appendix), it suffices to show the result for the case $p=4$. We first make a dyadic decomposition: $B^{\lambda}=\sum_{k}T_{k}$ and matters are reduced to proving the $L_{4}$ norm of $T_{k}$ has enough decay in $k$. We next make a microlocal decomposition: $T_{k}=\sum_{l}T_{k,l}$ where the support of each $T_{k,l}$ lies in a small piece (denoted by $\Gamma_{k,l}$) of annulus with the major direction $e^{2\pi il2^{-\frac{k}{2}}}$. Notice that the $L_{4}$ norm of $T_{k}(f)$ has an expression

If the major directions of these pieces are closed to each other (i.e. $|l-l^{\prime}|\leq C$), then the above term is bounded by a column square function norm $\big{\|}\big{(}\sum_{l}|T_{k,l}(f)|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L_{4}({\mathcal{N}})}$. If $|l-l^{\prime}|\geq C$, then we need a very important geometric observation: $\{\Gamma_{k,l}-\Gamma_{k,l^{\prime}}\}_{l,l^{\prime}}$ is finite overlapped. With this geometric estimate, the $L_{4}$ norm of $T_{k}f$ is bounded by a row square function norm $\big{\|}\big{(}\sum_{l}|T_{k,l}(f)^{*}|^{2}\big{)}^{\frac{1}{2}}\big{\|}_{L_{4}({\mathcal{N}})}$. Consequently we use both column and row square function norms to control the $L_{4}$ norm of $T_{k}(f)$, which is consistent with the theory of noncommutative square functions. To estimate this column/row square functions, we should do more analysis for the kernel of $T_{k,l}$. Roughly speaking, a key fact in our proof is that $T_{k,l}$ can be bounded by a sum of Kakeya average operators $\mathcal{K}_{R}$s where orientations of $R$s are just in a fixed direction—the major direction of $\Gamma_{k,l}$. By the dual theory between $L_{p^{\prime}}({\mathcal{M}};\ell_{\infty})$ and $L_{p}({\mathcal{M}};\ell_{1})$, our estimates for noncommutative square functions can be reduced to that of Kakeya maximal functions, which will be systematically studied in Section 3.

The methods above heavily rely on the geometry of the plane. For the higher dimensional case, to get some nontrivial boundedness of Bochner-Riesz means, some more new tools in harmonic analysis should be transferred to the noncommutative setting. We hope to work this problem in the future.

This paper is organized as follows. First we give some preliminaries of noncommutative $L_{p}$ spaces, noncommutative maximal/square functions and related lemmas in Section 2. In Section 3, we investigate the noncommutative Kakeya maximal function and establish its sharper estimate there. In Section 4, we obtain the full $L_{p}$ boundedness of the operator-valued Bochner-Riesz means on ${\mathbb{R}}^{2}$. The proof is based on sharper estimates of noncommutative Kakeya maximal functions and a square function inequality studied in the previous sections. Section 5 is devoted to the study of Bochner-Riesz means on quantum tori. In this section, we first establish the full estimates of the operator-valued Bochner-Riesz means on usual tori $\mathbb{T}^{2}$ and then transfer this result to that on two-dimensional quantum tori (i.e. Theorem 1.1). Finally for the reader’s convenience, we give a proof of the noncommutative analytic interpolation theorem in the appendix which may be known to experts.

Notation. Throughout this paper, the letter $C$ stands for a positive finite constant which is independent of the essential variables, not necessarily the same one in each occurrence. $A\lesssim B$ means $A\leq CB$ for some constant $C$. By the notation $C_{\varepsilon}$ we mean that the constant depends on the parameter $\varepsilon$. $A\approx B$ means that $A\lesssim B$ and $B\lesssim A$. For any measurable set $A\subset{\mathbb{R}}^{d}$, we denote the Lebesgue measure by $|A|$. ${\mathbb{Z}}_{+}$ denotes the set of all nonnegative integers and ${\mathbb{Z}}_{+}^{d}={{\mathbb{Z}}_{+}\times\cdots\times{\mathbb{Z}}_{+}}$ with $d$-tuples product. For $\alpha\in{\mathbb{Z}}_{+}^{d}$ and $x\in{\mathbb{R}}^{d}$, $x^{\alpha}=x_{1}^{\alpha_{1}}\cdots x_{d}^{\alpha_{d}}$. Set ${\mathbb{R}}_{+}=(0,\infty)$. $\forall s\in{\mathbb{R}}_{+}$, $\lfloor s\rfloor$ denotes the integer part of $s$. We use LHS to represent left hand side of an expression. Given a function $f$ on $\mathbb{T}^{d}$, the Fourier transform of $f$ is defined by $\widehat{f}(k)=\int_{\mathbb{T}^{d}}f(x)e^{-2\pi ix\cdot k}dx$. For a function $f$ on ${\mathbb{R}}^{d}$, define $\mathcal{F}f$ (or $\hat{f}$) and $\mathcal{F}^{-1}f$ (or $\check{f}$) the Fourier transform and the inversion Fourier transform of $f$ by

In this section, we introduce some basic knowledge of noncommutative harmonic analysis including noncommutative $L_{p}$ spaces, maximal functions, square functions and many operator-valued inequalities which are useful in this paper.

