Weyl multipliers for $(L^p, L^q)$

Abstract

We discuss the Weyl multipliers for pairs of Lebesgue spaces, and identify some sub classes of $B(L^2({\mathbb {R}}^n))$ which form Weyl multipliers for $(L^p, L^q), \, 1\le p,q \le \infty$. Similar results are also obtained for Lorentz spaces.

1 Introduction

The Weyl transform of an integrable function f on $\mathbb {C}^n$, is given by

$$\begin{aligned} W(f)= \int \limits _{z \in \mathbb {C}^n}f(z) \, \pi (z) \, dz, \end{aligned}$$

(1.1)

where $\pi (z)$ is the unitary operator on $L^2(\mathbb {R}^n)$ defined by

$$\begin{aligned} \pi (z) \varphi \, (\xi )=e^{i \, x \cdot (\xi + \frac{y}{2})}\varphi (\xi +y), ~ z =x+iy, ~ x, y \in {\mathbb {R}}^n. \end{aligned}$$

(1.2)

The operator valued integral on the right hand side of (1.1) is to be interpreted in the sense of Bochner: i.e., W(f) is the operator on $L^2(\mathbb {R}^n)$ given by

$$\begin{aligned} \langle {W(f) \varphi , \psi }\rangle := \int \limits _{z \in \mathbb {C}^n} f(z) \langle {\pi (z)\varphi , \psi }\rangle \, dz, \end{aligned}$$

(1.3)

for all $\varphi , \psi \in L^2(\mathbb {R}^n)$. Clearly for $f \in L^1({\mathbb {C}}^n)$, the above integral is absolutely convergent as $|\langle {\pi (z)\varphi , \psi }\rangle | \le \Vert \varphi \Vert _{L^2(\mathbb {R}^n)} \Vert \psi \Vert _{L^2(\mathbb {R}^n)},$ $\pi (z)$ being unitary. It follows that W(f) is a bounded operator on $L^2({\mathbb {R}}^n)$ with norm $\Vert W(f)\Vert _{op} \le \Vert f\Vert _{L^1({\mathbb {C}}^n)}$ in this case.

The expression for W(f) given by (1.3), allows the definition of $W(\Lambda )$ for any tempered distribution $\Lambda$ on ${\mathbb {C}}^n$ by replacing the integral by the pairing $(\,\,, \, )$ of a tempered distribution and a Schwartz class function: For $\varphi , \psi \in \mathcal {S}({\mathbb {R}}^n )$ setting

$$\begin{aligned} \langle W(\Lambda ) \varphi , \psi \rangle := ( \Lambda , \langle {\pi (z)\varphi , \psi }\rangle ) \end{aligned}$$

(1.4)

we see that $W(\Lambda ): \mathcal {S}({\mathbb {R}}^n) \rightarrow \mathcal {S}'({\mathbb {R}}^n)$ is a continuous linear map. Note that the pairing on the right hand side of (1.4) makes sense as the function $z \rightarrow \langle {\pi (z)\varphi , \psi }\rangle$ is in $\mathcal {S}({\mathbb {C}}^n)$ for $\varphi , \psi \in \mathcal {S}({\mathbb {R}}^n )$, and the continuity follows from the fact that $\langle \pi (z)\varphi _n, \psi \rangle \rightarrow 0$ in $\mathcal {S}({\mathbb {C}}^n)$ whenever $\varphi _n \rightarrow 0$ in $\mathcal {S}({\mathbb {R}}^n)$. Thus W(f) is defined for any $f \in L^p(\mathbb {C}^n), ~ 1 \le p \le \infty$ and moreover, W(f) is a Hilbert–Schmidt operator for $f \in L^2(\mathbb {C}^n)$, see [1, 2, 10]. Since any $f \in L^p(\mathbb {C}^n), 1\le p \le 2$ can be written as $f_1 + f_2$ with $f_r \in L^r({\mathbb {C}}^n), r=1,2$, it follows that W(f) defines a bounded operator on $L^2({\mathbb {R}}^n)$ whenever $f \in L^p(\mathbb {C}^n), 1\le p \le 2$.

