$L_p$–$L_q$ Fourier Multipliers on Locally Compact Quantum Groups

Abstract

Let $\mathbb {G}$ be a locally compact quantum group with dual $\widehat{\mathbb {G}}$. Suppose that the left Haar weight $\varphi$ and the dual left Haar weight $\widehat{\varphi }$ are tracial, e.g. $\mathbb {G}$ is a unimodular Kac algebra. We prove that for $1<p\le 2 \le q<\infty$, the Fourier multiplier $m_{x}$ is bounded from $L_p(\widehat{\mathbb {G}},\widehat{\varphi })$ to $L_q(\widehat{\mathbb {G}},\widehat{\varphi })$ whenever the symbol x lies in $L_{r,\infty }(\mathbb {G},\varphi )$, where $1/r=1/p-1/q$. Moreover, we have

$$\begin{aligned} \Vert m_{x}:L_p(\widehat{\mathbb {G}},\widehat{\varphi })\rightarrow L_q(\widehat{\mathbb {G}},\widehat{\varphi })\Vert \le c_{p,q} \Vert x\Vert _{L_{r,\infty }(\mathbb {G},\varphi )}, \end{aligned}$$

where $c_{p,q}$ is a constant depending only on p and q. This was first proved by Hörmander (Acta Math 104:93–140, 1960) for $\mathbb {R}^n$, and was recently extended to more general groups and quantum groups. Our work covers all these results and the proof is simpler. In particular, this also yields a family of $L_p$-Fourier multipliers over discrete group von Neumann algebras. A similar result for $\mathcal {S}_p$-$\mathcal {S}_q$ Schur multipliers is also proved.

1 Introduction

For a nice function (e.g. Schwartz function) $f:\mathbb {R}^n\rightarrow \mathbb {C}$, let $\widehat{f}$ denote its Fourier transform:

$$\begin{aligned} \widehat{f}(\xi )=\int _{\mathbb {R}^n}f(x)e^{-2\pi i\langle x,\xi \rangle }dx, \end{aligned}$$

where $\langle \cdot ,\cdot \rangle$ denotes the Euclidean inner product on $\mathbb {R}^n$. Given $p,q>0$ and $\phi :\mathbb {R}^n\rightarrow \mathbb {C}$, the operator $m_{\phi }$ defined via

$$\begin{aligned} \widehat{m_{\phi } f}(\xi )=\phi (\xi )\widehat{f}(\xi ),~~\xi \in \mathbb {R}^n, \end{aligned}$$

is called an $L_p$-$L_q$ Fourier multiplier if it is bounded from $L_p(\mathbb {R}^n)$ to $L_q(\mathbb {R}^n)$. When $p=q$, it is called an $L_p$-Fourier multiplier for short. The function $\phi$ is called the symbol of the Fourier multiplier $m_\phi$.

Hörmander proved the following $L_p$-$L_q$ Fourier multipliers theorem:

Theorem 1.1

[14, Theorem 1.11] Let $1<p\le 2\le q<\infty$ and $1/r=1/p-1/q$. Then we have

$$\begin{aligned} \Vert m_{\phi }:L_p(\mathbb {R}^n)\rightarrow L_q(\mathbb {R}^n)\Vert \precsim _{p,q}\Vert \phi \Vert _{L_{r,\infty }(\mathbb {R}^n)}. \end{aligned}$$

Here $L_{p,\infty }(\mathbb {R}^n)$ denotes the usual weak $L_p$-space. Throughout this paper, $C_1\precsim C_2$ always means $C_1\le c C_2$ for some positive constant $c<\infty$. We write $C_1\precsim _p C_2$ if the constant $c=c_p$ is dependent of p. To prove Theorem 1.1, Hörmander used the following Paley-type inequalities.

Theorem 1.2

[14, Theorem 1.10] For $1<p\le 2$ and $1/s=2/p-1$, we have

$$\begin{aligned} \Vert f\widehat{g}\Vert _{L_{p}(\mathbb {R}^n)}\precsim _{p} \Vert f\Vert _{L_{s,\infty }(\mathbb {R}^n)}\Vert g\Vert _{L_{p}(\mathbb {R}^n)}. \end{aligned}$$

Both theorems have been generalized to compact Lie groups by Akylzhanov, Nursultanov and Ruzhansky [2], to locally compact separable unimodular groups by Akylzhanov and Ruzhansky [3], and to compact quantum groups of Kac type by Akylzhanov, Majid and Ruzhansky [1]. Theorem 1.2 for compact quantum groups of Kac type was also shown by Youn [26]. All their proofs go back to Hörmander [14].

Our first result is a generalization of Theorem 1.1 to locally compact quantum groups $\mathbb {G}$ whose left Haar weight $\varphi$ and the dual left Haar weight $\widehat{\varphi }$ are both tracial. Our proof is slightly simpler and does not require Paley-type inequalities.

Theorem 1.3

Let $1<p\le 2\le q<\infty$. Let $\mathbb {G}=(\mathcal {M},\Delta ,\varphi ,\psi )$ be a locally compact quantum group with its dual $\widehat{\mathbb {G}}=(\widehat{\mathcal {M}},\widehat{\Delta },\widehat{\varphi },\widehat{\psi })$. Suppose that $\varphi$ and $\widehat{\varphi }$ are both tracial. Then for each $x\in L_{r,\infty }(\mathbb {G},\varphi )$ with $1/r=1/p-1/q$, $m_x$ is an $L_p$-$L_q$ Fourier multiplier such that

$$\begin{aligned} \Vert m_x:L_p(\widehat{\mathbb {G}},\widehat{\varphi })\rightarrow L_q(\widehat{\mathbb {G}},\widehat{\varphi })\Vert \precsim _{p,q}\Vert x\Vert _{L_{r,\infty }(\mathbb {G},\varphi )}. \end{aligned}$$

See Sects. 2 and 3 for the corresponding definitions. We will not deduce Theorem 1.3 from Paley-type inequalities, but we may still extend Theorem 1.2 to locally compact quantum groups with a slightly simpler proof. This is our second result.

Theorem 1.4

Let $1<p\le 2$. Let $\mathbb {G}=(\mathcal {M},\Delta ,\varphi ,\psi )$ be a locally compact quantum group with its dual $\widehat{\mathbb {G}}=(\widehat{\mathcal {M}},\widehat{\Delta },\widehat{\varphi },\widehat{\psi })$. Suppose that $\varphi$ and $\widehat{\varphi }$ are both tracial. Then we have

$$\begin{aligned} \Vert a\mathcal {F}(x)\Vert _{L_p(\widehat{\mathbb {G}},\widehat{\varphi })} \precsim \Vert a\Vert _{L_{s,\infty }(\widehat{\mathbb {G}},\widehat{\varphi })}\Vert x\Vert _{L_p(\mathbb {G},\varphi )}, \end{aligned}$$

for all $a\in L_{s,\infty }(\widehat{\mathbb {G}},\widehat{\varphi })$ and $x\in L_p(\mathbb {G},\varphi )$, where $1/s=2/p-1$.

Here $\mathcal {F}$ denotes the Fourier transform; see Sect. 3 for the definition. If furthermore, the dual quantum group $\widehat{\mathbb {G}}$ is compact, then Theorem 1.3 gives a sufficient condition for $L_p$-Fourier multipliers on $\mathbb {G}$. This is our third result.

Theorem 1.5

Fix $1<p<\infty$ and $1/p^*=|1/2-1/p|$. Let $\mathbb {G}$ be a compact quantum group of Kac type with Haar state h. Let $\widehat{\mathbb {G}}$ be its dual with dual Haar weight $\widehat{h}$. Let $\widehat{\mathcal {F}}$ be the Fourier transform over $\widehat{\mathbb {G}}$. Then for any $a=(a_{\pi })_{\pi \in \textrm{Irr}(\mathbb {G})}\in \widehat{\mathbb {G}}$, the Fourier multiplier $m_a:\widehat{\mathcal {F}}(b)\mapsto \widehat{\mathcal {F}}(ab)$ satisfies

$$\begin{aligned} \Vert m_a:L_p(\mathbb {G},h)\rightarrow L_p(\mathbb {G},h)\Vert \precsim _{p}\Vert a\Vert _{\ell _{p^*,\infty }(\widehat{\mathbb {G}},\widehat{h})}. \end{aligned}$$

Remark 1.6

Clearly, the Fourier multiplier with the symbol $c\varvec{1}$ is bounded on $L_p(\mathbb {G},h)$ with the norm |c|, where $c\in \mathbb {C}$ and $\varvec{1}$ is the unit element in $\widehat{\mathbb {G}}$. However, $c\varvec{1}$ may not lie in $\ell _{p^*,\infty }(\widehat{\mathbb {G}},\widehat{h})$ in general. Since the family of $L_p$-Fourier multipliers is a vector space, one can replace the upper bound by $|c|+c_p\Vert a-c\varvec{1}\Vert _{\ell _{p^*,\infty }(\widehat{\mathbb {G}},\widehat{h})}$.

An interesting family of such examples is obtained by choosing $\mathbb {G}=\widehat{G}$ as the group von Neumann algebra of a discrete group G.

