Hermite Multipliers on Modulation Spaces 3

Home , Modulation space

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i ξ α the operator (See Definition 4.4 below) with Fourier multiplier e | | (α > 2) is bounded on Lp(Rd) if and only if p =2. The study of Fourier multiplier operators, which are of the form m( ∆) in the context of modulation space M 2,1(Rd) was initiated in the works of Wang-Zhao-Guo− [19]. In fact, in the consequent year Bényi-Gröchenig-Okoudjou-Rogers [1] have shown that the Fourier i ξ α p,q d multiplier operator with multiplier e | | (α [0, 2]) is bounded on M (R ) for all 1 p, q . The cases α = 1 and α = 2 are particularly∈ interesting and have been studied intensively≤ ≤ in∞ PDE, because they occur in the time evolution of the wave equation (α = 1) and the free Schrödinger operator (α = 2). Thus, the Schödinger and wave propagators are not Lp(p = 2)- bounded but M p,q- bounded for all 1 p, q . In fact, this leads to fixed-time estimates6 for Schödinger and wave propagators and≤ some≤∞ of their applications to well-posedness results on modulation spaces M p,q(Rd). Modulation spaces have turned out to be very fruitful in numerous applications in various problems in analysis and PDE. And yet there has been a lot of ongoing interest in these spaces from the harmonic analysis and PDE points of view. We refer to the recent survey [12] and the references therein. Coming back to the Hermite operator, we note that Thangavelu [18] (See also [15, The- orem 4.2.1]) has proved an analogue of the Hörmander-Mikhlin type multiplier theorem for Hermite expansions on Lp(Rd). Specifically, he showed that under certain conditions on m, the operator m(H) is bounded on Lp(Rd)(1

m L2 = sup ψm(t ) L2 . k k β,sloc t>0 k · k β We then have the following theorem.

Theorem 1.1. Let 1

(2d + 1)/2. Then the operator m(H) is bounded on M q,p(Rd). We remark that this theorem is not sharp. For m to be an Lp multiplier it is sufficient to assume the condition on m with β > d/2. We believe the same is true in the case of multipliers on modulation spaces though our method of proof requires a stronger assumption on m. However, it is worth noting that M p,p(Rd) for p> 2 is a much wider class than Lp(Rd) (See Lemma 2.6(3) below). Thus, Theorem 1.1 shows that the Hörmander-Mikhlin multiplier type theorem is true for a much wider class than Lp(Rd). We deduce Theorem 1.1 from a corresponding result on the polarised Heisenberg group Hd = Rd Rd R which is of Euclidean dimension (2d +1). Let ˜ stand for the sublapla- pol × × L cian on Hd and define m( ˜) using spectral theorem (See Subsection 2.2 below for precise pol L definitions). We then have the following transference result. Theorem 1.2. Let 1

The above theorem allows us to deduce some interesting corollaries for Hermite multipliers. ∂ 1/2 For example, let Rj =( + ξj)H− , j =1, 2, ..., d be the Riesz transforms associated to − ∂ξj H. It is well known that these Riesz transforms are bounded on Lp(Rd), 1

Corollary 1.3. Let 1 < p q 2 or 2 q p < . Then the Riesz transforms Rj are bounded on M q,p(Rd). ≤ ≤ ≤ ≤ ∞ Another interesting corollary is the following result about solutions of the wave equation associated to H. Consider the following Cauchy problem:

2 ∂t u(x, t)= Hu(x, t), u(x, 0)=0, ∂tu(x, 0) = f(x) − 1/2 1/2 whose solution is given by u(x, t)= H− sin(tH )f(x). Corollary 1.4. Let u be the solution of the above Cauchy problem for H. Then for 1 < p q 2 or 2 q p < we have the estimate u( , t) M q,p Ct f M q,p provided 1 ≤ 1 ≤< 1 . ≤ ≤ ∞ k · k ≤ k k | p − 2 | 2d When p = q Theorem 1.2 as well as Corollary 1.4 can be improved. This will be achieved using transference as before, but now the transference is from multiplier theorems for multiple p d Fourier series. Given a function m on R we deﬁne a Fourier multiplier Tm on L (T ) where Td is the d dimensional torus by − iµ x T f(x)= m( µ )fˆ(µ)e · . m | | µ Zd X∈ Here fˆ(µ) are the Fourier coeﬃcients of f and µ = d µ . Using the connection (See | | j=1 | j| Proposition 4.3 below) between Fourier multipliers T on Lp(Td) and m(H) on M p,p we Pm prove

Theorem 1.5. Let 1 d/2. Then the ∞ k k β,sloc ∞ operator m(H) is bounded on M p,p(Rd). We also have the following improvement of Corollary 1.4. More generally, we consider multipliers of the form

ei(2k+d)γ (1.1) m(2k + d)= , (β > 0,γ > 0). (2k + d)β Theorem 1.6. Let 1 p< , and 1 1 < β , and let m be given by (1.1). Then m(H) ≤ ∞ | p − 2 | dγ is bounded on M p,p(Rd). In particular, Corollary 1.4 is valid on the bigger range 1 1 < 1 | p − 2 | d when p = q. We now turn our attention to a multiplier which occurs in the time evolution of the free Schrödinger equation associated to H. Specifically, we consider the Cauchy problem for the Schrödinger equation associated to H : i∂ u(x, t) Hu(x, t)=0, u(x, 0) = f(x) t − whose solution is given by u(x, t)= eitH f(x). 4 D.G.BHIMANI,R.BALHARA,ANDS.THANGAVELU