One of the most interesting properties of the Weyl transform is its behaviour under twisted convolution:

$$\begin{aligned} W(f\times g)=W(f) \, W(g), \end{aligned}$$

(1.5)

where the twisted convolution of two functions f and g is defined by

$$\begin{aligned} f \times g(z)= \int \limits _{\mathbb {C}^n} f(z-w) \, g (w) \, e^{\frac{i}{2}Im (z. \overline{w})} dw \end{aligned}$$

(1.6)

whenever the integral converges. For $f,g \in \mathcal {S}({\mathbb {C}}^n)$, (1.5) follows from (1.6) in view of the relation

$$\begin{aligned} \pi (w+z)= e^{\frac{i}{2} \text {Im} z \cdot {\bar{w}} } \pi (w) \pi (z). \end{aligned}$$

(1.7)

The identities (1.5) and (1.6) can be extended to the case when f is a tempered distribution on ${\mathbb {C}}^n$ and $g \in \mathcal {S}({\mathbb {C}}^n)$, see Sect. 2.

Now we proceed to define the Weyl multipliers. As mentioned above, W(f) is a Hilbert–Schmidt operator for $f\in L^2({\mathbb {C}}^n)$. Since the space of Hilbert–Schmidt operators form a two sided ideal in $B(L^2({\mathbb {R}}^n))$, we have both MW(f) and W(f)M are Hilbert–Schmidt operators, for any $M \in B( L^2({\mathbb {R}}^n) )$. Since every Hilbert–Schmidt operator on $L^2({\mathbb {R}}^n)$ is given by an $L^2$ kernel, by Plancherel theorem for the Weyl transform, see [2] page 12, this gives rise to a bounded linear operator T on $L^2({\mathbb {C}}^n)$ via $M W(f)=W(Tf)$. This leads to the notion of (left) Weyl multipliers for $L^p$ studied by Mauceri [3].

More generally, we can also consider the Weyl multipliers for $(L^p, L^q )$ for $1 \le p,q\le \infty$.

Definition 1.1

An operator $M \in B(L^2({\mathbb {R}}^n))$ is said to be a left Weyl multiplier for the pair $(L^p, L^q),~ 1 \le p, q \le \infty$, if the linear operator T defined by

$$\begin{aligned} W(Tf) = M W(f) ~ \text {for}~ f \in L^2\cap L^p({\mathbb {C}}^n) \end{aligned}$$

(1.8)

satisfies the inequality $\Vert Tf\Vert _{L^q({\mathbb {C}}^n)} \le C \, \Vert f\Vert _{L^p({\mathbb {C}}^n)}$ with some constant C valid for all $f \in L^2\cap L^p({\mathbb {C}}^n)$. In this case we write $M \in \mathcal {M}_p^q$, the space of all left Weyl multiplier operators for the pair $(L^p, L^q)$. When $p=q$ we write $\mathcal {M}_p$ instead of $\mathcal {M}_p^p$. The right Weyl multipliers are defined similarly by the condition $W(Tf) = W(f)M$.

If M is a Weyl multiplier for the pair $(L^p,L^q)$ with $p<\infty$, then $T|_{L^2({\mathbb {C}}^n) \cap L^p({\mathbb {C}}^n) }$ extends uniquely as a bounded operator from $L^p ({\mathbb {C}}^n)$ to $L^q ({\mathbb {C}}^n)$, by density of $L^2\cap L^p({\mathbb {C}}^n)$ in $L^p({\mathbb {C}}^n)$, for $1 \le p < \infty$.

