Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}–Lq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_q$$\end{document} Fourier Multipliers on Locally Compact Quantum Groups

Let G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {G}$$\end{document} be a locally compact quantum group with dual G^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\mathbb {G}}$$\end{document}. Suppose that the left Haar weight φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varphi $$\end{document} and the dual left Haar weight φ^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{\varphi }$$\end{document} are tracial, e.g. G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {G}$$\end{document} is a unimodular Kac algebra. We prove that for 1<p≤2≤q<∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1<p\le 2 \le q<\infty $$\end{document}, the Fourier multiplier mx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$m_{x}$$\end{document} is bounded from Lp(G^,φ^)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p(\widehat{\mathbb {G}},\widehat{\varphi })$$\end{document} to Lq(G^,φ^)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_q(\widehat{\mathbb {G}},\widehat{\varphi })$$\end{document} whenever the symbol x lies in Lr,∞(G,φ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_{r,\infty }(\mathbb {G},\varphi )$$\end{document}, where 1/r=1/p-1/q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/r=1/p-1/q$$\end{document}. Moreover, we have ‖mx:Lp(G^,φ^)→Lq(G^,φ^)‖≤cp,q‖x‖Lr,∞(G,φ),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \Vert m_{x}:L_p(\widehat{\mathbb {G}},\widehat{\varphi })\rightarrow L_q(\widehat{\mathbb {G}},\widehat{\varphi })\Vert \le c_{p,q} \Vert x\Vert _{L_{r,\infty }(\mathbb {G},\varphi )}, \end{aligned}$$\end{document}where cp,q\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c_{p,q}$$\end{document} is a constant depending only on p and q. This was first proved by Hörmander (Acta Math 104:93–140, 1960) for Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document}, 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 Lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_p$$\end{document}-Fourier multipliers over discrete group von Neumann algebras. A similar result for Sp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}_p$$\end{document}-Sq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {S}_q$$\end{document} Schur multipliers is also proved.


Introduction
For a nice function (e.g.Schwartz function) f : R n → C, let f denote its Fourier transform: where •, • denotes the Euclidean inner product on R n .Given p, q > 0 and φ : R n → C, the operator m φ defined via is called an L p -L q Fourier multiplier if it is bounded from L p (R n ) to L q (R n ).When p = q, it is called an L p -Fourier multiplier for short.The function φ is called the symbol of the Fourier multiplier m φ .
Hörmander proved the following L p -L q Fourier multipliers theorem: Theorem 1.1 [14,Theorem 1.11] Let 1 < p ≤ 2 ≤ q < ∞ and 1/r = 1/ p − 1/q.Then we have Here L p,∞ (R n ) denotes the usual weak L p -space.Throughout this paper, C 1 C 2 always means C 1 ≤ cC 2 for some positive constant c < ∞.We write C 1 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 ≤ 2 and 1/s = 2/ p − 1, we have 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 G whose left Haar weight ϕ and the dual left Haar weight ϕ are both tracial.Our proof is slightly simpler and does not require Paley-type inequalities., , ϕ, ψ) be a locally compact quantum group with its dual G = ( M, , ϕ, ψ).Suppose that ϕ and ϕ are both tracial.Then for each x ∈ L r ,∞ (G, ϕ) with 1/r = 1/ p − 1/q, m x is an L p -L q Fourier multiplier such that m x : L p ( G, ϕ) → L q ( G, ϕ) p,q x L r ,∞ (G,ϕ) .
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.
Here F denotes the Fourier transform; see Sect. 3 for the definition.If furthermore, the dual quantum group G is compact, then Theorem 1.3 gives a sufficient condition for L p -Fourier multipliers on G.This is our third result.  G, h) .
Remark 1. 6 Clearly, the Fourier multiplier with the symbol c1 is bounded on L p (G, h) with the norm |c|, where c ∈ C and 1 is the unit element in G.However, c1 may not lie in p * ,∞ ( G, 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 a − c1 p * ,∞ ( G, h) .
An interesting family of such examples is obtained by choosing G = G as the group von Neumann algebra of a discrete group G.

Corollary 1.7 For any discrete group G, let G be the group von Neumann algebra equipped with the canonical tracial state τ . Then for any
where λ 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 S p (H ) to denote the Schatten p-classes L p (B(H )).See [21] and references therein for more related results.
where on the right hand side a = (a i j ) i, j∈X is identified as an element in C X ×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.