Corollary 1.7

For any discrete group G, let $\widehat{G}$ be the group von Neumann algebra equipped with the canonical tracial state $\tau$. Then for any $\phi :G\rightarrow \mathbb {C}$ such that $\phi -c1\in \ell _{p^*,\infty }(G)$ for some $c\in \mathbb {C}$ with $1/p^*=|1/2-1/p|$, the Fourier multiplier $m_{\phi }:L_p(\widehat{G},\tau )\rightarrow L_{p}(\widehat{G},\tau ),\lambda _g\mapsto \phi (g)\lambda _g$ extends to a bounded map satisfying

$$\begin{aligned} \Vert m_{\phi }:L_p(\widehat{G},\tau )\rightarrow L_{p}(\widehat{G},\tau )\Vert \le |c|+c_p\Vert \phi -c1\Vert _{\ell _{p^*,\infty }(G)}, \end{aligned}$$

where $\lambda$ is the left regular representation of G.

An analogue of $L_p$-$L_q$ Fourier multipliers theorem is also valid for Schur multipliers. We use $\mathcal {S}_p(H)$ to denote the Schatten p-classes $L_p(B(H))$. See [21] and references therein for more related results.

Theorem 1.8

Let $1< p\le 2\le q<\infty$ and $1/r=1/p-1/q$. Let X be a set. The Schur multiplier $A:(x_{ij})_{i,j\in X}\mapsto (a_{ij}x_{ij})_{i,j\in X}$ satisfies

$$\begin{aligned} \Vert A:\mathcal {S}_p(\ell _{2}(X))\rightarrow \mathcal {S}_q(\ell _{2}(X))\Vert \precsim _{p,q} \Vert a\Vert _{\ell _{r,\infty }(X\times X)}, \end{aligned}$$

where on the right hand side $a=(a_{ij})_{i,j\in X}$ is identified as an element in $\mathbb {C}^{X\times X}$.

The paper is organized as follows. In Sect. 2 we recall basic knowledge of locally compact quantum groups and noncommutative ($L_p$- and) Lorentz spaces. Section 3 presents the Fourier transform on locally compact quantum groups and the (complex, real) Hausdorff–Young inequalities. In Sect. 4, we prove the main results and give some examples.

2 Preliminaries

In this section, we collect some necessary preliminaries of locally compact quantum groups, noncommutative $L_p$-spaces and noncommutative Lorentz spaces.

2.1 Noncommutative $L_p$-Spaces and Lorentz Spaces Associated with a Semifinite von Neumann Algebra

We concentrate ourselves on noncommutative $L_p$-spaces associated with semifinite von Neumann algebras, which were first laid out in the early 50’s by Segal [20] and Dixmier [9]. The noncommutative Lorentz spaces will be treated at the same time. We refer to [18] for more discussions.

Let $\mathcal {M}$ be a semifinite von Neumann algebra equipped with a normal semifinite faithful (n.s.f.) trace $\tau$. Denote by $\mathcal {M}^+$ the positive cone of $\mathcal {M}$. Let $\mathcal {S}^+$ denote the set of all $x\in \mathcal {M}^+$ such that $\tau (\text {supp}(x))<\infty$, where $\text {supp}(x)$ denotes the support of x. Let $\mathcal {S}$ be the linear span of $\mathcal {S}^+$. Then $\mathcal {S}$ is a weak*-dense *-subalgebra of $\mathcal {M}$. Given $0<p<\infty$, we define

$$\begin{aligned} \Vert x\Vert _p:=[\tau (|x|^p)]^{\frac{1}{p}},~~x\in \mathcal {S}, \end{aligned}$$

where $|x|=(x^*x)^{\frac{1}{2}}$ is the modulus of x. Then $(\mathcal {S},\Vert \cdot \Vert _p)$ is a normed (or quasi-normed for $p<1$) space. Its completion is called noncommutative $L_p$-space associated with $(\mathcal {M},\tau )$, denoted by $L_p(\mathcal {M},\tau )$ or simply by $L_p(\mathcal {M})$. As usual, we set $L_\infty (\mathcal {M},\tau )=\mathcal {M}$ equipped with the operator norm.

For $1\le p<\infty$, the dual space of $L_p(\mathcal {M})$ is $L_{p'}(\mathcal {M})$ with respect to the duality

$$\begin{aligned} \langle x,y\rangle :=\tau (xy),~~,x\in L_p(\mathcal {M}),y\in L_{p'}(\mathcal {M}). \end{aligned}$$

In particular, $L_1(\mathcal {M})$ is identified with $\mathcal {M}_*$ via the map $j(x):=\tau (x\cdot ),x\in L_1(\mathcal {M})$.

The elements in $L_p(\mathcal {M})$ can be viewed as closed densely defined operators on H (H being the Hilbert space on which $\mathcal {M}$ acts). A linear closed operator x is said to be affiliated with $\mathcal {M}$ if it commutes with all unitary elements in $\mathcal {M}'$, i.e. $xu=ux$ for any unitary $u\in \mathcal {M}'$. Note that x can be unbounded on H. An operator x affiliated with $\mathcal {M}$ is said to be measurable with respect to $(\mathcal {M},\tau )$, or simply measurable if for any $\varepsilon >0$, there exists a projection $e\in \mathcal {M}$ such that

$$\begin{aligned} e(H)\subset \mathcal {D}(x) \text { and } \tau (e^\perp )\le \varepsilon , \end{aligned}$$

where $e^\perp =1-e$. We denote by $L_0(\mathcal {M},\tau )$, or simply $L_0(\mathcal {M})$ the family of measurable operators. For $x\in L_0(\mathcal {M},\tau )$, we define the distribution function of x

$$\begin{aligned} \lambda _s(x):=\tau (\chi _{(s,\infty )}(|x|)),~~s\ge 0, \end{aligned}$$

where $\chi _{(s,\infty )}(|x|)$ is the spectral projection of |x| corresponding to the interval $(s,\infty )$, and define the generalized singular numbers of x

$$\begin{aligned} \mu _t(x):=\inf \{s>0:\lambda _s(x)< t\},~~t\ge 0. \end{aligned}$$

Similar to the classical case, for $0<p<\infty , 0<q\le \infty$, the noncommutative Lorentz space $L_{p,q}(\mathcal {M})$ is defined as the collection of all measurable operators x such that

$$\begin{aligned} \Vert x\Vert _{p,q}:=\left( \int _{0}^{\infty }(t^{\frac{1}{p}}\mu _t(x))^q\frac{dt}{t}\right) ^{\frac{1}{q}}<\infty . \end{aligned}$$

Clearly, $L_{p,p}(\mathcal {M})=L_p(\mathcal {M})$ with $\Vert \cdot \Vert _{p,p}=\Vert \cdot \Vert _{p}$. The space $L_{p,\infty }(\mathcal {M})$ is usually called the weak $L_p$-space, $0<p<\infty$, and one defines

$$\begin{aligned} \Vert x\Vert _{p,\infty }:=\sup _{t>0}t^{\frac{1}{p}}\mu _t(x). \end{aligned}$$

Like the classical $L_p$-spaces, noncommutative $L_p$-spaces behave well with respect to interpolation. Our reference for interpolation theory is [4]. Let $1\le p_0\le p_1\le \infty$, $1\le q\le \infty$ and $0<\theta <1$. Suppose

$$\begin{aligned} \frac{1}{p}=\frac{1-\theta }{p_0}+\frac{\theta }{p_1}. \end{aligned}$$

Then it is well-known that [18, Sect. 2]

$$\begin{aligned} (L_{p_0}(\mathcal {M}),L_{p_1}(\mathcal {M}))_\theta =L_p(\mathcal {M}) \text { (with equal norms) } \end{aligned}$$

and

$$\begin{aligned} (L_{p_0}(\mathcal {M}),L_{p_1}(\mathcal {M}))_{\theta ,q}=L_{p,q}(\mathcal {M})\text { (with equivalent quasi-norms)}, \end{aligned}$$

(2.1)

where $(\cdot ,\cdot )_\theta$ and $(\cdot ,\cdot )_{\theta ,q}$ denote respectively the complex and real interpolation methods.

We formulate here some properties that we will use in this paper. For the proofs we refer to [11] and [12].

Lemma 2.1

Let $\mathcal {M}$ be a von Neumann algebra equipped with an n.s.f. trace $\tau$. We have

• (1) $\mu _{s+t}(xy)\le \mu _s(x)\mu _t(y)$ for all $s,t\ge 0$ and $x,y\in L_0(\mathcal {M})$;

• (2) for any $1< p,q<\infty$ and $q<r\le \infty$,

  $$\begin{aligned} \Vert x\Vert _{p,r}\precsim _{p,q,r}\Vert x\Vert _{p,q},~~x\in L_{p,q}(\mathcal {M}), \end{aligned}$$

  (2.2)

  where the constant is $c_{p,q,r}=(q/p)^{\frac{1}{q}-\frac{1}{r}}$.

Hölder type inequalities hold on noncommutative Lorentz space. We only present here a special case that is enough for our use. We give a proof here for reader’s convenience.