Theorem 1.7. The Schrödinger propagator m(H) = eitH is bounded on M p,p(Rd) for all 1 p< . ≤ ∞ Recently Cordero-Nicola [2, Section 5.1] have studied the operator m(H)= eitH on Wiener amalgam spaces (closely related to modulation spaces). Later Kato-Kobayashi-Ito [8] have given a refinement of Cordero-Nicola’s results in the context of Wiener amalgam spaces. We have studied (Theorem 1.7) the boundeness of m(H) = eitH in the context of modulation spaces. Here we would like to point out that our method of proof is completely different from the method used in the context of Wiener amalgam spaces. We also believe that our method of proof is much simpler than the proofs available in the literature. Our proof relies on properties of Hermite and special Hermite functions and illustrates the importance of these functions in the study of such problems. Finally, we note that modulation spaces have been used as regularity classes for initial data associated to Cauchy problems for nonlinear dispersive equations (eg., NLS, NLW, etc..) but so far mainly for the nonlinear dispersive equations associated to the Laplacian without potential (D = ∆) [12, 19, 20]. There is also an ongoing interest to use harmonic analysis tools (specifically, multiplier results) to solve modern nonlinear PDE problems (See e.g., [5]). Thus, we strongly believe that our results will be useful in the future for studying nonlinear dispersive equations associated to H in the realm of modulation spaces. The paper is organized as follows. In Section 2, we introduce notations and preliminaries which will be used in the sequel. Specifically, in Subsection 2.2, we introduce the Heisen- berg and polarised Heisenberg groups, the sublaplacians corresponding to these groups, and spectral multipliers associated to these sublaplacians. In Subsection 2.4, we prove the transference result which connects spectral multiplier on polarised Heisenberg groups and reduced polarised Heisenberg groups. In Subsection 2.5, we introduce modulation spaces and recall some of their basic properties. In Section 3, we prove Theorems 1.1 and 1.2. In Section 4, we prove Theorems 1.5 and 1.6. In Section 5, we prove Theorem 1.7.

2. Notation and Preliminaries

2.1. Notations. The notation A . B means A cB for some constant c > 0. The ≤ symbol A1 ֒ A2 denotes the continuous embedding of the topological linear space A1 → d d into A . If α = (α , ..., α ) Nd is a multi-index, we set α = α ,α! = α !. 2 1 d ∈ | | j=1 j j=1 j If z = (z , ..., z ) Cd, we put zα = d zαj . We denote the d dimensional torus by 1 d ∈ j=1 j P− Q Td [0, 2π)d, and the Lp(Td) norm by ≡ − Q 1/p p f Lp(Td) = f(t) dt . k k d | | Z[0,2π) The class of trigonometric polynomials on Td is denoted by (Td). The mixed Lp(Rd, Lq(Rd)) norm is denoted by P

p/q 1/p q f Lp,q = f(x, y) dy dx (1 p,q < ), k k Rd Rd | | ≤ ∞ Z Z ! Rd Rd the L∞( ) norm is f L∞ = ess.supx Rd f(x) . The Schwartz class is denoted by ( ) k k ∈ | | dS (with its usual topology), and the space of tempered distributions is denoted by ′(R ). For S HERMITE MULTIPLIERS ON MODULATION SPACES 5

Rd d Rd Rd x = (x1, , xd),y = (y1, ,yd) , we put x y = i=1 xiyi. Let : ( ) ( ) be the Fourier··· transform defined··· by∈ · F S → S P d/2 ix ξ d f(ξ)= f(ξ)=(2π)− f(x)e− · dx, ξ R . F Rd ∈ Z Then is a bijection and the inverse Fourier transform is given by F b 1 d/2 ix ξ d − f(x)= f ∨(x)=(2π)− f(ξ) e · dξ, x R . F Rd ∈ Z d d It is well known that the Fourier transform can be uniquely extended to : ′(R ) ′(R ). F S → S 2.2. Heisenberg group and Fourier multipliers. We consider the Heisenberg group Hd = Cd R with the group law × 1 (z, t)(w,s)=(z + w, t + s + Im(z w¯)). 2 · Sometimes we use real coordinates (x, y, t) instead of (z, t) if z = x + iy, x, y Rd. In order to define the (group) Fourier transform on the Heisenberg group we briefly recall∈ the following family of irreducible unitary representations. For each non-zero real λ, we have a 2 d representation πλ realised on L (R ) as follows: iλt i(x ξ+ 1 x y) πλ(z, t)ϕ(ξ)= e e · 2 · ϕ(ξ + y) where ϕ L2(Rd). For f L1(Hd), one defines the operator ∈ ∈

fˆ(λ)= f(z, t)πλ(z, t)dzdt. Hd Z The operator valued function f fˆ(λ) is called the group Fourier transform of f on Hd. We refer to [16] for more about the→ group Fourier transform. The Fourier transform initially deﬁned on L1(Hd) L2(Hd) can be extended to the whole of L2(Hd) and we have a version of Plancherel theorem.∩ Speciﬁcally, when f L1 L2(Hd), ∈ ∩ it can be shown that fˆ(λ) is a Hilbert-Schmidt operator and the Plancherel theorem holds: d 1 2 2 − ∞ ˆ 2 d f(z, t) dzdt = d+1 f(λ) HS λ dλ Hd | | π k k | | Z Z−∞ 2 where is the Hilbert-Schmidt norm given by T = tr(T ∗T ), for T a bounded k·kHS k kHS operator, T ∗ being the adjoint operator of T. Under the assumption that fˆ(λ) is of trace class we have the inversion formula

d 1 ∞ d f(z, t)=(2π)− − tr(π (z, t)∗fˆ(λ)) λ dλ. λ | | Z−∞ Let f λ stand for the inverse Fourier transform of f in the central variable t

∞ (2.1) f λ(z)= f(z, t)eiλtdt. Z−∞ By taking the Euclidean Fourier transform of f λ(z) in the variable λ, we obtain

1 ∞ iλt λ (2.2) f(z, t)= e− f (z) dλ. 2π Z−∞ 6 D.G.BHIMANI,R.BALHARA,ANDS.THANGAVELU

Recalling the deﬁnition of πλ we see that we can write the group Fourier transform as

λ fˆ(λ)= πλ(f ) λ λ Cd where πλ(f ) = Cd f (z)πλ(z, 0)dz. The operator which takes a function g on into the operator R g(z)πλ(z, 0)dz Cd Z λ is called the Weyl transform of g and is denoted by Wλ(g). Thus f(λ) = Wλ(f ). With these notations we can rewrite the inversion formula as b d 1 ∞ iλt λ d f(z, t)=(2π)− − e− tr(π (z, 0)∗π (f )) λ dλ. λ λ | | Z−∞ On the Heisenberg group, we have the vector ﬁelds