In this section, we study the boundedness of noncommutative Kakeya maximal functions. Before that we give several definitions and lemmas. We only consider $d=2$. Set ${\mathcal{N}}=L_{\infty}({\mathbb{R}}^{2})\overline{\otimes}{\mathcal{M}}$ and $\varphi=\int_{{\mathbb{R}}^{2}}dx\otimes\tau$. Denote $L_{p}({\mathcal{N}})$ the noncommutative $L_{p}$ space associated with pairs $({\mathcal{N}},\varphi)$.

The main preliminaries of noncommutative maximal functions have been given in Subsection 2.2. We introduce the noncommutative Kakeya maximal function as follows. Define the eccentricity of a rectangle by the ratio of the length of its long side to that of its short side. Let $N$ be a positive integer. Define the set $\mathcal{R}_{N}$ as rectangles in the plane of arbitrary orientation whose center is the origin and eccentricity is $N$. For $f\in L_{p}({\mathcal{N}})$, we define the Kakeya average operator as follows

where $R$ is a rectangle belonging to $\mathcal{R}_{N}$. We are mainly interested in the maximal $L_{2}$ norm of noncommutative Kakeya average operator, since it is a crucial estimate in the study of noncommutative Bochner-Riesz means.

Let us first give a trivial bound of its $L_{2}$ norm. Recall that the Hardy-Littlewood average operator is defined by

where $Q$ is a cube in ${\mathbb{R}}^{2}$ with center zero and arbitrary orientation. For any rectangle $R\in\mathcal{R}_{N}$, there exists a cube $Q$ such that $R\subset Q$ and $l(Q)$ equals to the length of long side of $R$. Consider $f$ as a positive function in ${\mathcal{N}}$. Then $\mathcal{K}_{R}f(x)\leq NM_{Q}f(x)$. Using the noncommutative Hardy-Littlewood maximal operator is of weak type $(1,1)$ (see [36]), we get the Kakeya maximal operator is of weak type $(1,1)$ with bound $N$. Applying the noncommutative Marcinkiewicz interpolation theorem in [30], together with the fact that the maximal operator of $\mathcal{K}_{R}$ is of $(\infty,\infty)$, we get that the maximal operator of $\mathcal{K}_{R}$ is of strong $(2,2)$ type with bound $N^{\frac{1}{2}}$. However this bound is quite rough and not sufficient for our later application. The following improved bound is our main result in this section.

Combining the noncommutative Marcinkiewicz interpolation theorem (see [30]), together with a trivial weak type $(1,1)$ bound and a strong $(\infty,\infty)$ bound of $\mathcal{K}_{R}$, we immediately get the following corollary.

It should be pointed out that the bound $\log N$ in Theorem 3.1 is sharp even in the commutative case, see [18, Proposition 5.3.4]. A. Córdoba [9] first obtained the boundedness of the Kakeya maximal function on $L_{2}({\mathbb{R}}^{2})$ with norm $(\log N)^{2}$. The sharp bound $\log N$ was later established by J. O. Strömberg [49] where he used several estimates of distribution functions and some geometric constructions. S. Wainger [55] also obtained the sharp bound $\log N$ without a proof but he mentioned that the idea given by A. Nagel, E. M. Stein, S. Wainger [39] can be modified to his setting. It is well-known that the distribution function is difficult to deal with in the noncommutative setting (for example the weak $(1,1)$ boundedness problem is a challenge problem). Hence the method from J. O. Strömberg [49] may be difficult to be applied in the noncommutative setting. The strategies used in our proof below are the Fourier transform, square and maximal function theories and the microlocal decomposition, which are mainly motivated by [39] and [55].

Before giving the proof of Theorem 3.1, we introduce the noncommutative directional Hardy-Littlewood average operator defined by

where $e$ is a unit vector in ${\mathbb{R}}^{2}$. By using the standard method of rotation, the definition of maximal norm in (2.3) and the fact one dimensional noncommutative Hardy-Littlewood maximal operator is of strong type $(p,p)$ for $1<p\leq\infty$ (see [36]), the author and his collaborators recently established the following result in [21, Lemma 6.3].

Now we are in a position to prove Theorem 3.1.