Lemma 2.2

Let $0<p_0,p_1<\infty$, $0<q<\infty$, and $1/p=1/p_0+1/p_1$. Let $\mathcal {M}$ be a von Neumann algebra equipped with an n.s.f. trace $\tau$. Then we have

$$\begin{aligned} \Vert xy\Vert _{p,q}\precsim _{p}\Vert x\Vert _{p_0,\infty }\Vert y\Vert _{p_1,q},~~x\in L_{p_0,\infty }(\mathcal {M}),y\in L_{p_1,q}(\mathcal {M}), \end{aligned}$$

(2.3)

where the constant is $c_p=2^{\frac{1}{p}}$.

Proof

From Lemma 2.1(1) and the definition of $\Vert \cdot \Vert _{p_0,\infty }$, it follows that

$$\begin{aligned} \Vert xy\Vert _{p,q}&=\left( \int _{0}^{\infty }\left( t^{\frac{1}{p}}\mu _t(xy)\right) ^q\frac{dt}{t}\right) ^{\frac{1}{q}}\\&=2^{\frac{1}{p}}\left( \int _{0}^{\infty }\left( t^{\frac{1}{p}}\mu _{2t}(xy)\right) ^q\frac{dt}{t}\right) ^{\frac{1}{q}}\\&\le 2^{\frac{1}{p}}\left( \int _{0}^{\infty }\left( t^{\frac{1}{p_0}}\mu _t(x)\cdot t^{\frac{1}{p_1}}\mu _t(y)\right) ^q\frac{dt}{t}\right) ^{\frac{1}{q}}\\&\le 2^{\frac{1}{p}}\Vert x\Vert _{p_0,\infty }\Vert y\Vert _{p_1,q}. \end{aligned}$$

$\square$

2.2 Locally Compact Quantum Groups

In this subsection we recall the definition of locally compact quantum groups in the sense of Kustermans and Vaes [16, 17]. See also the notes [6]. We shall mainly work with the von Neumann algebraic version. For any n.s.f. weight $\varphi$ on a von Neumann algebra $\mathcal {M}$, we set

$$\begin{aligned} \mathfrak {n}_{\varphi }:=\{x\in \mathcal {M}:\varphi (x^*x)<\infty \},~~\mathfrak {m}_{\varphi }:=\mathfrak {n}_{\varphi }^{*}\mathfrak {n}_{\varphi }. \end{aligned}$$

A locally compact quantum group $\mathbb {G}=(\mathcal {M},\Delta ,\varphi ,\psi )$ consists of

• (1) a von Neumann algebra $\mathcal {M}$;

• (2) a normal, unital, $*$-homomorphism $\Delta :\mathcal {M}\rightarrow \mathcal {M}\overline{\otimes }\mathcal {M}$ such that

  $$\begin{aligned} (\Delta \otimes \text {id})\Delta =(\text {id}\otimes \Delta )\Delta ; \end{aligned}$$

• (3) an n.s.f. weight $\varphi$ which is left invariant

  $$\begin{aligned} \varphi [(\omega \otimes \text {id})\Delta (x)]=\varphi (x)\omega (1),~~\omega \in \mathcal {M}_*^+, x\in \mathfrak {m}^+_\varphi ; \end{aligned}$$

• (4) an n.s.f. weight $\psi$ which is right invariant

  $$\begin{aligned} \psi [(\text {id}\otimes \omega )\Delta (x)]=\psi (x)\omega (1),~~\omega \in \mathcal {M}_*^+,x\in \mathfrak {m}^+_{\psi }; \end{aligned}$$

where $\overline{\otimes }$ denotes the von Neumann algebra tensor product and $\text {id}$ denotes the identity map. The normal, unital, $*$-homomorphism $\Delta$ is called comultiplication on $\mathcal {M}$, $\varphi$ is called left Haar weight and $\psi$ is called right Haar weight.

Example 2.3

Let G be a locally compact group. Then $(L_\infty (G,\mu ),\Delta ,\mu ,\nu )$ is a locally compact quantum group, where $\Delta :L_\infty (G,\mu )\rightarrow L_\infty (G,\mu )\overline{\otimes }L_\infty (G,\mu )\simeq L_\infty (G\times G,\mu \times \mu )$ is given by $\Delta (f)(s,t)=f(st),s,t\in G$, and $\mu ,\nu$ are the left and right Haar measures on G, respectively.

Given a locally compact quantum group $\mathbb {G}=(\mathcal {M},\Delta ,\varphi ,\psi )$, we now define its dual $\widehat{\mathbb {G}}=(\widehat{\mathcal {M}},\widehat{\Delta },\widehat{\varphi },\widehat{\psi })$ that is also a locally compact quantum group. For this, we equip $\mathfrak {n}_{\varphi }$ with the inner product

$$\begin{aligned} \langle x,y\rangle =\varphi (y^*x), \end{aligned}$$

and denote by $H_{\varphi }$ the induced Hilbert space after completion. For any $x\in \mathfrak {n}_{\varphi }\subset \mathcal {M}$ we write $\Lambda _{\varphi }(x)$ for the corresponding element in $H_{\varphi }.$ For any $x\in \mathcal {M}$, $\pi _{\varphi }(x)$ denotes the bounded operator over $H_{\varphi }$ such that $\pi _{\varphi }(x)\Lambda _{\varphi }(y)=\Lambda _{\varphi }(xy)$. So $(H_{\varphi },\pi _{\varphi },\Lambda _{\varphi })$ is the GNS representation of $\varphi$. We omit the subscript $\varphi$ in the sequel whenever there is no ambiguity. Assume that $\mathcal {M}$ acts on $H_\varphi$ with its predual $\mathcal {M}_*$. The multiplicative unitary of $\mathbb {G}$ is the unitary operator W on $H_\varphi \otimes H_\varphi$ such that

$$\begin{aligned} W^*(\Lambda (x)\otimes \Lambda (y))=(\Lambda \otimes \Lambda )(\Delta (y)(x\otimes 1)),~~x,y\in \mathfrak {n}_\varphi . \end{aligned}$$

It implements the comultiplication:

$$\begin{aligned} \Delta (x)=W^*(1\otimes x)W,~~x\in \mathcal {M}. \end{aligned}$$

For any $\omega \in \mathcal {M}_*$, define

$$\begin{aligned} \lambda (\omega ):=(\omega \otimes \text {id})W. \end{aligned}$$

(2.4)

Then the underlying von Neumann algebra of $\widehat{\mathbb {G}}$ is defined as $\widehat{\mathcal {M}}:=\lambda (\mathcal {M}_*)''\subset B(H_\varphi )$. The comultiplication of $\widehat{\mathbb {G}}$ is given by

$$\begin{aligned} \widehat{\Delta }(x)=\widehat{W}(1\otimes x)\widehat{W}^*,~~x\in \widehat{\mathcal {M}}, \end{aligned}$$

where $\widehat{W}=\Sigma W^*\Sigma$ is the multiplicative unitary on $\widehat{\mathbb {G}}$ with $\Sigma$ being the flip on $H_\varphi \otimes H_\varphi$, i.e. $\Sigma (\xi \otimes \eta )=\eta \otimes \xi$.

To define the dual left Haar weights, set

$$\begin{aligned} \mathcal {I}:=\left\{ \omega \in \mathcal {M}_*:\exists C>0 \text { such that } |\omega (x^*)|\le C\Vert \Lambda (x)\Vert ,~x\in \mathfrak {n}_\varphi \right\} . \end{aligned}$$

By the Riesz representation theorem, there exists unique $\xi (\omega )\in H_\varphi$ such that

$$\begin{aligned} \omega (x^*)=\langle \xi (\omega ),\Lambda (x)\rangle ,~~x\in \mathfrak {n}_\varphi . \end{aligned}$$

Then the dual left Haar weight $\widehat{\varphi }$ is defined to be the unique n.s.f. weight on $\widehat{\mathcal {M}}$ with the GNS representation $(H,\iota ,\widehat{\Lambda })$ such that $\lambda (\mathcal {I})$ is a $\sigma$-strong*-norm core for $\widehat{\Lambda }$ and $\widehat{\Lambda }(\lambda (\omega ))=\xi (\omega )$ for all $\omega \in \mathcal {I}$. Thus we have

$$\begin{aligned} \widehat{\varphi }(\lambda (\omega )^*\lambda (\omega )) =\langle \widehat{\Lambda }(\lambda (\omega )),\widehat{\Lambda }(\lambda (\omega ))\rangle ,~~\omega \in \mathcal {I}. \end{aligned}$$

(2.5)

The dual right Haar weight $\widehat{\psi }$ can be defined in a similar way, which we will not do here. Then $\widehat{\mathbb {G}}=(\widehat{\mathcal {M}},\widehat{\Delta },\widehat{\varphi },\widehat{\psi })$ forms a locally compact quantum group. Constructing the dual $\widehat{\widehat{\mathbb {G}}}$ of $\widehat{\mathbb {G}}$, the Pontryagin duality says $\widehat{\widehat{\mathbb {G}}}=\mathbb {G}$. Furthermore, we have $\widehat{\widehat{\Lambda }}=\Lambda$.

In this paper we are interested in locally compact quantum groups $\mathbb {G}$ on which both left Haar weight $\varphi$ and dual left Haar weight $\widehat{\varphi }$ are tracial. We close this subsection with some examples of locally compact quantum groups of this type.