∂ ∂ 1 ∂ ∂ 1 ∂ T = , Xj = + yj ,Yj = xj , j =1, ..., d, ∂t ∂xj 2 ∂t ∂yj − 2 ∂t and they form a basis for the Lie algebra of left invariant vector ﬁelds on the Heisenberg group. The second-order operator

d = (X2 + Y 2) L − j j j=1 X is called the sublaplacian which is self-adjoint and nonnegative and hence admits a spectral decomposition ∞ = λ dE . L λ Z0 Given a bounded function m defined on (0, ) one can define the operator m( ) formally by setting ∞ L ∞ m( )f = m(λ) dE f. L λ Z0 \ It can be shown that m( )f(λ)= fˆ(λ)m(H(λ)) where H(λ)= ∆+ λ2 x 2 are the scaled Hermite operators. Hence,L the operators m( )f are examples of− Fourier| | multipliers| on the Heisenberg group. More generally, (right) FourierL multipliers on the Heisenberg group are operators defined by TM f(λ) = fˆ(λ)M(λ) where M(λ), called the multiplier is a family of bounded linear operators on L2(Rd). The boundedness propertiesd of these operators have been studied by several authors, see R+ e.g. [6, 11, 7]. We make use of the following result. Let 0 = ψ C0∞( ) be a fixed cut-off 1 6 ∈ function with support contained in the interval [ 2 , 1], and define the scale invariant localized Sobolev norm of order β of m by

m L2 = sup ψm(t ) L2 . k k β,sloc t>0 k · k β

Theorem 2.1 (M¨uller-Stein, Hebisch [11, 7]). If m L2 < for some β > (2d + 1)/2, k k β,sloc ∞ then m( ) is bounded on Lp(Hd) for 1

We make use of this theorem in proving our main result. Actually we need an analogue of Hd Rd Rd R the above result in the context of polarised Heisenberg group pol which is just with the group law × ×

(x, y, t)(x′,y′, t′)=(x + x′,y + y′, t + t′ + x′ y). · A basis for the algebra of left invariant vectors fields on this group are given by ∂ ∂ ∂ ∂ T = , X˜j = + yj , Y˜j = , j =1, ..., d. ∂t ∂xj ∂t ∂yj The sublaplacian is then defined as the second-order operator d ˜ = (X˜ 2 + Y˜ 2). L − j j j=1 X Hd Hd Hd The group pol is isomorphic to and the isomorphism is given by the map Φ : Hd 1 → pol, Φ(x, y, t)=(x, y, t + 2 x y). Note that Φ is measure preserving and it is easy to check that · 1 (f Φ− )= ˜f Φ L ◦ L ◦ Hd for reasonable functions f on pol. In view of this, an analogue of Theorem 2.1 is true for m( ˜). This also follows from the fact that the above theorem is valid in a more general contextL of H-type groups, see [10, 7, 6]. Hd Using the isomorphism Φ we can define the following family of representations for pol. 1 For each non zero real λ, the representations ρλ = πλ Φ− are irreducible and unitary. We ◦ Hd can use them to define Fourier transform on the group pol. 2.3. Hermite and special Hermite functions. The spectral decomposition of H = ∆+ x 2 is given by the Hermite expansion. Let Φ (x), α Nd be the normalized Hermite− | | α ∈ functions which are products of one dimensional Hermite functions. More precisely, Φα(x)= d Πj=1hαj (xj) where k k 1/2 k 1 x2 d x2 h (x)=(√π2 k!)− ( 1) e 2 e− . k − dxk The Hermite functions Φα are eigenfunctions of H with eigenvalues (2 α + d) where α = 2 d | | | | α1+...+αd. Moreover, they form an orthonormal basis for L (R ). The spectral decomposition of H is then written as ∞ H = (2k + d)P , P f(x)= f, Φ Φ k k h αi α k=0 α =k X |X| where , is the inner product in L2(Rd). Given a function m defined and bounded on the set of allh· ·i natural numbers we can use the spectral theorem to define m(H). The action of m(H) on a function f is given by

∞ m(H)f = m(2 α + d) f, Φ Φ = m(2k + d)P f. | | h αi α k α Nd k=0 X∈ X This operator m(H) is bounded on L2(Rd). This follows immediately from the Plancherel theorem for the Hermite expansions as m is bounded. On the other hand, the mere boundedness of m is not sufficient to imply the Lp boundedness of m(H) for p = 2. So, we need to impose some conditions m to ensure that m(H) 6 8 D.G.BHIMANI,R.BALHARA,ANDS.THANGAVELU is bounded on Lp(Rd). The boundedness results of m(H) on Lp(Rd) have been studied by several authors, see e.g. [18, 15]. Our aim in this paper is to study the boundedness of m(H) on modulation spaces. In the sequel, we make use of some properties of special Hermite functions Φα,β which are defined as follows. Let π(z)= π1(z, t) and define d/2 (2.3) Φ (z)=(2π)− π(z)Φ , Φ . α,β h α βi Then it is well known that these so called special Hermite functions form an orthonormal basis for L2(Cd). Moreover, they are eigenfunctions of the special Hermite operator L which is defined by the equation (f(z)eit)= eitLf(z). L Indeed, we can show ([15, Theorem 1.3.3]) that LΦ = (2 α + d)Φ . α,β | | α,β For a function f L2(Cd) we have the eigenfunction expansion ∈ f(z)= f, Φ Φ (z) h α,βi α,β α Nd β Nd X∈ X∈ which is called the special Hermite expansion. Let f and g be two measurable functions on Cd. We recall that the λ twisted convolution (0 = λ R) of f and g is the function f g defined by − 6 ∈ ∗λ i λ Im(z w¯) f λ g(z)= f(z w)g(w)e 2 · dw ∗ Cd − Z for all z such that the integral exits. When λ =1, we simply call them twisted convolution and denote them by f g. It is well known that the twisted convolution of special Hermite functions satisfies [15, Proposition× 1.3.2] the following relation (2.4) Φ Φ = (2π)d/2δ Φ α,β × µ,v β,µ α,v where δβ,µ = 1 if β = µ, otherwise it is 0. Using the identity (2.4), we can show that the special Hermite expansions can be written ([15, Section 2.1]) in the compact form as follows:

d/2 f(z) = (2π)− f Φ (z) × α,α α Nd X∈ ∞ d/2 = (2π)− f Φ (z) ×  α,α  k=0 α =k X |X| ∞   d = (2π)− f φ (z) × k Xk=0 where ϕ are the Laguerre functions of type (d 1): k − d/2 d 1 1 2 1 z 2 ϕ (z)=(2π) Φ (z)= L − ( z )e− 4 | | . k α,α k 2| | α =k |X| We note that f Φα,α is an eigenfunction of the operator L with the eigenvalue (2 α + d). d × | | Hence (2π)− f φ is the projection of f onto the eigenspace corresponding to the eigenvalue × k HERMITE MULTIPLIERS ON MODULATION SPACES 9