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

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 M be a semifinite von Neumann algebra equipped with a normal semifinite faithful (n.s.f .)trace τ .Denote by M + the positive cone of M. Let S + denote the set of all x ∈ M + such that τ (supp(x)) < ∞, where supp(x) denotes the support of x.Let S be the linear span of S + .Then S is a weak*-dense *-subalgebra of M. Given 0 < p < ∞, we define where |x| = (x * x) 1 2 is the modulus of x.Then (S, • p ) is a normed (or quasi-normed for p < 1) space.Its completion is called noncommutative L p -space associated with (M, τ ), denoted by L p (M, τ ) or simply by L p (M).As usual, we set L ∞ (M, τ ) = M equipped with the operator norm.
For 1 ≤ p < ∞, the dual space of L p (M) is L p (M) with respect to the duality In particular, L 1 (M) is identified with M * via the map j(x) The elements in L p (M) can be viewed as closed densely defined operators on H (H being the Hilbert space on which M acts).A linear closed operator x is said to be affiliated with M if it commutes with all unitary elements in M , i.e. xu = ux for any unitary u ∈ M .Note that x can be unbounded on H .An operator x affiliated with M is said to be measurable with respect to (M, τ ), or simply measurable if for any ε > 0, there exists a projection e ∈ M such that e(H ) ⊂ D(x) and τ (e ⊥ ) ≤ ε, where e ⊥ = 1−e.We denote by L 0 (M, τ ), or simply L 0 (M) the family of measurable operators.For x ∈ L 0 (M, τ ), we define the distribution function of x λ s (x) := τ (χ (s,∞) (|x|)), s ≥ 0, where χ (s,∞) (|x|) is the spectral projection of |x| corresponding to the interval (s, ∞), and define the generalized singular numbers of x μ t (x) := inf{s > 0 : λ s (x) < t}, t ≥ 0.
Similar to the classical case, for 0 < p < ∞, 0 < q ≤ ∞, the noncommutative Lorentz space L p,q (M) is defined as the collection of all measurable operators x such that Clearly, L p, p (M) = L p (M) with • p, p = • p .The space L p,∞ (M) is usually called the weak L p -space, 0 < p < ∞, and one defines 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 ≤ p 0 ≤ p 1 ≤ ∞, 1 ≤ q ≤ ∞ and 0 < θ < 1. Suppose Then it is well-known that [18, Sect.2] and where (•, •) θ and (•, •) θ,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 M be a von Neumann algebra equipped with an n.s.f .trace τ .We have (1) μ s+t (x y) ≤ μ s (x)μ t (y) for all s, t ≥ 0 and x, y ∈ L 0 (M); (2) for any 1 < p, q < ∞ and q < r ≤ ∞, x p,r p,q,r x p,q , x ∈ L p,q (M), (2.2) where the constant is c p,q,r = (q/ p) 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.
Let M be a von Neumann algebra equipped with an n.s.f .trace τ .Then we have ) Proof From Lemma 2.1(1) and the definition of • p 0 ,∞ , it follows that

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 ϕ on a von Neumann algebra M, we set A locally compact quantum group G = (M, , ϕ, ψ) consists of (1) a von Neumann algebra M; (2) a normal, unital, * -homomorphism : M → M⊗M such that (4) an n.s.f .weight ψ which is right invariant where ⊗ denotes the von Neumann algebra tensor product and id denotes the identity map.The normal, unital, * -homomorphism is called comultiplication on M, ϕ is called left Haar weight and ψ is called right Haar weight.