Example 2.4

(Unimodular Kac algebras) We refer to [10] for more about Kac algebras. Here we only remark that unimodular Kac algebras are locally compact quantum groups $\mathbb {G}=(\mathcal {M},\Delta ,\varphi ,\psi )$ for which $\varphi =\psi$ is tracial. If a Kac algebra $\mathbb {G}$ is unimodular, then so is its dual $\widehat{\mathbb {G}}$ [10, Proposition 6.1.4]. We give more concrete examples in the following.

Example 2.5

(Locally compact unimodular groups) Let G be a locally compact unimodular group with $\mu$ being left (also right) Haar measure. Then $\mathbb {G}=(L_{\infty }(G,\mu ),\Delta ,\mu ,\mu )$ is a locally compact quantum group, as we have seen in Example 2.3. Clearly, its left Haar weight is tracial. According to [15], on its dual quantum group $\widehat{\mathbb {G}}=(\widehat{G},\widehat{\Delta },\widehat{\mu },\widehat{\mu })$, the left (right) dual Haar weight is tracial. Here $\widehat{G}$ is the von Neumann algebra acting on $L_2(G,\mu )$ generated by all $\lambda (f),f\in L_1(G,\mu )$, where $\lambda (f)$ is the convolution operator: $\lambda (f)g=f*g,g\in L_2(G,\mu )$. The multiplicative unitary W acts on $L_2(G,\mu )\otimes L_2(G,\mu )\simeq L_2(G\times G,\mu \times \mu )$ as

$$\begin{aligned} WF(s,t)=F(s,s^{-1}t). \end{aligned}$$

Example 2.6

(Compact quantum groups of Kac type) A compact quantum group is a locally compact quantum group $\mathbb {G}$ such that the left Haar weight is finite, i.e. $\varphi (\varvec{1})<\infty$. This agrees with Woronowicz’s definition of compact quantum groups [25], which we shall now recall. A compact quantum group consists of a pair $\mathbb {G}=(A,\Delta )$, where A is a unital C*-algebra and $\Delta$ is a unital $*$-homomorphism from A to $A\otimes A$ such that

• (1) $(\Delta \otimes \text {id})\Delta =(\text {id}\otimes \Delta )\Delta$;

• (2) $\{\Delta (a)(1\otimes b):a,b\in A\}$ and $\{\Delta (a)(b\otimes 1):a,b\in A\}$ are linearly dense in $A\otimes A$.

Here $A\otimes A$ is the minimal C*-algebra tensor product. Any compact quantum group admits a unique Haar state, i.e. a state h on A that is both left and right invariant:

$$\begin{aligned} (h\otimes \text {id})\Delta (a)=h(a)\varvec{1}=(\text {id}\otimes h)\Delta (a),~~a\in A. \end{aligned}$$

Consider an element $u\in A\otimes B(H)$ with $\dim H=n$. By identifying $A\otimes B(H)$ with $M_n(A)$ we can write $u=[u_{ij}]_{i,j=1}^{n}$, where $u_{ij}\in A$. The matrix u is called an n-dimensional representation of $\mathbb {G}$ if we have

$$\begin{aligned} \Delta (u_{ij})=\sum _{k=1}^{n}u_{ik}\otimes u_{kj},~~i,j=1,\dots ,n. \end{aligned}$$

A representation u is called unitary if u is unitary as an element in $M_n(A)$, and irreducible if the only matrices $T\in M_n(\mathbb {C})$ such that $uT=Tu$ are multiples of identity matrix. Two representations $u,v\in M_n(A)$ are said to be equivalent if there exists an invertible matrix $T\in M_n(\mathbb {C})$ such that $Tu=vT$. Denote by $\textrm{Irr}(\mathbb {G})$ the set of equivalence classes of irreducible unitary representations of $\mathbb {G}$. For each $\pi \in \textrm{Irr}(\mathbb {G})$, denote by $u^\pi \in A\otimes B(H_\pi )$ a representative of the class $\pi$, where $H_\pi$ is the finite-dimensional Hilbert space on which $u^\pi$ acts. In the sequel we write $n_\pi =\dim H_\pi$. Denote $\textrm{Pol}(\mathbb {G})=\text {span} \left\{ u^\pi _{ij}:1\le i,j\le n_\pi ,\pi \in \textrm{Irr}(\mathbb {G})\right\}$. This is a dense subalgebra of A.

The dual of a compact quantum group $\mathbb {G}$ is a discrete quantum group $\widehat{\mathbb {G}}=(\widehat{A},\widehat{\Delta },\widehat{h}_{\text {L}},\widehat{h}_{\text {R}})$, where $\widehat{A}$ is the $c_0$-direct sum of matrix algebras

$$\begin{aligned} \widehat{A}=\bigoplus _{\pi \in \textrm{Irr}(\mathbb {G})}B(H_{\pi }). \end{aligned}$$

The dual left Haar weight $\widehat{h}_{\text {L}}$ and dual right Haar weight $\widehat{h}_{\text {R}}$ are not the same in general. A compact quantum group $\mathbb {G}$ is of Kac type if the Haar state h is tracial. In this case $\widehat{h}_{\text {L}}$ and $\widehat{h}_{\text {R}}$ coincide, which we denote by $\widehat{h}$ for short. It takes the following form

$$\begin{aligned} \widehat{h}(a)=\sum _{\pi \in \textrm{Irr}(\mathbb {G})}d_{\pi }\text {Tr}(a_{\pi }). \end{aligned}$$

The multiplicative unitary W of $\mathbb {G}$ is

$$\begin{aligned} W:=\bigoplus _{\pi \in \textrm{Irr}(\mathbb {G})} u^{\pi }. \end{aligned}$$

Then the Fourier transform $\mathcal {F}$ over $\textrm{Pol}(\mathbb {G})$ is given by

$$\begin{aligned} \mathcal {F}(x)=(h(\cdot x) \otimes \text {id})W=(\widehat{x}(\pi ))_{\pi \in \textrm{Irr}(\mathbb {G})}, \end{aligned}$$

where $\widehat{x}(\pi )=(h(\cdot x)\otimes \text {id})(u^{\pi })$.

Classical compact groups are certainly compact quantum groups of Kac type (the commutative case). In the next, we give another family of such quantum groups (the cocommutative case). There are also compact quantum groups of Kac type which are neither commutative nor cocommutative, e.g. free orthogonal quantum groups $O_N^+$ [23] and free permutation quantum groups $S_N^+$ [24]. We will not explain here in detail.

Example 2.7

(Discrete group von Neumann algebras) Let G be a discrete group. Then $\mathbb {G}=(\ell _{\infty }(G),\Delta ,\mu ,\mu )$ is a locally compact quantum group with $\mu$ being the counting measure. Suppose that $\{\delta _g\}_{g\in G}$ is the canonical basis of $\ell _2(G)$. Then the left regular representation of G is given through $\lambda :G\rightarrow B(\ell _2(G)),\lambda _g(\delta _h)=\delta _{gh}$. The group von Neumann algebra $\widehat{G}$ is the von Neumann algebra generated by $\lambda (g),g\in G$ in $B(\ell _2(G))$. Thus the dual quantum group of G is $\widehat{\mathbb {G}}=(\widehat{G},\widehat{\Delta },\tau ,\tau )$, where $\tau$ is a normal faithful tracial state defined by $\tau (x)=\langle \delta _e,x\delta _e\rangle$, where e is the unit of G and $\langle \cdot ,\cdot \rangle$ is the inner product on $\ell _2(G)$.

3 Fourier Transform on Locally Compact Quantum Groups

In the remaining part of the paper, unless otherwise stated, for any $1<p<\infty$, $p'$ always denotes the conjugate number of p, i.e. $1/p+1/p'=1$. $\mathbb {G}=(\mathcal {M},\Delta ,\varphi ,\psi )$ always denotes a locally compact quantum group with dual $\widehat{\mathbb {G}}=(\widehat{\mathcal {M}},\widehat{\Delta },\widehat{\varphi },\widehat{\psi })$, where $\varphi$ and $\widehat{\varphi }$ are both tracial. We shall use $L_{p}(\mathbb {G},\varphi )$ and $L_{p,q}(\mathbb {G},\varphi )$ to denote $L_{p}(\mathcal {M},\varphi )$ and $L_{p,q}(\mathcal {M},\varphi )$, respectively. The same goes to $L_{p}(\widehat{\mathbb {G}},\widehat{\varphi })$ and $L_{p,q}(\widehat{\mathbb {G}},\widehat{\varphi })$.

3.1 A Brief History

In this section we briefly recall the history of Fourier transform on locally compact quantum groups and its definition in our setting.