By the discussion before Definition 1.1, it is clear that the space of (left/right) Weyl multipliers for the pair $(L^2({\mathbb {C}}^n), L^2({\mathbb {C}}^n))$ is precisely $B(L^2({\mathbb {R}}^n))$, in analogy with the Fourier multipliers. Similarly, the Weyl multipliers for the pair $(L^1({\mathbb {C}}^n), L^1({\mathbb {C}}^n))$ are characterised as operators of the form $M =W(\mu )$ for some finite measure $\mu$ on ${\mathbb {C}}^n$. This also follows from the Riesz representation theorem as in the Fourier multiplier case, see [4]. These are well known facts, already observed by Mauceri in [3], where he established an analogue of Hörmander’s multiplier theorem for the Weyl multipliers for $(L^p, L^p)$, via singular integral theory.

In this article, we discuss the Weyl multipliers for pairs of Lebesgue spaces $(L^p, L^q), 1 \le p, q \le \infty ,$ and also for pairs of Lorentz spaces. It turns out that the Weyl multipliers for $(L^p, L^q)$ are also of the form W(K) for some $K \in \mathcal {S}'({\mathbb {C}}^n)$, see Proposition 4.1.

The main results in this article are Theorems 3.6, 4.2 and 5.2. Theorem 3.6 describes all Weyl multipliers for $(L^p, L^q)$ as twisted convolution operators and in Theorem 4.2, we identify pairs (p, q) for which operators of the form $M=W(K) \in B(L^2 ({\mathbb {R}}^n))$, with $K \in L^{p_0}({\mathbb {C}}^n), p_0 \in [1, \infty ]$, is a Weyl multiplier for $(L^p, L^q)$. Theorem 5.2 is a similar result in the Lorentz space setting.

2 Basic properties of Weyl transform

We start with reviewing some basic properties of the Weyl transform. As mentioned above, for $f \in L^2({\mathbb {C}}^n)$, W(f) is a Hilbert–Schmidt operator and the Plancherel identity

$$\begin{aligned} \Vert W(f) \Vert _{HS} = (2\pi )^{n/2} \, \Vert f \Vert _{L^2({\mathbb {C}}^n)}, ~ f\in {\mathcal {S}}({\mathbb {C}}^n) \end{aligned}$$

(2.1)

holds, where $\Vert \cdot \Vert _{HS}$ denotes the Hilbert–Schmidt norm, see [2] page 12.

The Weyl transform has many interesting properties like the Fourier transform. For $z \in {\mathbb {C}}^n$, consider the twisted translate of a function f defined by

$$\begin{aligned} \tau _zf (w) = f (w-z)\, e^{\frac{i}{2} \text {Im} (w \cdot {\bar{z}})}. \end{aligned}$$

(2.2)

The convolution associated with this translation is the twisted convolution defined in (1.6):

$$\begin{aligned} f \times g(z) = \int \limits _{\mathbb {C}^n}f(w) \, g(z-w)\, e^{-\frac{i}{2}Im (z. \overline{w})} \, dw = \int \limits _{\mathbb {C}^n} f(w) \, \tau _z\tilde{g} (w) \, dw, \end{aligned}$$

(2.3)

where $\tilde{g}(w)= g(-w)$.

A general result on the mapping properties of the bi-linear twisted convolution operator $B: (f,g) \rightarrow f \times g$ between Lebesgue spaces is obtained in [5], (Theorem 1.1) which we restate as

Theorem A. Let $f \in L^{p_1}({\mathbb {C}}^n), g \in L^{p_2}({\mathbb {C}}^n), 1\le p_1,p_2\le \infty$. The inequality

$$\begin{aligned} \Vert f \times g\Vert _{L^q(\mathbb {C}^n)} \le C \Vert f \Vert _{L^{p_1}(\mathbb {C}^n)} \Vert g\Vert _{L^{p_2}(\mathbb {C}^n)} \end{aligned}$$

holds with a constant C independent of f, g, for a triple $(p_1, p_2, q), ~1 \le p_1, p_2,q \le \infty$, if and only if $0 \le \frac{1}{p_1}+ \frac{1}{p_2}-1 \le \frac{1}{q} \le \text {min} \{ \frac{1}{p_1}, \frac{1}{p_2} \}$.