Example 2.3
Let G be a locally compact group.Then (L ∞ (G, μ), , μ, ν) is a locally compact quantum group, where : Given a locally compact quantum group G = (M, , ϕ, ψ), we now define its dual G = ( M, , ϕ, ψ) that is also a locally compact quantum group.For this, we equip n ϕ with the inner product and denote by H ϕ the induced Hilbert space after completion.For any x ∈ n ϕ ⊂ M we write ϕ (x) for the corresponding element in H ϕ .For any x ∈ M, π ϕ (x) denotes the bounded operator over H ϕ such that π ϕ (x) ϕ (y) = ϕ (x y).So (H ϕ , π ϕ , ϕ ) is the GNS representation of ϕ.We omit the subscript ϕ in the sequel whenever there is no ambiguity.Assume that M acts on H ϕ with its predual M * .The multiplicative unitary of G is the unitary operator W on H ϕ ⊗ H ϕ such that It implements the comultiplication: (2.4) Then the underlying von Neumann algebra of G is defined as where W = W * is the multiplicative unitary on G with being the flip on To define the dual left Haar weights, set By the Riesz representation theorem, there exists unique ξ(ω) ∈ H ϕ such that Then the dual left Haar weight ϕ is defined to be the unique n.s.f .weight on M with the GNS representation (H , ι, ) such that λ(I) is a σ -strong*-norm core for and (λ(ω)) = ξ(ω) for all ω ∈ I. Thus we have The dual right Haar weight ψ can be defined in a similar way, which we will not do here.Then G = ( M, , ϕ, ψ) forms a locally compact quantum group.Constructing the dual G of G, the Pontryagin duality says G = G.Furthermore, we have = .
In this paper we are interested in locally compact quantum groups G on which both left Haar weight ϕ and dual left Haar weight ϕ 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 G = (M, , ϕ, ψ) for which ϕ = ψ is tracial.If a Kac algebra G is unimodular, then so is its dual 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 μ being left (also right) Haar measure.Then G = (L ∞ (G, μ), , μ, μ) 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 G = ( G, , μ, μ), the left (right) dual Haar weight is tracial.Here G is the von Neumann algebra acting on Example 2.6 (Compact quantum groups of Kac type) A compact quantum group is a locally compact quantum group G such that the left Haar weight is finite, i.e. ϕ(1) < ∞.This agrees with Woronowicz's definition of compact quantum groups [25], which we shall now recall.A compact quantum group consists of a pair G = (A, ), where A is a unital C*-algebra and is a unital * -homomorphism from A to A ⊗ A such that (1) Here A ⊗ 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: , where u i j ∈ A. The matrix u is called an n-dimensional representation of G if we have A representation u is called unitary if u is unitary as an element in M n (A), and irreducible if the only matrices T ∈ M n (C) such that uT = T u are multiples of identity matrix.Two representations u, v ∈ M n (A) are said to be equivalent if there exists an invertible matrix T ∈ M n (C) such that T u = vT .Denote by Irr(G) the set of equivalence classes of irreducible unitary representations of G.For each π ∈ Irr(G), denote by u π ∈ A ⊗ B(H π ) a representative of the class π , where H π is the finitedimensional Hilbert space on which u π acts.In the sequel we write . This is a dense subalgebra of A.
The dual of a compact quantum group G is a discrete quantum group G = ( A, , h L , h R ), where A is the c 0 -direct sum of matrix algebras

B(H π ).
The dual left Haar weight h L and dual right Haar weight h R are not the same in general.A compact quantum group G is of Kac type if the Haar state h is tracial.In this case h L and h R coincide, which we denote by h for short.It takes the following form The multiplicative unitary W of G is Then the Fourier transform F over Pol(G) is given by 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.

Fourier Transform on Locally Compact Quantum Groups
In the remaining part of the paper, unless otherwise stated, for any 1 < p < ∞, p always denotes the conjugate number of p, i.e. 1/ p + 1/ p = 1.G = (M, , ϕ, ψ) always denotes a locally compact quantum group with dual G = ( M, , ϕ, ψ), where ϕ and ϕ are both tracial.We shall use L p (G, ϕ) and L p,q (G, ϕ) to denote L p (M, ϕ) and L p,q (M, ϕ), respectively.The same goes to L p ( G, ϕ) and L p,q ( G, ϕ).

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 μ, then the Fourier transform of f ∈ L 1 (G, μ) takes the form: By choosing the dual Haar measure μ on G suitably, the map G, μ) is isometric and can be extended to an isometry between L 2 (G, μ) and L 2 ( G, μ).This defines the Fourier transform of f ∈ L 2 (G, μ).The definition of the Fourier transform of f ∈ L p (G, μ) follows from the the famous Hausdorff-Young inequality, which states that for any 1 ≤ p ≤ 2 we have 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 λ( f ) denote the left regular representation of f ∈ L 1 (G, μ) on L 2 (G, μ), which is an operator given by Denote by L f the operator on L 2 (G, μ) given by multiplying f .Since F turns convolution into multiplication, we have Recall that F is unitary on L 2 (G, μ), so λ( f ) is unitarily equivalent to the operator L F ( f ) .This suggests us to use λ( f ) as a substitute of F( f ).From this Kunze defined the Fourier transform on locally compact unimodular groups (G, μ) and generalized Hausdorff-Young inequalities (3.1) to locally compact unimodular groups.The dual of G, still denoted by G, is no longer a group, but can be studied via the von Neumann algebra generated by λ(L 1 (G, μ)) in B(L 2 (G, μ)).It turns out that there is a canonical trace μ on G, so L p ( G, μ) 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 ∈ L p (G, μ) is the operator on L 2 (G, μ) given by λ( f ) 1 p , where μ is the left Haar measure and is the modular function on G. Remark that here is understood as a multiplication operator by .The dual G of G is not necessarily equipped with a trace.In this context we also have Hausdorff-Young inequalities, where L p ( 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 (G, ϕ) (1 < p < 2) into L 1 (G, ϕ) + L 2 (G, ϕ) in a natural way.So we will not recall their approaches here.