Let G be a locally compact abelian group with Haar measure $\mu$, then the Fourier transform of $f\in L_{1}(G,\mu )$ takes the form:

$$\begin{aligned} \mathcal {F}(f)(\xi )=\widehat{f}(\xi )=\int _{G}f(s)\overline{\xi (s)}d\mu (s),~~\xi \in \widehat{G}. \end{aligned}$$

By choosing the dual Haar measure $\widehat{\mu }$ on $\widehat{G}$ suitably, the map $L_{1}(G,\mu )\cap L_{2}(G,\mu )\ni f\mapsto \widehat{f}\in L_{2}(\widehat{G},\widehat{\mu })$ is isometric and can be extended to an isometry between $L_{2}(G,\mu )$ and $L_{2}(\widehat{G},\widehat{\mu })$. This defines the Fourier transform of $f\in L_{2}(G,\mu )$. The definition of the Fourier transform of $f\in L_{p}(G,\mu )$ follows from the the famous Hausdorff–Young inequality, which states that for any $1\le p\le 2$ we have

$$\begin{aligned} \Vert \widehat{f}\Vert _{L_{p'}(\widehat{G},\widehat{\mu })}\le \Vert f\Vert _{L_{p}(G,\mu )},~~f\in L_{p}(G,\mu ). \end{aligned}$$

(3.1)

It is natural to ask what the Fourier transform looks like for general locally compact groups and whether we still have (3.1) or not. The first breakthrough is due to Kunze [15], who observed the following fact. Let G be a locally compact abelian group as above. Let $\lambda (f)$ denote the left regular representation of $f\in L_{1}(G,\mu )$ on $L_{2}(G,\mu )$, which is an operator given by

$$\begin{aligned} (\lambda (f)g)(s):=f*g(s)=\int _{G}f(t)g(t^{-1}s)d\mu (t),~~g\in L_{2}(G,\mu ). \end{aligned}$$

Denote by $L_{f}$ the operator on $L_{2}(G,\mu )$ given by multiplying f. Since $\mathcal {F}$ turns convolution into multiplication, we have

$$\begin{aligned} \mathcal {F}(\lambda (f)g)=\mathcal {F}(f)\mathcal {F}(g)=L_{\mathcal {F}(f)}\mathcal {F}(g),~~f\in L_{1}(G,\mu ),~g\in L_{2}(G,\mu ). \end{aligned}$$

Recall that $\mathcal {F}$ is unitary on $L_{2}(G,\mu )$, so $\lambda (f)$ is unitarily equivalent to the operator $L_{\mathcal {F}(f)}$. This suggests us to use $\lambda (f)$ as a substitute of $\mathcal {F}(f)$. From this Kunze defined the Fourier transform on locally compact unimodular groups $(G,\mu )$ and generalized Hausdorff–Young inequalities (3.1) to locally compact unimodular groups. The dual of G, still denoted by $\widehat{G}$, is no longer a group, but can be studied via the von Neumann algebra generated by $\lambda (L_{1}(G,\mu ))$ in $B(L_{2}(G,\mu ))$. It turns out that there is a canonical trace $\widehat{\mu }$ on $\widehat{G}$, so $L_{p'}(\widehat{G},\widehat{\mu })$ is constructed in the sense of Diximier and Segal. Terp [22] extended this approach to locally compact non-unimodular groups G. Her Fourier transform for $f\in L_{p}(G,\mu )$ is the operator on $L_{2}(G,\mu )$ given by $\lambda (f)\Delta ^{\frac{1}{p'}}$, where $\mu$ is the left Haar measure and $\Delta$ is the modular function on G. Remark that $\Delta$ here is understood as a multiplication operator by $\Delta$. The dual $\widehat{G}$ of G is not necessarily equipped with a trace. In this context we also have Hausdorff–Young inequalities, where $L_{p'}(\widehat{G})$ is the noncommutative $L_p$-space in the sense of Hilsum [13] and Connes [7]. Finally the Hausdorff–Young inequalities were extended to locally compact quantum groups by Cooney [8] and Caspers [5].

In this paper we are concerned with the locally compact quantum group case, but the associated left Haar weight and dual left Haar weight are both tracial. This makes the definition of $L_p$-Fourier transform much simpler than those of Cooney and Caspers. Indeed, we can embed our noncommutative $L_p$-space $L_p(\mathbb {G},\varphi )$ ($1<p<2$) into $L_1(\mathbb {G},\varphi )+L_2(\mathbb {G},\varphi )$ in a natural way. So we will not recall their approaches here.

3.2 Fourier Transform and Hausdorff–Young Inequalities

This subsection does not contain any new results. See for example [5]. We collect the proofs here for reader’s convenience.

Proposition 3.1

We have $L_1(\mathbb {G},\varphi )\cap L_2(\mathbb {G},\varphi )=\mathcal {I}$.

This holds for general locally compact quantum groups and should be understood under suitable embedding of $\mathcal {I}$, $L_1(\mathbb {G},\varphi )$ and $L_2(\mathbb {G},\varphi )$ into some Banach space [5, Theorem 3.3]. We give a proof here when $\varphi$ is tracial, which is the case this paper concerns with. In such case, $\mathcal {I}$ should be understood as $j^{-1}(\mathcal {I})$, where $j:L_1(\mathbb {G},\varphi )\rightarrow \mathcal {M}_*, x\mapsto \varphi (x\cdot )$ is the isometry map.

Proof of Proposition 3.1 when $\varphi$ is tracial

By definition,

$$\begin{aligned} j^{-1}(\mathcal {I})=\{y\in L_1(\mathbb {G},\varphi ):\exists C<\infty \text { such that }|\varphi (x^*y)|\le C\Vert x\Vert _{L_2(\mathbb {G},\varphi )}, x\in \mathfrak {n}_{\varphi } \}. \end{aligned}$$

Note that $\mathfrak {n}_{\varphi }$ is dense in $L_2(\mathbb {G},\varphi )$, by duality of $L_p$-spaces, we have

$$\begin{aligned} j^{-1}(\mathcal {I})&=\{y\in L_1(\mathbb {G},\varphi ):\exists C<\infty \text { such that }|\varphi (x^*y)|\\&\le C\Vert x\Vert _{L_2(\mathbb {G},\varphi )}, x\in L_2(\mathbb {G},\varphi ) \}\\&=\{y\in L_1(\mathbb {G},\varphi ):\Vert y\Vert _{L_2(\mathbb {G},\varphi )}<\infty \}\\&=L_1(\mathbb {G},\varphi )\cap L_2(\mathbb {G},\varphi ). \end{aligned}$$

$\square$

Recall that $L_1(\mathbb {G},\varphi )$ is identified with $\mathcal {M}_*$ via the map $j(x)= \varphi _x$, where $\varphi _x:=\varphi (x\cdot ).$ Since W is unitary, from (2.4) we have

$$\begin{aligned} \Vert \lambda (\varphi _x)\Vert _{L_\infty (\widehat{\mathbb {G}},\widehat{\varphi })} \le \Vert \varphi _x\Vert _{\mathcal {M}_*}=\Vert x\Vert _{L_1(\mathbb {G},\varphi )},~~x\in L_1(\mathbb {G},\varphi ). \end{aligned}$$

We define the $L_1$-Fourier transform as $\mathcal {F}_1:=\lambda \circ j:L_1(\mathbb {G},\varphi )\rightarrow L_\infty (\widehat{\mathbb {G}},\widehat{\varphi }),x\mapsto \lambda (\varphi _x)$, then it is a contraction:

$$\begin{aligned} \Vert \mathcal {F}_1(x)\Vert _{L_\infty (\widehat{\mathbb {G}},\widehat{\varphi })} \le \Vert x\Vert _{L_1(\mathbb {G},\varphi )},~~x\in L_1(\mathbb {G},\varphi ). \end{aligned}$$

For the $L_2$-Fourier transform, we firstly define it as $\mathcal {F}_1$ on the intersection of $L_1(\mathbb {G},\varphi )$ and $\mathfrak {n}_{\varphi }$. By Proposition 3.1, for any $x\in L_1(\mathbb {G},\varphi )\cap \mathfrak {n}_{\varphi }$, $\varphi _x$ belongs to $\mathcal {I}$. Note that by definition of $\widehat{\Lambda }$, we have

$$\begin{aligned} \langle \Lambda (x),\Lambda (y)\rangle =\varphi (y^*x) =\varphi _x(y^*) =\langle \widehat{\Lambda }(\lambda (\varphi _x)),\Lambda (y) \rangle ,~~y\in \mathfrak {n}_{\varphi }. \end{aligned}$$

Since $\mathfrak {n}_{\varphi }$ is dense in $L_2(\mathbb {G},\varphi )$, we have

$$\begin{aligned} \widehat{\Lambda }(\lambda (\varphi _x))=\Lambda (x),~~x\in L_1(\mathbb {G},\varphi )\cap \mathfrak {n}_{\varphi }. \end{aligned}$$

(3.2)

From (2.5) it follows that

$$\begin{aligned} \widehat{\varphi }(\lambda (\varphi _x)^*\lambda (\varphi _x)) =\langle \widehat{\Lambda }(\lambda (\varphi _x)),\widehat{\Lambda }(\lambda (\varphi _x))\rangle =\varphi (x^*x),~~x\in L_1(\mathbb {G},\varphi )\cap \mathfrak {n}_{\varphi }. \end{aligned}$$

So we have

$$\begin{aligned} \Vert \mathcal {F}_2(x)\Vert _{L_2(\widehat{\mathbb {G}},\widehat{\varphi })}=\Vert x\Vert _{L_2(\mathbb {G},\varphi )},~~x\in L_1(\mathbb {G},\varphi )\cap \mathfrak {n}_{\varphi }. \end{aligned}$$

Since $L_1(\mathbb {G},\varphi )\cap \mathfrak {n}_{\varphi }$ is dense in $L_2(\mathbb {G},\varphi )$, $\mathcal {F}_2$ can be extended to an isometry from $L_2(\mathbb {G},\varphi )$ to $L_2(\widehat{\mathbb {G}},\widehat{\varphi })$, which we still denote by $\mathcal {F}_2$.

