Sharp Estimates of Noncommutative Bochner–Riesz Means on Two-Dimensional Quantum Tori

Abstract

In this paper, we establish the full $L_p$ boundedness of noncommutative Bochner–Riesz means on two-dimensional quantum tori, which completely resolves an open problem raised in Chen et al. (Commun Math Phys 322(3):755–805, 2013) in the sense of the $L_p$ convergence for two dimensions. The main ingredients are sharp estimates of noncommutative Kakeya maximal functions and geometric estimates in the plane. We make the most of noncommutative theories of maximal/square functions, together with microlocal decompositions in both proofs of sharper estimates of Kakeya maximal functions and Bochner–Riesz means.

Appendix A. Interpolation of Analytic Families of Operators on Noncommutative \(L_p\) Spaces

In this appendix, we state precisely an analytic interpolation theorem which may be known to experts. Let ${\mathcal {S}}$ be the linear span of all $x\in {{\mathcal {M}}}_{+}$ whose support projections have finite trace. Suppose that $T_z$ is a linear operator mapping ${\mathcal {S}}$ to itself for every z in the closed strip ${\bar{S}}=\{z\in {{\mathbb {C}}}:0\le \text {Re} z\le 1\}$. We say the family $\{T_z\}_z$ is analytic if the function

$$\begin{aligned} z\rightarrow \tau (gT_z(f)) \end{aligned}$$

is analytic in the open strip $S=\{z\in {{\mathbb {C}}}:0<\text {Re}z<1\}$ and continuous on ${\bar{S}}$ for any functions f and g in ${\mathcal {S}}$. Moreover we say the analytic family $\{T_z\}_z$ is of admissible growth if there exists a constant $0<a<\pi$ such that

$$\begin{aligned} e^{-a|\text {Im}z|}\log |\tau (gT_z(f))|<\infty \end{aligned}$$

for all $z\in {\bar{S}}$. Now we can state the following analytic interpolation theorem.

Theorem 1

Suppose that $T_z$ is an analytic family of linear operators of admissible growth. Let $p_0,p_1,q_0,q_1\in (0,\infty )$ and assume that $M_0,M_1$ are positive functions on ${{\mathbb {R}}}$ such that

$$\begin{aligned} \sup _{y\in {{\mathbb {R}}}}e^{-b|y|}\log M_j(y)<\infty \end{aligned}$$

(A.1)

for $j=0,1$ and some $b\in (0,\pi )$. Let $p,\theta$ satisfy $\frac{1}{p}=\frac{1-\theta }{p_0}+\frac{\theta }{p_1}$ and $\frac{1}{q}=\frac{1-\theta }{q_0}+\frac{\theta }{q_1}$. Suppose that

$$\begin{aligned} \begin{aligned} \Vert T_{iy}(f)\Vert _{L_{q_0}({{\mathcal {M}}})}&\le M_0(y)\Vert f\Vert _{L_{p_0}({{\mathcal {M}}})},\quad \Vert T_{1+iy}(f)\Vert _{L_{q_1}({{\mathcal {M}}})}\le M_1(y)\Vert f\Vert _{L_{p_1}({{\mathcal {M}}})} \end{aligned}\nonumber \\ \end{aligned}$$

(A.2)

hold for all $f\in {\mathcal {S}}$. Then for any $\theta \in (0,1)$, we have

$$\begin{aligned} \Vert T_{\theta }(f)\Vert _{L_q({{\mathcal {M}}})}\le M(\theta )\Vert f\Vert _{L_p({{\mathcal {M}}})}, \end{aligned}$$

where for $0<t<1$,

$$\begin{aligned} M(t)=\exp \Big \{\frac{\sin (\pi t)}{2}\int _{-\infty }^{\infty }\Big [\frac{\log M_0(y)}{\cosh (\pi y)-\cos (\pi t)}+\frac{\log M_1(y)}{\cosh (\pi y)+\cos (\pi t)}\Big ]dy\Big \}. \end{aligned}$$

To prove this theorem, we need an extension of the three lines theorem which could be found in [48, Page 206, Lemma 4.2].

Lemma A.2

Suppose that F is analytic on the open strip S and continuous on its closure such that

$$\begin{aligned} \sup _{z\in {\bar{S}}}e^{-a|\text {Im}z|}\log |F(z)|<\infty \end{aligned}$$

for some $a\in (0,\pi )$. Then for any $0<x<1$, we have

$$\begin{aligned} |F(x)|\le \exp \Big \{\frac{\sin (\pi x)}{2}\int _{-\infty }^{\infty }\Big [\frac{\log |F(iy)|}{\cosh (\pi y)-\cos (\pi x)}+\frac{\log |F(1+iy)|}{\cosh (\pi y)+\cos (\pi x)}\Big ]dy\Big \}. \end{aligned}$$