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.
This holds for general locally compact quantum groups and should be understood under suitable embedding of I, L 1 (G, ϕ) and L 2 (G, ϕ) into some Banach space [5,Theorem 3.3].We give a proof here when ϕ is tracial, which is the case this paper concerns with.In such case, I should be understood as j −1 (I), where j :

Proof of Proposition 3.1 when ϕ is tracial By definition,
Note that n ϕ is dense in L 2 (G, ϕ), by duality of L p -spaces, we have Recall that L 1 (G, ϕ) is identified with M * via the map j(x) = ϕ x , where ϕ x := ϕ(x•).Since W is unitary, from (2.4) we have We define the L 1 -Fourier transform as For the L 2 -Fourier transform, we firstly define it as F 1 on the intersection of L 1 (G, ϕ) and n ϕ .By Proposition 3.1, for any x ∈ L 1 (G, ϕ) ∩ n ϕ , ϕ x belongs to I. Note that by definition of , we have Since n ϕ is dense in L 2 (G, ϕ), we have From (2.5) it follows that So we have ), which we still denote by F 2 .Now we may define an operator . By complex interpolation, we have the Hausdorff-Young inequality: where 1 ≤ p ≤ 2 and 1/ p + 1/ p = 1.If we use real interpolation instead of complex interpolation, we get Compared with (3.3), the constant c p in (3.4) is worse, but the space L p, p (G, ϕ) is larger than L p (G, ϕ) when 1 ≤ p < 2. Definition 3.2 For any x ∈ L 0 (G, ϕ), we call m x an L p -L q Fourier multiplier if the map F(y) → F(x y) is well-defined and extends to bounded map from L p ( G, ϕ) to L q ( G, ϕ).One may also consider the map F(y) → F(yx) that is similar.

The Dual/Inverse Fourier Transform
On the dual quantum group one can also define the Fourier transform Proposition 3.3 Let G = (M, , ϕ, ψ) be a locally compact quantum group with dual G = ( M, , ϕ, ψ).Then we have (1) Proof Note that the inclusion map : n ϕ → H ϕ = L 2 (G, ϕ) can be extended to the whole Hilbert space L 2 (G, ϕ).We shall still use to denote its extension.The same goes to .Recall that Therefore, F(F(x)) = x.This proves (1).The proof of (2) is similar.
Since F is the Fourier transform on G, we have This, together with Proposition 3.3, yields for all x ∈ L 2 (G, ϕ) such that F(x) ∈ L 1 ( G, ϕ), or equivalently, for all x such that Combining (3.5), (3.6), and applying real interpolation, we get for all x such that F(x) ∈ L p ( G, ϕ).

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 • p,q to denote • L p,q (G,ϕ) or • L p,q ( G, ϕ) whenever no ambiguity can occur.

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 x r instead of c p,q x r ,∞ .Details are provided for the Schur multipliers.See Remark 4.5.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 x p ≤ x q whenever x ∈ L q (G, h) and p ≤ q.Thus for any 1 < p ≤ 2 ≤ q < ∞ we have

Proof of
for all a ∈ r ,∞ ( G, h) and F(b) ∈ L q (G, h).The first two inequalities of (4.1) imply that m a is an L p -Fourier multiplier: while the last two inequalities of (4.1) imply that m a is an L q -Fourier multiplier: We may choose q = 2 in (4.2) and p = 2 in (4.3).So for any 1 < p < ∞ we have where 1/r = 1/ p − 1/q.Indeed, by [ we have By (4.5), f ∈ L p (T) while m φ ( f ) / ∈ L q (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 = Z and G = T. Then there exists φ : Z → R such that φ ∈ 2,∞ (Z) while the Fourier multiplier m φ is unbounded on L 1 (T).To see this, take Suppose that the Fourier multiplier m φ is bounded over L 1 (T).Then there exist a measure μ on T such that m φ ( f ) = μ * f , with * being the convolution.Since for some constant K > 0. However, the left hand side is unbounded as φ / ∈ 2 (Z).This leads to a contradiction.Therefore, Corollary 1.7 fails when p = 1.Therefore, and we have φ ∈ p * ,∞ (G), whence m φ is an L p -Fourier multiplier on L p ( G, τ ).For free group on N generators 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 φ : G → C such that |φ(g)| ≤ C|g| − k p * , we have φ ∈ p * ,∞ (G), and then m φ is an L p -Fourier multiplier on L p ( G, τ ).