Now we may define an operator $\mathcal {F}$ on $L_1(\mathbb {G},\varphi )+ L_2(\mathbb {G},\varphi )$ as $\mathcal {F}(x)=\mathcal {F}_1(x_1)+\mathcal {F}_2(x_2)$, where $x=x_1+x_2$ with $x_i\in L_i(\mathbb {G},\varphi ),i=1,2$. One can check that it is well-defined and $\mathcal {F}|_{L_i(\mathbb {G},\varphi )}=\mathcal {F}_i,i=1,2$. Thus the general $L_p$-Fourier transform $\mathcal {F}_p,1<p<2$, is defined to be the restriction of $\mathcal {F}$ to $L_p(\mathbb {G},\varphi )\subset L_1(\mathbb {G},\varphi )+ L_2(\mathbb {G},\varphi )$. By complex interpolation, we have the Hausdorff–Young inequality:

$$\begin{aligned} \Vert \mathcal {F}_p(x)\Vert _{L_{p'}(\widehat{\mathbb {G}},\widehat{\varphi })}\le \Vert x\Vert _{L_p(\mathbb {G},\varphi )},~~x\in L_p(\mathbb {G},\varphi ), \end{aligned}$$

(3.3)

where $1\le p\le 2$ and $1/p+1/p'=1$. If we use real interpolation instead of complex interpolation, we get

$$\begin{aligned} \Vert \mathcal {F}(x)\Vert _{L_{p'}(\widehat{\mathbb {G}},\widehat{\varphi })} \precsim _{p} \Vert x\Vert _{L_{p,p'}(\mathbb {G},\varphi )},~~x\in L_{p,p'}(\mathbb {G},\varphi ). \end{aligned}$$

(3.4)

Compared with (3.3), the constant $c_p$ in (3.4) is worse, but the space $L_{p,p'}(\mathbb {G},\varphi )$ is larger than $L_{p}(\mathbb {G},\varphi )$ when $1\le p<2$.

Definition 3.2

For any $x\in L_0(\mathbb {G},\varphi )$, we call $m_x$ an $L_p$-$L_q$ Fourier multiplier if the map $\mathcal {F}(y)\mapsto \mathcal {F}(xy)$ is well-defined and extends to bounded map from $L_p(\widehat{\mathbb {G}},\widehat{\varphi })$ to $L_q(\widehat{\mathbb {G}},\widehat{\varphi })$. One may also consider the map $\mathcal {F}(y)\mapsto \mathcal {F}(yx)$ that is similar.

3.3 The Dual/Inverse Fourier Transform

On the dual quantum group one can also define the Fourier transform $\widehat{\mathcal {F}}:L_1(\widehat{\mathbb {G}},\widehat{\varphi })+L_2(\widehat{\mathbb {G}},\widehat{\varphi })\rightarrow L_\infty (\mathbb {G},\varphi )+L_2(\mathbb {G},\varphi )$, whose restriction to $L_1(\widehat{\mathbb {G}},\widehat{\varphi })$ is $\widehat{\lambda }\circ \widehat{j}^{-1}$, where $\widehat{j}:L_1(\widehat{\mathbb {G}},\widehat{\varphi })\rightarrow \widehat{\mathcal {M}}_*,x\mapsto \widehat{\varphi }(\cdot x)$. Then $\widehat{\mathcal {F}}_2$ is the inverse of $\mathcal {F}_2$.

Proposition 3.3

Let $\mathbb {G}=(\mathcal {M},\Delta ,\varphi ,\psi )$ be a locally compact quantum group with dual $\widehat{\mathbb {G}}=(\widehat{\mathcal {M}},\widehat{\Delta },\widehat{\varphi },\widehat{\psi })$. Then we have

• (1) $\widehat{\mathcal {F}}(\mathcal {F}(x))=x,~~x\in L_2(\mathbb {G},\varphi )$;

• (2) $\mathcal {F}(\widehat{\mathcal {F}}(a))=a,~~a\in L_2(\widehat{\mathbb {G}},\widehat{\varphi })$.

Proof

Note that the inclusion map $\Lambda :\mathfrak {n}_{\varphi }\rightarrow H_{\varphi }=L_2(\mathbb {G},\varphi )$ can be extended to the whole Hilbert space $L_2(\mathbb {G},\varphi )$. We shall still use $\Lambda$ to denote its extension. The same goes to $\widehat{\Lambda }$. Recall that

$$\begin{aligned} \langle \Lambda (x),\Lambda (y)\rangle =\langle \widehat{\Lambda }(\mathcal {F}(x)),\Lambda (y) \rangle ,~~x\in L_1(\mathbb {G},\varphi )\cap \mathfrak {n}_{\varphi },y\in \mathfrak {n}_{\varphi }. \end{aligned}$$

Since $L_1(\mathbb {G},\varphi )\cap \mathfrak {n}_{\varphi }$ is dense in $L_2(\mathbb {G},\varphi )$, for any $x\in L_2(\mathbb {G},\varphi )$ we may choose a net $\{x_{\alpha }\}_{\alpha }\subset L_1(\mathbb {G},\varphi )\cap \mathfrak {n}_{\varphi }$ such that $\lim \limits _{\alpha }x_{\alpha }=x$ in $\mathfrak {n}_{\varphi }$. $\mathcal {F}|_{\mathfrak {n}_{\varphi }}$ is an isometry, so we obtain

$$\begin{aligned}{} & {} \langle \Lambda (x),\Lambda (y)\rangle =\lim \limits _{\alpha }\langle \Lambda (x_{\alpha }),\Lambda (y)\rangle \\{} & {} \quad =\lim \limits _{\alpha }\langle \widehat{\Lambda }(\mathcal {F}(x_{\alpha })),\Lambda (y) \rangle =\langle \widehat{\Lambda }(\mathcal {F}(x)),\Lambda (y) \rangle ,~~y\in \mathfrak {n}_{\varphi }. \end{aligned}$$

Hence $\widehat{\Lambda }(\mathcal {F}(x))=\Lambda (x), x\in L_2(\mathbb {G},\varphi ).$ Since $\mathcal {F}(x)\in L_2(\widehat{\mathbb {G}},\widehat{\varphi })$ for all $x\in L_2(\mathbb {G},\varphi )$, we have

$$\begin{aligned} \Lambda (\widehat{\mathcal {F}}(\mathcal {F}(x)))=\widehat{\widehat{\Lambda }}(\widehat{\mathcal {F}}(\mathcal {F}(x)))=\widehat{\Lambda }(\mathcal {F}(x))=\Lambda (x),~~x\in L_2(\mathbb {G},\varphi ). \end{aligned}$$

Therefore, $\widehat{\mathcal {F}}(\mathcal {F}(x))=x$. This proves (1). The proof of (2) is similar. $\square$

Since $\widehat{\mathcal {F}}$ is the Fourier transform on $\widehat{\mathbb {G}}$, we have

$$\begin{aligned} \Vert \widehat{\mathcal {F}}(a)\Vert _{L_{\infty }(\mathbb {G},\varphi )} \le \Vert a\Vert _{L_1(\widehat{\mathbb {G}},\widehat{\varphi })},~~a\in L_1(\widehat{\mathbb {G}},\widehat{\varphi }), \end{aligned}$$

This, together with Proposition 3.3, yields

$$\begin{aligned} \Vert x\Vert _{L_{\infty }(\mathbb {G},\varphi )} \le \Vert \mathcal {F}(x)\Vert _{L_1(\widehat{\mathbb {G}},\widehat{\varphi })}, \end{aligned}$$

(3.5)

for all $x\in L_2(\mathbb {G},\varphi )$ such that $\mathcal {F}(x)\in L_1(\widehat{\mathbb {G}},\widehat{\varphi })$, or equivalently, for all x such that $\mathcal {F}(x)\in L_1(\widehat{\mathbb {G}},\widehat{\varphi })\cap L_2(\widehat{\mathbb {G}},\widehat{\varphi })$. Since $L_1(\widehat{\mathbb {G}},\widehat{\varphi })\cap L_2(\widehat{\mathbb {G}},\widehat{\varphi })$ is dense in $L_1(\widehat{\mathbb {G}},\widehat{\varphi })$, the map $\mathcal {F}(x)\mapsto x$ can be extended to a contraction from $L_1(\widehat{\mathbb {G}},\widehat{\varphi })$ to $L_{\infty }(\mathbb {G},\varphi )$. Recall that

$$\begin{aligned} \Vert x\Vert _{L_2(\mathbb {G},\varphi )}=\Vert \mathcal {F}(x)\Vert _{L_2(\widehat{\mathbb {G}},\widehat{\varphi })},~~x\in L_2(\mathbb {G},\varphi ). \end{aligned}$$

(3.6)

Combining (3.5), (3.6), and applying real interpolation, we get

$$\begin{aligned} \Vert x\Vert _{L_{p',p}(\mathbb {G},\varphi )} \precsim _{p} \Vert \mathcal {F}(x)\Vert _{L_p(\widehat{\mathbb {G}},\widehat{\varphi })}, \end{aligned}$$

(3.7)

for all x such that $\mathcal {F}(x)\in L_p(\widehat{\mathbb {G}},\widehat{\varphi })$.