(2k + d). The spectral decomposition of the operator L is given by the special Hermite functions and we can write the same in a compact form as

∞ d Lf(z)=(2π)− (2k + d)f ϕ (z). × k Xk=0 As in the case of the Hermite operators, one can deﬁne and study special Hermite multipliers m(L), see [15]. The functions Φα,β can be expressed in terms of Laguerre functions. In particular, we have ([15, Theorem 1.3.5])

α d/2 1/2 i | | α 1 z 2 (2.5) Φ (z)=(2π)− (α!)− z¯ e− 4 | | . α,0 √ 2 We also have to deal with the family of operators H(λ)= ∆+λ2 x 2 whose eigenfunctions − d| | are given by scaled Hermite functions. For λ R∗ and each α N , we define the family of scaled Hermite functions ∈ ∈ λ d d Φ (x)= λ 4 Φ ( λ x), x R . α | | α | | ∈ They form an orthonormal basis for L2(Rd). Thep spectral decomposition of the scaled Hermite operator H(λ)= ∆+ λ 2 x 2 is then written as − | | | | ∞ (2.6) H(λ)= (2k + d) λ P (λ), | | k Xk=0 2 d for λ R∗, where Pk(λ) are the (finite-dimensional) orthogonal projections defined on L (R ) by ∈ P (λ)f = f, Φλ Φλ, k h αi α α =k |X| where f L2(Rd). We can now define scaled special Hermite functions ∈ λ d/2 d/2 λ λ (2.7) Φ (z)=(2π)− λ π (z)Φ , Φ . α,β | | h λ α βi They form a complete orthonormal system in L2(Cd). For every f L2(Cd), we have the special Hermite expansion ∈ (2.8) f = f, Φλ Φλ . h α,βi α,β α Nd β Nd X∈ X∈ We now define the (scaled) special Hermite operator (twisted Laplacian) by the relation (2.9) (eiλtf(z)) = eiλtL f(z). L λ We note that the operators Lλ and H(λ) are related via the Weyl transform:

πλ(Lλf)= πλ(f)H(λ).

In fact, the scaled Hermite functions are eigenfunctions of the operator Lλ : (2.10) L Φλ (z)=(2 α + d) λ Φλ (z). λ α,β | | | | α,β th Note that the eigenvalues of Lλ are of the form (2k + d) λ ,k = 0, 1, 2, ..., and the k eigenspace corresponding to the eigenvalue (2k + d) λ is inﬁnite-dimensional| | being the span | | 10 D.G.BHIMANI,R.BALHARA,ANDS.THANGAVELU

λ Nd of Φα,β : α = k, β . It is well known that the special Hermite expansion can be written{ in the| | compact∈ form} as follows:

∞ d d λ (2.11) f(z)=(2π)− λ f ϕ (z) | | ∗λ k Xk=0 where ϕλ is the scaled Laguerre functions of type (d 1) k − λ n 1 1 2 1 λ z 2 (2.12) ϕ (z)= L − λ z e− 4 | || | . k k 2| || | The spectral decomposition of Lλ is then written as

∞ d d λ L f(z)=(2π)− λ (2k + d) λ f ϕ (z). λ | | | | ∗λ k Xk=0 In particular, recalling f λ (see(2.1)), we have

∞ λ d d λ λ (2.13) L f (z)=(2π)− λ (2k + d) λ f ϕ (z). λ | | | | ∗λ k Xk=0 Taking the Fourier transform in λ variable in equation (2.13), and using (2.9) and (2.2), we obtain the spectral decomposition of the sublaplacian on Hd as follows: L ∞ d 1 ∞ λ λ iλt d f(z, t)=(2π)− − (2k + d) λ f ϕ (z) e− λ dλ. L | | ∗λ k | | Z−∞ Xk=0 For more on spectral theory of sublaplacian on Heisenberg group, we refer to [17, Chapter 2]. 2.4. A transference theorem for m( ). Let Γ be the subgroup of Hd consisting of L 0 elements of the form (0, 0, 2πk) where k Z. Then Γ0 is a central subgroup and the quotient d d ∈ d H /Γ0 whose underlying manifold is R R T is called the Heisenberg group with compact × × d center.(See [17, Chapter 4]) Let 0 be the sublaplacian on H /Γ0. Note that 0f = f, considering f as a function on HdL. L L Note that on functions g which are independent of t, 0g(z) = ∆g(z) where ∆ is the d L − Euclidean Laplacian on C . The spectral decomposition of the sublaplacian 0 on this group is given by L

∞ 0 d 1 ikt k k d f(z, t)=( ∆)f (z)+(2π)− − e− ((2j + d) k )f ϕ (z) k L0 − | | ∗k j | | k Z 0 j=0 ∈X\{ } X k k k ikt k k where ϕ (z) = ϕ ( k z) and f is deﬁned by the equation f e (z, t) = e− f ϕ (z), j j | | ∗ j ∗k j ek(z, t)= eiktϕk(z). More generally, j j p ∞ 0 d 1 ikt k k d m( )f(z, t)= m( ∆)f (z)+(2π)− − e− m((2j + d) k )f ϕ (z) k . L0 − | | ∗k j | | k Z 0 j=0 ∈X\{ } X d d There is a transference theorem which connects m( ) on H with m( 0) on H /Γ0 which is the analogue of the classical de Leeuw’s theorem (SeeL Theorem 4.5) onL the real line. Theorem 2.2. Let 1