Proof of Theorem 1

The proof is quite similar to that in the commutative case. Let $f,g\in {\mathcal {S}}$ with polar decompositions $f=u|f|$ and $g=v|g|$. Without loss of generality, we may suppose that $\Vert f\Vert _{L_p({{\mathcal {M}}})}=1=\Vert g\Vert _{L_{q'}({{\mathcal {M}}})}$. By the duality, to prove our theorem, it suffices to show

$$\begin{aligned} |\tau (gT_{\theta }(f))|\le M({\theta }). \end{aligned}$$

For $z\in {\bar{S}}$, define $f(z)=u|f|^{\frac{p(1-z)}{p_0}+\frac{pz}{p_1}}$ and $g(z)=v|g|^{\frac{q'(1-z)}{q'_0}+\frac{q'z}{q'_1}}$ where the continuous functional calculus is defined by complex powers of positive operators. By the density argument, we could suppose that |f| and |g| are linear combinations of mutually orthogonal projections of finite trace, i.e.

$$\begin{aligned} |f|=\sum _{j=1}^n\alpha _je_j,\quad |g|=\sum _{k=1}^m\beta _k{\tilde{e}}_k \end{aligned}$$

where $\alpha _j$s, $\beta _k$s are real and $e_j$s, ${\tilde{e}}_k$s are mutually orthogonal basis. Then

$$\begin{aligned} f(z)=\sum _{j=1}^n\alpha _j^{\frac{p(1-z)}{p_0}+\frac{pz}{p_1}}ue_j. \end{aligned}$$

Therefore the function $z\rightarrow f(z)$ is an analytic function on ${{\mathbb {C}}}$ taking values in ${{\mathcal {M}}}$. Similar properties hold for the function $z\rightarrow g(z)$. Define

$$\begin{aligned} F(z)=\tau (g(z)T_z(f(z))). \end{aligned}$$

Then we have

$$\begin{aligned} F(z)=\sum _{j=1}^n\sum _{k=1}^m\alpha _j^{\frac{p(1-z)}{p_0}+\frac{pz}{p_1}} \beta _k^{\frac{q'(1-z)}{q'_0}+\frac{q'z}{q'_1}}\tau (v{\tilde{e}}_kT_z(ue_j)). \end{aligned}$$

By our assumption, $\tau (v{\tilde{e}}_kT_z(ue_j))$ is analytic. Hence F(z) is an analytic function satisfying the hypothesis of Lemma A.2. Recall a property of polar decomposition: $|f|=|f|u^*u=u^*u|f|$, then by the continuous functional calculus of |f|, we obtain

$$\begin{aligned} \omega (|f|)=\omega (|f|)u^*u=u^*uw(|f|), \end{aligned}$$

(A.3)

where $\omega$ is a continuous function on ${{\mathbb {R}}}_+$. Since

$$\begin{aligned} f(iy)=u|f|^{iyp(\frac{1}{p_1}-\frac{1}{p_0})+\frac{p}{p_0}}, \end{aligned}$$

then by (A.3), we get

$$\begin{aligned} \begin{aligned} |f(iy)|^2=f^*(iy)f(iy)=|f|^{-iyp(\frac{1}{p_1}-\frac{1}{p_0})+\frac{p}{p_0}} u^*u|f|^{iyp(\frac{1}{p_1}-\frac{1}{p_0})+\frac{p}{p_0}} =|f|^{\frac{2p}{p_0}}. \end{aligned} \end{aligned}$$

Therefore we get $\Vert f(iy)\Vert _{L_{p_0}({{\mathcal {M}}})}=1$. Similarly $\Vert f(1+iy)\Vert _{L_{p_1}({{\mathcal {M}}})}=1=\Vert g(iy)\Vert _{L_{q'_0}({{\mathcal {M}}})}=\Vert g(1+iy) \Vert _{L_{q'_1}({{\mathcal {M}}})}$. Hölder’s inequality and our assumption show that for all $y\in {{\mathbb {R}}}$,

$$\begin{aligned} \begin{aligned} |F(iy)|&\le \Vert T_{iy}(f(iy))\Vert _{L_{q_0}({{\mathcal {M}}})}\Vert g(iy)\Vert _{L_{q'_0}({{\mathcal {M}}})}\\&\le M_0(y)\Vert f(iy)\Vert _{L_{p_0}({{\mathcal {M}}})}\Vert g(iy)\Vert _{L_{q'_0}({{\mathcal {M}}})}=M_0(y). \end{aligned} \end{aligned}$$

Similarly for all $y\in {{\mathbb {R}}}$, $|F(1+iy)|\le M_1(y)$. Now applying Lemma A.2 with the preceding two estimates and notice that $\cosh (\pi y)=\frac{1}{2}(e^{\pi y}+e^{-\pi y})\ge 1\ge \cos (\pi x)$, we get

$$\begin{aligned} |\tau (gT_\theta (f))| =|F(\theta )|\le M(\theta ), \end{aligned}$$

which implies the required estimate. $\square$