4 The Proofs and Examples

4.1 Fourier Multipliers

This subsection is devoted to the proofs of our results for Fourier multipliers. Some examples will also be presented. In the following we shall simply use $\Vert \cdot \Vert _{p,q}$ to denote $\Vert \cdot \Vert _{L_{p,q}(\mathbb {G},\varphi )}$ or $\Vert \cdot \Vert _{L_{p,q}(\widehat{\mathbb {G}},\widehat{\varphi })}$ whenever no ambiguity can occur.

Proof of Theorem 1.3

Note that $1/q'=1/r+1/p'$. Then for any $x\in L_{r,\infty }(\mathbb {G},\varphi )$ and $y\in L_1(\mathbb {G},\varphi )+L_2(\mathbb {G},\varphi )$ such that $\mathcal {F}(y)\in L_p(\widehat{\mathbb {G}},\widehat{\varphi })$, we have

$$\begin{aligned}{} & {} \Vert \mathcal {F}(xy)\Vert _q {\mathop {\precsim _{q}}\limits ^{(3.4)}} \Vert xy\Vert _{q',q} {\mathop {\precsim _{q}}\limits ^{(2.3)}}\\{} & {} \displaystyle \Vert x\Vert _{r,\infty }\Vert y\Vert _{p',q} {\mathop {\precsim _{p,q}}\limits ^{(2.2)}} \Vert x\Vert _{r,\infty }\Vert y\Vert _{p',p} {\mathop {\precsim _{p,q}}\limits ^{(3.7)}} \Vert x\Vert _{r,\infty }\Vert \mathcal {F}(y)\Vert _{p}. \end{aligned}$$

$\square$

Remark 4.1

From the proof, one can see that the result can be extended to the boundedness of Fourier multipliers between more general Lorentz spaces, which is beyond the aim of this paper. Also, if we use complex interpolation instead of real interpolation, i.e. the usual Hausdorff–Young inequalities, then one may get an upper bound of $\Vert x\Vert _r$ instead of $c_{p,q}\Vert x\Vert _{r,\infty }$. Details are provided for the Schur multipliers. See Remark 4.5.

Proof of Theorem 1.4

Note that $1/p=1/p'+1/s$. For any $a\in L_{s,\infty }(\widehat{\mathbb {G}},\widehat{\varphi })$ and any $x\in L_p(\mathbb {G},\varphi )$, we have

$$\begin{aligned} \Vert a\mathcal {F}(x)\Vert _{p} =\Vert a\mathcal {F}(x)\Vert _{p,p} {\mathop {\precsim _{p}}\limits ^{(2.3)}} \Vert a\Vert _{s,\infty }\Vert \mathcal {F}(x)\Vert _{p',p} {\mathop {\precsim _{p}}\limits ^{(3.4)}} \Vert a\Vert _{s,\infty }\Vert x\Vert _{p}. \end{aligned}$$

$\square$

Proof of Theorem 1.5

This is a direct consequence of Theorem 1.3. Indeed, since h is a state, we have by Hölder’s inequality that $\Vert x\Vert _{p}\le \Vert x\Vert _{q}$ whenever $x\in L_q(\mathbb {G},h)$ and $p\le q$. Thus for any $1<p\le 2\le q<\infty$ we have

$$\begin{aligned} \Vert \widehat{\mathcal {F}}(ab)\Vert _{p} \le \Vert \widehat{\mathcal {F}}(ab)\Vert _{q} \precsim _{p,q} \Vert a\Vert _{r,\infty }\Vert \widehat{\mathcal {F}}(b)\Vert _{p} \precsim _{p,q} \Vert a\Vert _{r,\infty }\Vert \widehat{\mathcal {F}}(b)\Vert _{q}, \end{aligned}$$

(4.1)

for all $a\in \ell _{r,\infty }(\widehat{\mathbb {G}},\widehat{h})$ and $\widehat{\mathcal {F}}(b)\in L_q(\mathbb {G},h)$. The first two inequalities of (4.1) imply that $m_a$ is an $L_p$-Fourier multiplier:

$$\begin{aligned} \Vert m_a:L_p(\mathbb {G},h)\rightarrow L_p(\mathbb {G},h)\Vert \precsim _{p,q} \Vert a\Vert _{r,\infty }, \end{aligned}$$

(4.2)

while the last two inequalities of (4.1) imply that $m_a$ is an $L_q$-Fourier multiplier:

$$\begin{aligned} \Vert m_a:L_q(\mathbb {G},h)\rightarrow L_q(\mathbb {G},h)\Vert \precsim _{p,q} \Vert a\Vert _{r,\infty }. \end{aligned}$$

(4.3)

We may choose $q=2$ in (4.2) and $p=2$ in (4.3). So for any $1<p<\infty$ we have

$$\begin{aligned} \Vert m_a:L_p(\mathbb {G},h)\rightarrow L_p(\mathbb {G},h)\Vert \precsim _{p} \Vert a\Vert _{p^*,\infty }, \end{aligned}$$

with $1/p^*=|1/2-1/p|$.

$\square$

Remark 4.2

The index r in Theorem 1.3 is sharp in general. To see this, take $\mathbb {G}=\mathbb {Z}$ with $\widehat{\mathbb {G}}=\mathbb {T}$. By Theorem 1.3 we have for $1<p\le 2\le q<\infty$ that

$$\begin{aligned} \Vert m_{\phi }:L_p(\mathbb {T})\rightarrow L_q(\mathbb {T})\Vert \precsim _{p,q} \Vert \phi \Vert _{\ell _{r,\infty }(\mathbb {Z})}, \end{aligned}$$

(4.4)

where $1/r=1/p-1/q$. Indeed, by [27, Lemma 6.6, page 129, Vol. II], for any $1<p<\infty$ and Fourier series

$$\begin{aligned} f(x):=\sum _{n=1}^{\infty }a_n \cos (nx)=\frac{1}{2}\sum _{n\in \mathbb {Z}}a_{|n|}e^{inx}, \end{aligned}$$

such that $a_n\downarrow 0$ as $n\rightarrow \infty$, we have

$$\begin{aligned} f\in L_p(\mathbb {T})\text { if and only if } \sum _{n\ge 1}n^{p-2}a^p_n<\infty . \end{aligned}$$

(4.5)

Now suppose that r in (4.4) can be replaced by some $s>r$. Consider $\phi (n):=|n|^{-\frac{1}{s}},n\ne 0$ and $\phi (0):=0$. It is easy to see that $\phi \in \ell _{s,\infty }(\mathbb {Z})\setminus \ell _{r,\infty }(\mathbb {Z})$. Set $\alpha :=1/r-1/s>0$ and $a_n:=n^{\frac{1}{p}-1-\alpha }$. Since

$$\begin{aligned} p-2+p\left( \frac{1}{p}-1-\alpha \right) =-1-p\alpha <-1, \end{aligned}$$

$$\begin{aligned} q-2+q\left( \frac{1}{p}-1-\alpha -\frac{1}{s}\right) =q-2+q\left( \frac{1}{q}-1\right) =-1, \end{aligned}$$

we have

$$\begin{aligned} \sum _{n\ge 1}n^{p-2}a^p_n=\sum _{n\ge 1}n^{-1-p\alpha }<\infty , \end{aligned}$$

while

$$\begin{aligned} \sum _{n\ge 1}n^{q-2}(a_n \phi (n))^q=\sum _{n\ge 1}n^{-1}=\infty . \end{aligned}$$

By (4.5), $f\in L_p(\mathbb {T})$ while $m_{\phi }(f)\notin L_q(\mathbb {T})$, which leads to a contradiction. So r is sharp.

Remark 4.3

The result of Corollary 1.7 may fail in the endpoint case $p=1$. I am very grateful to Éric Ricard for pointing this out to me, and for allowing me to include his proof here. Take $G=\mathbb {Z}$ and $\widehat{G}=\mathbb {T}$. Then there exists $\phi :\mathbb {Z}\rightarrow \mathbb {R}$ such that $\phi \in \ell _{2,\infty }(\mathbb {Z})$ while the Fourier multiplier $m_\phi$ is unbounded on $L_1(\mathbb {T})$. To see this, take

$$\begin{aligned} \phi (n)= {\left\{ \begin{array}{ll} \frac{1}{\sqrt{k}}&{}n=2^k, k\ge 1\\ 0&{}\text {otherwise} \end{array}\right. }. \end{aligned}$$

Clearly $\phi \in \ell _{2,\infty }(\mathbb {Z})\setminus \ell _2(\mathbb {Z})$. Suppose that the Fourier multiplier $m_\phi$ is bounded over $L_1(\mathbb {T})$. Then there exist a measure $\mu$ on $\mathbb {T}$ such that $m_\phi (f)=\mu *f$, with $*$ being the convolution. Since

$$\begin{aligned} \widehat{\mu }(n)=\phi (n)=0,~~n<0, \end{aligned}$$

by F. and M. Riesz theorem [19, Theorem 17.13, page 341], $\mu$ is absolutely continuous with respect to the Lebesgue measure $d\theta$. So $m_\phi$ is a convolution operator, i.e. $m_\phi (f)=h*f$ for some $h\in L_1(\mathbb {T})$ such that $\widehat{h}=\phi$. By construction, $\phi$ is supported on a Lacunary set {$2^k,k\ge 1$}. Hence we have [12, Theorem 3.6.4]

$$\begin{aligned} \Vert \phi \Vert _{\ell _2(\mathbb {Z})}=\Vert h\Vert _{L_2(\mathbb {T})}\le K\Vert h\Vert _{L_1(\mathbb {T})}, \end{aligned}$$

(4.6)

for some constant $K>0$. However, the left hand side is unbounded as $\phi \notin \ell _2(\mathbb {Z})$. This leads to a contradiction. Therefore, Corollary 1.7 fails when $p=1$.