This theorem has been proved and used in [13]. The idea is to realise m( ) ( m( 0) ) as an operator valued Fourier multiplier for R (resp. T.). Indeed, writing the FourierL inversionL as 1 ∞ iλt λ f(z, t)=(2π)− e− f (z)dλ Z−∞ and recalling that L is deﬁned by the equation (f(z)eiλt)= eiλtL f(z) we have λ L λ 1 ∞ iλt λ m( )f(z, t)=(2π)− e− m(L )f (z)dλ L λ Z−∞ where m(Lλ) is the multiplier transform associated to the elliptic operator Lλ which has an explicit spectral decomposition. By identifying Lp(Hd) with Lp(R, Lp(Cd)) we see that, in view of the above equation, m( ) can be considered as an operator valued Fourier multiplier L for the Euclidean Fourier transform on R. Similarly, m( 0) can be thought about as an operator valued multiplier for the Fourier series on T. Now,L it is an easy matter to imitate the proof of de Leeuw’s theorem to prove the above result. We need the following analogue of Theorem 2.2 for the reduced polarised Heisenberg group. ˜ d Hd ˜ ˜ Let 0 be the sublaplacian on G = pol/Γ0. We note that 0f = f considering f as a L Hd L L function on pol. Theorem 2.3. Let 1

q/p 1/q p f M p,q = Vgf(x, y) dx dy , k k Rd Rd | | Z Z ! for 1 p,q < . If p or q is inﬁnite, f M p,q is deﬁned by replacing the corresponding integral≤ by the essential∞ supremum. k k 12 D.G.BHIMANI,R.BALHARA,ANDS.THANGAVELU

Remark 2.5. The deﬁnition of the modulation space given above, is independent of the choice of the particular window function. See [4, Proposition 11.3.2(c), p.233]. Next, we shall see how the Fourier-Wigner transform and the STFT are related. Let π be the Schr¨odinger representation of the Heisenberg group with the parameter λ = 1 which is realized on L2(Rd) and explicitly given by

it i(x ξ+ 1 x y) π(x, y, t)φ(ξ)= e e · 2 · φ(ξ + y) where x, y Rd, t R,φ L2(Rd). The Fourier-Wigner transform of two functions f,g L2(Rd) is deﬁned∈ by∈ ∈ ∈ d/2 W f(x, y)=(2π)− π(x, y, 0)f,g . g h i For z = x + iy, we put π(x, y, 0) = π(z, 0) = π(z). We may rewrite the Fourier-Wigner transform as W f(x, y)= f, π∗(z)g , g h i where f,g denotes the inner product for L2 functions, or the action of the tempered dis- h i tribution f on the Schwartz class function g. We recall the representation ρ1 (See Section d 1 it 2 d 2.2) of H , ρ = ρ = π Φ− , ρ(x, y, e ) acting on L (R ) is given by pol 1 ◦ it it ix ξ 2 d ρ(x, y, e )φ(ξ)= e e · φ(ξ + y), φ L (R ). ∈ We now write the Fourier-Wigner transform in terms of the STFT: Speciﬁcally, we put ix ξ ρ(x, y)φ(ξ)= e · φ(ξ + y), and have

i x y i x y (2.14) π(x, y)f,g = e 2 · ρ(x, y)f,g = e− 2 · V f(y, x). h i h i g − This useful identity (2.14) reveals that the definition of modulation spaces we have introduced in the introduction and in the present section is essentially the same. The following basic properties of modulation spaces are well-known and for the proof we refer the reader to [4, 3]. Lemma 2.6. (1) The space M p,q(Rd)(1 p ) is a Banach space. . M p,p(Rd) ֒ Lp(Rd) for 1 p 2, and≤ L≤∞p(Rd) ֒ M p,p(Rd) for 2 p (2) → ≤ ≤p1,q1 d p2,q→2 d ≤ ≤∞ .( If q1 q2 and p1 p2, then M (R ) ֒ M (R (3) (4) (Rd≤) is dense in≤M p,q(Rd)(1 p,q < →). S ≤ ∞ We also refer to Gröchenig’s book [4] for the basic definitions and further properties of modulation spaces. Finally, we note that there is also an equivalent definition of modulation spaces using frequency-uniform decomposition techniques (which is quite similar in the spirit of Besov spaces), independently studied by Wang et al. in [19], which has turned out to be very fruitful in PDE, see [20].

3. Hermite multipliers via transference theorems As mentioned in the introduction we prove our main results via transference techniques. q,p d We ﬁrst investigate the connection between m(H) acting on M (R ) and m( 0) acting on Lp(Gd) where Gd = Hd /Γ . Recall that f M q,p(Rd) if and only if ρ(x,L y)f, Φ pol 0 ∈ h 0i ∈ Lp(Rd, Lq(Rd)). Moreover, for any left invariant vector ﬁled X on Gd we have

X ρ(x, y, t)f, Φ = ρ(x, y, t)ρ∗(X)f, Φ h 0i h 0i HERMITE MULTIPLIERS ON MODULATION SPACES 13

d where ρ∗(X)ϕ = dt t=0ρ(exp(tX))ϕ. A simple calculation shows that ρ∗(Xj) = iξj and ∂ | ρ∗(Yj) = for j = 1, 2, ..., d. Consequently, we get ρ(x, y, t)f, Φ0 = ρ(x, y, t)Hf, Φ0 ∂ξj Lh i h i which leads to the identity m( ˜ ) ρ( )f, Φ = ρ(x, y, t)m(H)f, Φ L0 h · 0i h 0i via spectral theorem (See [16, Section 2.3] for details). Thus we see that ˜ (3.1) m( 0) ρ( )f, Φ0 Lp(Rd,Lq(Rd T)) = m(H)f M q,p(Rd). k L h · ik × k k q,p d Consequently, the boundedness of m(H) on M (R ) is implied by the boundedness of m( ˜0) on Lp(Rd, Lq(Rd T)). We can now prove Theorem 1.2. L ×