For a given $g \in L^{p_2}({\mathbb {C}}^n), 1 \le p_2 \le \infty$, the above theorem shows that the operator $T_g: f \rightarrow f \times g$ maps $L^{p_1}({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n)$, with the boundedness estimate $\Vert T_gf\Vert _{L^q({\mathbb {C}}^n)} \le C \Vert f\Vert _{L^{p_1}({\mathbb {C}}^n)} \Vert g\Vert _{L^{p_2}({\mathbb {C}}^n)}$ when $\frac{1}{p_1}+ \frac{1}{p_2}\ge 1.$

In view of (2.3), we can also define $\Lambda \times g$ for $\Lambda \in S^{\prime }(\mathbb {C}^n)$ and $g \in S(\mathbb {C}^n)$ by setting

$$\begin{aligned} \Lambda \times g (z)= (\Lambda , \tau _z \tilde{g} ) \end{aligned}$$

(2.4)

and the identity

$$\begin{aligned} W(\Lambda \times g)= W(\Lambda ) W(g) \end{aligned}$$

(2.5)

holds, see [6]. Note that the right hand side of (2.4) is a bounded function on $\mathbb {C}^n$ and (2.4) coincides with (2.3) when $\Lambda$ is ‘nice’ a function, say locally integrable with at most polynomial growth at infinity.

3 Weyl multipliers and twisted convolution operators

In this section, we show that if M is a Weyl Multiplier for the pair $(L^p, L^q)$, then the associated operator T given by (1.8) is a twisted convolution operator. This is based on the observation that T is twisted translation invariant, see Proposition 3.3 below. We start with the following

Definition 3.1

An operator $T:L^p({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n), 1 \le p,q \le \infty ,$ is said to be twisted translation invariant, if for each $z \in {\mathbb {C}}^n$, the identity $T(\tau _z f) = \tau _z (Tf)$ holds for all $f \in \mathcal {S}({\mathbb {C}}^n),$ where $\tau _z$ is the twisted translation operator given by (2.2)

Remark 3.2

If $T: L^p({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n)$ is bounded with $1\le p<\infty$, then the above definition coincides with the commutativity of T with $\tau _z$ for each $z\in {\mathbb {C}}^n$, i.e., $T(\tau _z f) = \tau _z (Tf)$ for all $f \in L^p({\mathbb {C}}^n)$, $z\in {\mathbb {C}}^n$. This follows, from the density of $\mathcal {S}$ in $L^p({\mathbb {C}}^n)$ for $1\le p<\infty$ and the continuity of the twisted translation operators in $L^p$. For $p=\infty$, we only demand the commutativity of $T|_{\mathcal {S}({\mathbb {C}}^n)}$ with $\tau _z$. This definition will be found convenient in Sect. 4.

Proposition 3.3

Let $M \in B(L^2({\mathbb {R}}^n))$ and $T:L^p({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n), 1 \le p, q \le \infty$ be a continuous linear map that satisfy the identity $W(Tf) = M W(f)$ for all $f \in \mathcal {S}({\mathbb {C}}^n)$. Then T is twisted translation invariant.