Example 4.4

Let G be a finitely generated group with the unit e and a symmetric set S of generators. By saying symmetric we mean $x^{-1}\in S$ whenever $x\in S$. Then it has an exponential growth, i.e.,

$$\begin{aligned} |\{x\in G:d(x,e)\le n\}|\le M^n,~~n\ge 1, \end{aligned}$$

(4.7)

for some $M>1$, where d is the word metric on G with respect to S and $|\cdot |$ denotes the counting measure on G. Indeed, one can always choose M to be |S|. Then for any $\phi :G\rightarrow \mathbb {C}$ such that $|\phi (g)|\le C M^{-\frac{|g|}{p^*}}$, where $|g|:=d(g,e)$ and $C>0$ is a constant. So

$$\begin{aligned} |\phi (g)|\ge \alpha \text { implies }|g|\le -p^*\log _{M} \frac{\alpha }{C},~~\alpha >0. \end{aligned}$$

Therefore,

$$\begin{aligned} \alpha ^{p^*}|\{g\in G:|\phi (g)|\ge \alpha \}| \le \alpha ^{p^*} M^{-p^*\log _{M} \frac{\alpha }{C}} \le C^{p^*}<\infty ,~~\alpha >0, \end{aligned}$$

and we have $\phi \in \ell _{p^*,\infty }(G)$, whence $m_{\phi }$ is an $L_p$-Fourier multiplier on $L_p(\widehat{G},\tau )$. For free group on N generators $\mathbb {F}_N$, we may choose S as the set consisting of N generators with their inverses and let $M=2N$.

If moreover, G is of polynomial growth, i.e. the right hand side of (4.7) can be replaced by some polynomial p(n), or equivalently, $n^k$ for some $k>0$, then a similar argument yields that for any $\phi :G\rightarrow \mathbb {C}$ such that $|\phi (g)|\le C|g|^{-\frac{k}{p^*}}$, we have $\phi \in \ell _{p^*,\infty }(G)$, and then $m_{\phi }$ is an $L_p$-Fourier multiplier on $L_p(\widehat{G},\tau )$.

4.2 Schur Multipliers

In this subsection we prove Theorem 1.8 for $\mathcal {S}_p$-$\mathcal {S}_q$ Schur multipliers. Recall that the Schatten p-class $\mathcal {S}_p(H)$ is the noncommutative $L_p$-space $L_p(B(H),\text {Tr})$ with Tr being the usual trace. For any set X, any $a=(a_{ij})_{i,j\in X}$ induces a Schur multiplier A given by $A(x_{ij})=(a_{ij}x_{ij})$. Here we are interested in $\mathcal {S}_p$-$\mathcal {S}_q$ boundedness of A. In the following we use $\Vert \cdot \Vert _p$ to denote the Schatten p-norms. Note first that we have

$$\begin{aligned} \Vert x\Vert _{\infty }\le \Vert x\Vert _{\ell _1(X\times X)}, \end{aligned}$$

(4.8)

and

$$\begin{aligned} \Vert x\Vert _{2}=\Vert x\Vert _{\ell _2(X\times X)}. \end{aligned}$$

(4.9)

With (4.8) and (4.9), the complex interpolation gives

$$\begin{aligned} \Vert x\Vert _{p'}\le \Vert x\Vert _{\ell _{p}(X\times X)},~~1< p< 2, \end{aligned}$$

(4.10)

while the real interpolation implies

$$\begin{aligned} \Vert x\Vert _{p'}\precsim _{p} \Vert x\Vert _{\ell _{p,p'}(X\times X)},~~1< p< 2. \end{aligned}$$

(4.11)

Similarly, from

$$\begin{aligned} \Vert x\Vert _{\ell _\infty (X\times X)}\le \Vert x\Vert _{1},~~\Vert x\Vert _{\ell _2(X\times X)}=\Vert x\Vert _{2}, \end{aligned}$$

(4.12)

we have by complex interpolation that

$$\begin{aligned} \Vert x\Vert _{\ell _{p'}(X\times X)}\le \Vert x\Vert _{p},~~1< p< 2, \end{aligned}$$

(4.13)

and by real interpolation that

$$\begin{aligned} \Vert x\Vert _{\ell _{p',p}(X\times X)}\precsim _{p}\Vert x\Vert _{p},~~1< p< 2. \end{aligned}$$

(4.14)

Proof of Theorem 1.8

For Schur multipliers A induced by $a=(a_{ij})_{i,j\in X}$, we have for any $x=(x_{ij})_{i,j\in X}\in \mathcal {S}_p(\ell _{2}(X))$ that

$$\begin{aligned} \Vert Ax\Vert _q&{\mathop {\precsim _{q}}\limits ^{(4.11)}}\Vert (a_{ij}x_{ij})\Vert _{\ell _{q',q}(X\times X)}\\&{\mathop {\precsim _{q}}\limits ^{(2.3)}}\Vert a\Vert _{\ell _{r,\infty }(X\times X)}\Vert x\Vert _{\ell _{p',q}(X\times X)}\\&{\mathop {\precsim _{p,q}}\limits ^{(2.2)}}\Vert a\Vert _{\ell _{r,\infty }(X\times X)}\Vert x\Vert _{\ell _{p',p}(X\times X)}\\&{\mathop {\precsim _{p,q}}\limits ^{(4.14)}}\Vert a\Vert _{\ell _{r,\infty }(X\times X)}\Vert x\Vert _{p}. \end{aligned}$$

$\square$

Remark 4.5

If we use complex interpolation instead of real interpolation, we get

$$\begin{aligned} \Vert Ax\Vert _q {\mathop {\le }\limits ^{(4.10)}}\Vert (a_{ij}x_{ij})\Vert _{\ell _{q'}(X\times X)} {\mathop {\le }\limits ^{\text {H}\ddot{\text {o}}\text {lder}}}\Vert a\Vert _{\ell _{r}(X\times X)}\Vert x\Vert _{\ell _{p'}(X\times X)} {\mathop {\le }\limits ^{(4.13)}}\Vert a\Vert _{\ell _{r}(X\times X)}\Vert x\Vert _{p}. \end{aligned}$$

4.3 Remarks

Our proof uses the following interpolation result: for $1\le p_0<p_1\le \infty ,0<\theta <1$, and $1/p=(1-\theta )/p_0+\theta /p_1$, we have

$$\begin{aligned} (L_{p_0}(\mathcal {M},\varphi ),L_{p_1}(\mathcal {M},\varphi ))_{\theta ,p}=L_p(\mathcal {M},\varphi ) ~~(\text {with equivalent norms}), \end{aligned}$$

when $\varphi$ is a trace. However, when $\varphi$ is a weight, this fails in general [18, Sect. 3]. That is why we assume the left Haar weight $\varphi$ and its dual $\widehat{\varphi }$ to be tracial. If we use complex interpolation instead of real interpolation, then one can still get an upper bound of $\Vert x\Vert _{L_{r}(\mathbb {G},\varphi )}$ in Theorem 1.3 for general locally compact quantum groups. See Remarks 4.1 and 4.5. Certainly in this case the definition of Fourier multipliers is more involved.

We end with the following interesting question. Let $1<p\le 2\le q<\infty$ and $1/r=1/p-1/q$. Suppose that G is a locally compact non-unimodular group with $\mu$ being the left Haar measure. Let $\widehat{G}$ be the dual of G with $\widehat{\varphi }$ being the dual left Haar weight. Then for the Fourier multiplier $m_\phi$ with the symbol $\phi \in L_{r,\infty }(G,\mu )$, do we have

$$\begin{aligned} \Vert m_\phi : L_p(\widehat{G},\widehat{\varphi })\rightarrow L_q(\widehat{G},\widehat{\varphi })\Vert \precsim _{p,q}\Vert \phi \Vert _{L_{r,\infty }(G,\mu )} ? \end{aligned}$$

Here $\varphi =\mu$ is tracial, while $\widehat{\varphi }$ is not. One may choose various equivalent ways to define $L_p(\widehat{G},\widehat{\varphi })$, and the definition of Fourier multiplier $m_\phi$ needs to be suitably adapted accordingly.