Proof of Theorem 1.2. We do this in two steps. Assuming that m( ˜) is bounded on Lp(Hd ) L pol it follows from Theorem 2.3 that m( ˜ ) is bounded on Lp(Hd /Γ). As the underlying mani- L0 pol fold of Hd /Γ is Rd Rd T we have the boundedness of m( ˜ ) on the space Lp(Rd, Lp(Rd pol × × L0 × T)). If we can show that under the assumptions on p and q stipulated in Theorem 1.2, the operator m( ˜ ) is bounded on the mixed norm space Lp(Rd, Lq(Rd T)) then we are done L0 × simply by applying m( ˜0) to π(z, t)f, Φ0 (see (3.1)). To this end, we make use of the following result of HerzL and Rivi´ereh [9]. i Theorem 3.1 (Herz-Rivi`ere [9]). Let G = ΓH be the semi-direct product of an amenable group H by a locally compact group Γ. Assume that a bounded linear operator T : Lp(G) Lp(G) commutes with left-translations. Take q such that 1 p q 2 or 2 q p< →. Then for any complex-valued continuous function f of compact≤ ≤ support≤ on G,≤we≤ have ∞

T f p q . f p q . k kL (Γ,L (H)) k kL (Γ,L (H)) Consequently, T has a bounded extension to the mixed norm spaces Lp(Γ, Lq(H)). In view of the above theorem all we have to do is to realise Gd as a semidirect product of d d R with R T. As m( 0) is invariant under left translations, we can then apply the above theorem to arrive× at theL required conclusion. Let us briefly recall the definition of a semidirect product for the convenience of the readers. Let H be a topological group whose operation is written as + and Γ another topological group, written as multiplicatively, such that there is a continuous mapΓ H H: (σ, x) σx with σ(x + y)= σx + σy and τ(σx)=(τσ)x. In this situation, we say× that→ Γ acts on H.7→ The semi-direct product ΓH is then the topological space Γ H with the group operation × (σ, x) (τ,y)=(στ,τx + y). · Let Γ = (Rd, +) be the additive group and H = (Rd T, ) be the group with following group law: × · it it′ i(t+t′) (y, e ) (y′, e )=(y + y′, e ). · We define a map Γ H H as (x, (y, eit)) (y, ei(t+xy)). We note that via this map, Γ acts on H. If there is× no confusion,→ we write (x,7→(y, eit))=(x, y, eit) for (x, (y, eit)) Γ H. Forming the semi-direct product G =ΓH we see that the group law is given by ∈ × it it′ i(t+t′+x′.y) (x, y, e )(x′,y′, e )=(x + x′,y + y′, e ). 14 D.G.BHIMANI,R.BALHARA,ANDS.THANGAVELU

d Hd This is precisely the group G = pol/Γ. Hence, by Theorem 3.1 we get the boundedness of p d q d m( 0) on the mixed norm space L (R , L (R T)). This completes the proof of Theorem 1.2.L ×

Proof of Theorem 1.1. Using Theorems 2.1 and 2.3, we conclude that m( ˜0) is bounded on p Hd L L ( pol). We now apply Theorem 1.2, to complete the proof.

4. Hermite multipliers via Fourier multipliers on torus In the last section, we have proved the boundedness of Hermite multiplier on modulation spaces via transference results. Specifically, we have proved that the boundedness of multiplier operators on Heisenberg groups ensures the boundeness of corresponding Hermite multiplier operators on modulation spaces. In this section, we first we prove the transference result which connects Hermite multipliers on modulation spaces and Fourier multipliers on Lp(Td). Specifically, our result (Proposition 4.3) states that boundedness of Fourier multipliers on torus guarantees the boundedness of Hermite multipliers on modulation spaces. In order to find a fruitful application of our result, we need to know the boundedness of Fourier multiplier on Lp(Td). To this end, we prove(Proposition 4.6) boundedness of Fourier multiplier on Lp(Rd) and then use the celebrated theorem of de Leeuw to come back to the Fourier multiplier on −Lp(Td). Combining these results, finally in this section we prove Theorem 1.6. For the sake of convenience of the reader, we recall definitions: Definitions 4.1 (Hermite multipliers on M p,q(Rd)). Let m be a bounded function on Nd. We say that m is a Hermite multiplier on the space M p,q(Rd) if the linear operator defined by T f = m(α) f, Φ Φ , (f (Rd)) m h αi α ∈ S α Nd X∈ p,q d extends to a bounded linear operator from M (R ) into itself, that is, T f p,q . f p,q . k m kM k kM Definitions 4.2 (Fourier multiplier on Lp(Td)). Let m be a bounded measurable function d p d defined on Z . We say that m is a Fourier multiplier on L (T ) if the linear operator Tm defined by \ (T f)(α)= m(α)fˆ(α), (f (Td),α Zd) m ∈ P ∈ ˆ iθ α where f(α)= Td f(θ)e− · dθ are the Fourier coefficients of f, extends to a bounded linear p d operator from L (T ) into itself, that is, T f p Td . f p Td . R k m kL ( ) k kL ( ) Now we prove a transference result for Fourier multiplier on torus and Hermite multiplier on modulation spaces which is of interest in itself. Specifically, we have the following proposition. − Proposition 4.3. Let 1 p < . If m : Zd C is a Fourier multiplier on Lp(Td), then ≤ d ∞ → p,p d m Nd , the restriction of m to N , is a Hermite multiplier on M (R ). | Proof. Since (Rd) is dense (Lemma 2.6(4)) in M p,p(Rd), it would be sufficient to prove S (4.1) T f p,p . f p,p k m kM k kM d for f (R ). In fact, by density argument, it follows that Tm has a bounded extension to M p,p(R∈d S), that is, inequality (4.1) holds true for f M p,p(Rd). ∈ HERMITE MULTIPLIERS ON MODULATION SPACES 15

Let f (Rd). Then using special Hermite functions (2.3) and their property (2.5), we compute∈ the S modulation space norm (See Section 2.5 ) of Hermite multiplier operator (See Deﬁnition 4.1), and we obtain

p dp/2 p Tmf M p,p = (2π)− π(x, y)Tmf, Φ0 dy dx k k R2d |h i| Z p dp/2 = (2π)− m(α) f, Φα π(x, y)Φα, Φ0 dy dx R2d d h ih i Z α N X∈ p

By using polar coordinates z = r eiθj , r := z [0, ), z C and θ [0, 2π), we get j j j | j| ∈ ∞ j ∈ j ∈