Proof

For any continuous linear map $T: \mathcal {S}({\mathbb {C}}^n) \rightarrow \mathcal {S}'({\mathbb {C}}^n)$ that satisfies (1.8), this has been proved in [6]. Since T is bounded from $L^p({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n)$, we have $T|_{\mathcal {S}({\mathbb {C}}^n)}: \mathcal {S}({\mathbb {C}}^n) \rightarrow \mathcal {S}'({\mathbb {C}}^n)$ is continuous, in view of the continuity of the inclusions $\mathcal {S}({\mathbb {C}}^n) \hookrightarrow L^p({\mathbb {C}}^n)$ and $L^q({\mathbb {C}}^n) \hookrightarrow \mathcal {S}'({\mathbb {C}}^n)$. Hence the proof. $\square$

Next we characterize all twisted translation invariant operators from $L^p \rightarrow L^q$. The proof essentially follows by the approach of Hörmander in [4], on translation invariant operators. However, it requires an analogue of Lemma 1.1 in [4], expressed in terms of the differential operators

$$\begin{aligned} X_j= -\partial _{x_j} + \frac{i}{2}y_j, ~ Y_j= \partial _{y_j} +\frac{i}{2}x_j, ~j =1,2,\dots n. \end{aligned}$$

(3.1)

Note that $X_j$ and $Y_j$ are respectively, the infinitesimal generators of the one parameter groups of twisted translations $\{\tau _{tu_j} \}_{ t\in {\mathbb {R}}}$ and $\{\tau _{tv_j} \}_{ t \in {\mathbb {R}}},$ in the directions $u_j = (\textbf{0},\ldots , \textbf{1}, \dots , 0)$ and $v_j = (\textbf{0},\ldots ,\mathbf {-i}, \dots \mathbf{},\textbf{0})$ in $C^n$, with only the jth complex co-ordinate non zero.

Lemma 3.4

Let g be a measurable function on ${\mathbb {C}}^n$ which together with all it’s derivatives upto order $\le 2n$, with respect to $X_j$ and $Y_j, j=1,,2\dots ,n$ are in $L^p({\mathbb {C}}^n)$ for some $p \in [1,\infty )$. Then g is continuous after changing its values on a set of measure zero, and there is a constant C such that the inequality

$$\begin{aligned} |g(0)| \le C \sum _{|\alpha | \le n } \sum _{ |\beta | \le n } \left( \int _{|w|\le 1} |X^{\alpha }Y^{\beta } g|^p dw \right) ^{1/p} \end{aligned}$$

holds, where $X^{\alpha } = X_1^{\alpha _1}\dots ,X_n^{\alpha _n},$ $Y^{\beta }= Y_1^{\beta _1}\dots , Y_n^{\beta _n}$ for $\alpha , \beta \in \mathbb {N}_0 ^n$ with $\mathbb {N}_0=\mathbb {N} \cup \{0\}$.

Proof

Note that the above estimate does not follow from the estimate in [4], but the proof follows essentially by the same idea. We sketch a proof here for the sake of completeness.

We first assume that g is supported in the open ball $\{z \in {\mathbb {C}}^n: |z| < 1\}$. Writing $g = \delta \times g$, and expressing the Dirac measure $\delta$ as $\partial _{v_1} \dots \partial _{v_n} \partial _{u_1}\dots \partial _{u_n} H$, where for $w \in {\mathbb {C}}^n$, $H(w) = \Pi _{j=1}^n h(u_j) h(v_j)$ is the 2n fold tensor product of the Heaviside function h on ${\mathbb {R}}$ with $w_j = u_j +i v_j$, and an integration by parts yields,

$$\begin{aligned} g(z)= \int _{|z-w|\le 1} \partial _{u_n} \dots \partial _{u_1} \partial _{v_n} \dots \partial _{v_1} \left[ g (z-w) e^{-\frac{i}{2} \text {Im}(z \cdot \bar{w} )} \right] h(w)\, du dv \end{aligned}$$

(3.2)

for a.e $z\in {\mathbb {C}}^n$ such that $|z| <1$. Clearly for $|z|<1$, only the values of the integrand on $|w| \le 2$ matters in the right hand side of the above integral, as g is supported on $|w| \le 1$, by assumption.

Note that the right hand side of (3.2) is a continuous function of z. So by redefining on a set of measure 0, we can assume that g is continuous.