Schur Multipliers
In this subsection we prove Theorem 1.8 for S p -S q Schur multipliers.Recall that the Schatten p-class S p (H ) is the noncommutative L p -space L p (B(H ), Tr) with Tr being the usual trace.For any set X , any a = (a i j ) i, j∈X induces a Schur multiplier A given by A(x i j ) = (a i j x i j ).Here we are interested in S p -S q boundedness of A. In the following we use • p to denote the Schatten p-norms.Note first that we have (a i j x i j ) q ,q (X ×X ) (2.3) q a r ,∞ (X ×X ) x p ,q (X ×X ) p,q a r ,∞ (X ×X ) x p , p (X ×X ) (4.14) p,q a r ,∞ (X ×X ) x p .

Remark 4.5 If we use complex interpolation instead of real interpolation, we get
Ax q (4.10) ≤ (a i j x i j ) q (X ×X ) Hölder ≤ a r (X ×X ) x p (X ×X ) (4.13) ≤ a r (X ×X ) x p .

Remarks
Our proof uses the following interpolation result: for 1 ≤ p 0 < p 1 ≤ ∞, 0 < θ < 1, and 1/ p = (1 − θ)/ p 0 + θ/ p 1 , we have (L p 0 (M, ϕ), L p 1 (M, ϕ)) θ, p = L p (M, ϕ) (with equivalent norms), when ϕ is a trace.However, when ϕ is a weight, this fails in general [18,Sect. 3].That is why we assume the left Haar weight ϕ and its dual ϕ to be tracial.If we use complex interpolation instead of real interpolation, then one can still get an upper bound of x L r (G,ϕ) 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 ≤ 2 ≤ q < ∞ and 1/r = 1/ p − 1/q.Suppose that G is a locally compact non-unimodular group with μ being the left Haar measure.Let G be the dual of G with ϕ being the dual left Haar weight.Then for the Fourier multiplier m φ with the symbol φ ∈ L r ,∞ (G, μ), do we have m φ : L p ( G, ϕ) → L q ( G, ϕ) p,q φ L r ,∞ (G,μ) ?
Here ϕ = μ is tracial, while ϕ is not.One may choose various equivalent ways to define L p ( G, ϕ), and the definition of Fourier multiplier m φ needs to be suitably adapted accordingly.

Theorem 1 . 5
Fix 1 < p < ∞ and 1/ p * = |1/2 − 1/ p|.Let G be a compact quantum group of Kac type with Haar state h.Let G be its dual with dual Haar weight h.Let F be the Fourier transform over G. Then for any a = (a π ) π ∈Irr(G) ∈ G, the Fourier multiplier m a : F(b) → F(ab) satisfies and μ, ν are the left and right Haar measures on G, respectively.

Example 2 . 7 (
Discrete group von Neumann algebras) Let G be a discrete group.Then G = ( ∞ (G), , μ, μ) is a locally compact quantum group with μ being the counting measure.Suppose that {δ g } g∈G is the canonical basis of 2 (G).Then the left regular representation of G is given through λ : G → B( 2 (G)), λ g (δ h ) = δ gh .The group von Neumann algebra G is the von Neumann algebra generated by λ(g), g ∈ G in B( 2 (G)).Thus the dual quantum group of G is G = ( G, , τ, τ ), where τ is a normal faithful tracial state defined by τ (x) = δ e , xδ e , where e is the unit of G and •, • is the inner product on 2 (G).

Remark 4 . 2
m a : L p (G, h) → L p (G, h) p a p * ,∞ , with 1/ p * = |1/2 − 1/ p|.The index r in Theorem 1.3 is sharp in general.To see this, take G = Z with G = T.By Theorem 1.3 we have for 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 ∈ S whenever x ∈ S. Then it has an exponential growth, i.e.,|{x ∈ G : d(x, e) ≤ n}| ≤ M n , n ≥ 1,(4.7) for some M > 1, where d is the word metric on G with respect to S and | • | denotes the counting measure on G. Indeed, one can always choose M to be |S|.Then for any φ : G → C such that |φ(g)| ≤ C M − |g| p * , where |g| := d(g, e) and C > 0 is a constant.|φ(g)| ≥ α implies |g| ≤ −p * log M α C , α > 0.