α α iα θ (4.3) z = r e · and dz = r r r dθdr 1 2 ··· d where r =(r , ,r ), θ =(θ , , θ ), dr = dr dr , dθ = dθ dθ , r = d r2. 1 ··· d 1 ··· d 1 ··· d 1 ··· d | | j=1 j Cd R2d By writing the integral over = in polar coordinates in each time-frequencyqP pair and using (4.3), we have

i|α|rα 1 r 2 We put a = f, Φ e− 4 | | . Since f, Φ f 2 Φ 2 f , the series α h αi √α! 2|α|/2 |h αi| ≤ k kL k αkL ≤ k k2 p Td α Nd aα converges. Thus there exists a continuous function g L ( ) with Fourier ∈ | | d d d ∈ coeﬃcientsg ˆ(α)= aα for α N andg ˆ(α)=0 for α Z N . In fact, the Fourier series of Pthis g is absolutely convergent,∈ and therefore we may∈ write\

iθ iα θ (4.5) g(e )= aαe− · . α Zd X∈ Since m is a Fourier multiplier on Lp(Td), (4.5) gives

p p iα θ iα θ (4.6) m(α)aαe · dθ . aαe− · dθ. d d [0,2π] d [0,2π] d Z α N Z α N X∈ X∈

16 D.G.BHIMANI,R.BALHARA,ANDS.THANGAVELU

Now taking (4.2), (4.4), and (4.6) into account, we obtain

= f, Φα Φα,0(z) dz Cd h i Z α Nd ! X∈ p 1 p d/ 2 = (2π)− f, Φα π(x, y)Φα, Φ0 dy dx R2d h ih i d ! Z α N X∈ = f M p,p . k k d Thus, we conclude that Tmf M p,p . f M p,p for f (R ). This completes the proof of Proposition 4.3. k k k k ∈ S

Next we recall the celebrated theorem of de Leeuw, which gives the relation between Fourier multipliers on Euclidean spaces and tori. To do this, we start with Definitions 4.4 (Fourier multiplier on Lp(Rd)). Let m be a bounded measurable function d p d defined on R . We say that m is a Fourier multiplier on L (R ) if the linear operator Tm defined by \ (T f)(ξ)= m(ξ)fˆ(ξ), (f (Rd), ξ Rd) m ∈ S ∈ where fˆ is the Fourier transform, extends to a bounded linear operator from Lp(Rd) into itself, that is, T f p . f p . k m kL k kL Theorem 4.5 (de Leeuw). If m : Rd C is continuous and Fourier multiplier on Lp(Rd)(1 → d p d ≤ p< ), then m Zd , the restriction of m to Z , is a Fourier multiplier on L (T ). ∞ | In order prove Theorem 1.6, we also need the boundedness of a following Fourier multiplier

γ ei(2 ξ 1+d) (4.7) m(ξ)= | | (β > 0,γ > 0,ξ Rd) (2 ξ + d)β ∈ | |1 where ξ := d ξ , on Lp(Rd). This we shall prove in the next proposition. We note | |1 j=1 | j| that in [14, Theorem 1], it is proved that the function m(ξ) which is 0 near the origin and P β i ξ α d 2 p d ξ − e | | ( ξ = ξ ) outside a compact set is a Fourier multiplier on L (R ) for a | | | | j=1 j suitable choice of α,β,p,d. Our proof of Proposition 4.6 is motivated by this result. qP Proposition 4.6. Let m be given by (4.7). Then m is a Fourier multiplier on Lp(Rd) for 1 1 < β . | p − 2 | dγ To prove this proposition we need the following technical lemma. We will prove this lemma at the end. HERMITE MULTIPLIERS ON MODULATION SPACES 17

Lemma 4.7. Let σ> 0,γ > 0 and

γ γ 1 i(2 ξ 1+d) σ(2 ξ 1+d) ix ξ d | | − | | · R (4.8) kσ(x)= d/2 e e e dξ (x ). (2π) Rd ∈ Z Then d/2 1 σdγ k 1 . σ− e− 2 . k σkL Proof of Proposition 4.6. Performing a simple change of variables in the gamma function, we write

β 1 ∞ β 1 γ (4.9) (2 ξ + d)− = σ γ − exp( σ(2 ξ + d) ) dσ. | |1 Γ(β/γ) − | |1 Z0 In view of Deﬁnition 4.4, (4.7) and (4.9), we write

d/2 ix ξ Tmf(x) = (2π)− m(ξ)fˆ(ξ)e · dξ Rd Z 1 ∞ (β/γ) 1 (4.10) = σ − (k f)(x) dσ Γ(β/γ) σ ∗ Z0 where kσ is as deﬁned in Lemma 4.7. Using (4.10), we have

1 ∞ (β/γ) 1 (4.11) T f p 6 σ − k f p dσ. k m kL Γ(β/γ) k σ ∗ kL Z0 p The boundedness of the multiplier operator Tmf on L will follow if we could show that p the operator f kσ f is bounded on L . We shall achieve this by using Riesz-Thorin interpolation theorem→ ∗ and the standard duality argument. To this end, we use Lemma 4.7 and Young’s inequality, to obtain d/2 1 σdγ (4.12) k f 1 . σ− e− 2 f 1 . k σ ∗ kL k kL 1 σdγ Since kˆ (ξ) 6 e− 2 , Plancherel theorem gives | σ | 1 σdγ (4.13) k f 2 6 e− 2 f 2 . k σ ∗ kL k kL Taking (4.12), and (4.13) into account, Riesz-Thorin interpolation theorem gives

λ 1 σdγ (4.14) k f p 6 C σ− e− 2 f p k σ ∗ kL 1 k kL where λ = d 1 1 , for 1 p 2. Finally using (4.14) and (4.11), we see that T is p − 2 ≤ ≤ m bounded fromLp(Rd) Lp(Rd) if → ∞ β/γ 1 λ 1 σdγ σ − − e− 2 dσ < , ∞ Z0 which happens if and only if 1 1 < β . This proves the theorem for the case when 1 2 follows from the duality. Now we shall prove our Lemma 4.7.

Proof of Lemma 4.7. Since kσ (see (4.8)) is the inverse Fourier transform of the function exp ((i σ)(2 ξ + d)γ), we have − | |1 γ (i σ)(2 ξ 1+d) (4.15) kσ(ξ)= e − | | .

b 18 D.G.BHIMANI,R.BALHARA,ANDS.THANGAVELU

Nd d Let α = (α1, ,αd) be a multi-index of length l, that is, j=1 αj = l, and put α1 ··· ∈αd α ∂ ∂ d D = , ξ =(ξ1, ..., ξd) R . Then in view of (4.15), we have ξ ∂ξ1 ··· ∂ξd ∈ P 1 γ 1 γ α l(1 γ) σ(2 ξ 1+d) σd (4.16) D k (ξ) . (2 ξ + d)− − e− 2 | | e− 2 . | ξ σ | | |1 To obtain inequality (4.16), we have used these ideas: After taking partial derivatives of b higher order for kσ, we have estimated all the powers of (2 ξ 1 + d) by the highest appearing l(1 γ) | | power which is (2 ξ 1 + d)− − , this we could do because (2 ξ 1 + d) > 1. We have also 1 γ 1 γ − σ(2 ξ |1+|d) σd | | dominated e− 2 b| | by e− 2 , which is obvious. Next, by Plancherel’s theorem and (4.16), we get

1 2 γ 1 γ l 2l(1 γ) σ(2 ξ 1+d) σd x kσ L2 . (2 ξ 1 + d)− − e− | | dξ e− 2 . k| | k Rd | | Z Performing a change of variable, we get 1/2 l(1−γ) l d + 1 σdγ 1/γ x kσ L2 . σ− 2γ γ e− 2 g(2 ξ 1 + σ d) dξ , k| | k Rd | | Z 2l(1 γ) tγ 1/γ where g(t)= t− − e− (t> 0). Noting that g(2 ξ 1 + σ d) dξ < for all γ and l, we conclude | | ∞ l(1−γ) l dR+ 1 σdγ x k 2 . σ− 2γ γ e− 2 . k| | σkL Next we use the fact that for h L2(Rd) L1(Rd), l >d/2 and R> 0, we have ∈ ∩ d/2 l l h 1 . R ( h 2 + R− x h 2 ) k kL k kL k| | kL 1−γ for all R> 0. Taking h = kσ and R = σ γ , we obtain

d/2 1 σdγ k 1 6 Cσ− e− 2 . k σkL This completes the proof. We now use Proposition 4.6 and Theorem 4.5 to obtain the following corollary.

Corollary 4.8. Let 1 1 < β , β> 0,γ > 0, α = d α . Then the sequence p − 2 d γ | |1 j=1 | j| β γ p Td (2 α 1 + d)− exp(i(2 α 1 + d) ) α Zd deﬁnes a multiplier on L ( ). ∈ P { | | | | } Proof of Theorem 1.6. Using Corollary 4.8 and Proposition 4.6, we may deduce that m(H) is bounded on M p,p(Rd). This completes the proof.

5. Hermite multiplier for Schrodinger¨ propagator In this section, we prove the boundedness of Schr¨odinger propagator m(H) = eitH using the properties of Hermite and special Hermite functions. Our approach of proof illustrates how these functions nicely ﬁt into modulation spaces and prove useful estimate. − Proof of Theorem 1.7. Let f (Rd). Then we have the Hermite expansion of f as follows: ∈ S (5.1) f = f, Φ Φ . h αi α α Nd X∈ HERMITE MULTIPLIERS ON MODULATION SPACES 19

Now using (5.1) and (2.3), we obtain π(z)f, Φ = f, Φ π(z)Φ , Φ h 0i h αih α 0i α Nd X∈ (5.2) = f, Φ Φ (z). h αi α,0 α Nd X∈ Since Φ forms an orthonormal basis for L2(Rd), (5.2) gives { α} π(z)m(H)f, Φ = m(H)f, Φ Φ (z) h 0i h αi α,0 α Nd X∈ = m(2 α + d) f, Φ Φ (z). | | h αi α,0 α Nd X∈ Therefore, for m(H)= eitH , we have

α p 2it α 1/2 i | | α 1 z 2 (5.5) e | | f, Φα (α!)− z¯ e− 4 | | dz Cd h i √2 Z α Nd X∈ d α p | | d 1 2 1/2 i α i P (2t θj )αj r = f, Φα (α !)− r e j=1 − e− 4 | | rjdrj dθj. R+ √ [0,2π] dh i 2 j=1 Z Z α N Y X∈ By a simple change of variable (θ 2t) θ , we obtain j − → j d α p | | d 1 2 1/2 i α i P (2t θj )αj r (5.6) f, Φα (α!)− r e j=1 − e− 4 | | rjdrjdθj R+ [0,2π] h i √2 j=1 Z Z α Nd Y X∈ d α p 1/2 i | | α iθ α 1 r 2 = f, Φα (α!)− r e− · e− 4 | | rjdrjdθj. R+ [0,2π] h i √2 j=1 Z Z α Nd Y X∈ itH Combining (5.4), (5.5), (5.6), and Lemma 2.6(4), we have e f p,p = f p,p for f M M M p,p(Rd). This completes the proof of Theorem 1.7. k k k k ∈ Acknowledgment. The work leading to this article began while DGB was a project assis- tant with Professor Thangavelu in IISc. DGB is very grateful to Professor Thangavelu for providing the funding and arranging research facilities during his stay at IISc. DGB is also thankful to DST-INSPIRE and TIFR CAM for the current support. RB wishes to thank 20 D.G.BHIMANI,R.BALHARA,ANDS.THANGAVELU

UGC-CSIR for ﬁnancial support. ST is supported by J C Bose National Fellowship from D.S.T., Govt. of India.

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(D. G. Bhimani) Centre for Applicable Mathematics (CAM), Tata Institute of Fundamental Research, 560012 Bangalore, India E-mail address: [email protected]

(R. Balhara) Department of Mathematics, Indian Institute of Science, 560 012 Bangalore, India E-mail address: [email protected]

(S. Thangavelu) Department of Mathematics, Indian Institute of Science, 560 012 Banga- lore, India E-mail address: [email protected]