Using the relations $\partial _{u_j} = -X_j + \frac{i}{2} v_j$ and $\partial _{v_j} = Y_j + \frac{i}{2} u_j$ for $j=1,2,\dots n$, iteratively in (3.2) and the fact that $|u_j|\le 2$ and $|v_j|\le 2$, we get

$$\begin{aligned} |g(0)|\le & {} \tilde{C} _{n} \sum _{|\alpha | \le n } \sum _{ |\beta | \le n } \int _{{\mathbb {C}}^n} \left| \left( X^\alpha Y^\beta \tilde{g}\right) h (w) \right| dw \nonumber \\\le & {} C_{n} \sum _{|\alpha | \le n } \sum _{ |\beta | \le n } \left( \int _{|w| \le 1} \left| X^\alpha Y^\beta \, \tilde{g}\right| ^p dw \right) ^{1/p} \end{aligned}$$

(3.3)

by Hölder’s inequality, since g is supported in $|w| \le 1$.

If g is not supported in the unit ball around origin in ${\mathbb {C}}^n$, then it can be reduced to this case by considering $g_1 =\varphi g$ for some $\varphi \in C_c^\infty (|w| \le 1)$, with $\varphi (0)= 1.$ The only difference in the estimate is that the constant C will depend on the bound for $\varphi$ and its derivatives up to order 2n. $\square$

Theorem 3.5

Let $1 \le p,q \le \infty$ and $T:L^p ({\mathbb {C}}^n ) \rightarrow L^q({\mathbb {C}}^n)$ be a continuous linear map. Then T is invariant under twisted translations if and only if there exists a tempered distribution K on ${\mathbb {C}}^n$ such that $Tf = K \times f$ for all $f \in {{\mathcal {S}}} ({\mathbb {C}}^n).$

Proof

Since $K \times \tau _{z}f = \tau _z(K \times f)$, as can be easily verified for $K \in \mathcal {S}'({\mathbb {C}}^n)$ and $f \in \mathcal {S}({\mathbb {C}}^n)$, we see that operators of the form $T: f \rightarrow K \times f$ are clearly twisted translation invariant for any $K \in \mathcal {S}'({\mathbb {C}}^n)$. Hence one implication follows, including the case $q=\infty$ in view of Definition 3.1 and Remark 3.2.

To show that any twisted translation invariant operator T is of the form $T f = K \times f$ for some $K \in \mathcal {S}'({\mathbb {C}}^n)$, we use Lemma  3.4, and show that $K: f \rightarrow T\tilde{f}\, (0)$ is the required distribution.

We will show that Tf(0) makes sense for all $f \in \mathcal {S}({\mathbb {C}}^n),$ and that the map $f \rightarrow Tf \, (0)$ defines a continuous linear functional on $\mathcal {S}({\mathbb {C}}^n)$. This will show that the map $K: f \rightarrow T \tilde{f} \, (0)$ is a continuous linear functional on $\mathcal {S}({\mathbb {C}}^n)$ and hence $Tf(0):= \langle K, \tilde{f} \rangle = K \times f(0)$. By twisted translation invariance of T, and the identity $(\tau _{-z} f \tilde{)}= \tau _{z} \tilde{f}$, it also follows that $Tf(z) = K \times f (z)$ for $z \in {\mathbb {C}}^n$.

We show that for $f \in \mathcal {S}({\mathbb {C}}^n)$, the derivatives $X^ {\alpha } Y^{ \beta } Tf$ exists and are in $L^q$ for all multi indices $\alpha =(\alpha _1,\dots ,\alpha _n)$ and $\beta =(\beta _1,\dots ,\beta _n)$ with $|\alpha | \le n, |\beta | \le n$. Then by Lemma 3.4, the function $v=Tf$ is continuous for $f \in \mathcal {S}({\mathbb {C}}^n)$, and the inequality

$$\begin{aligned} |v(0)| \le C_n \sum _{|\alpha | \le n} \sum _{|\beta | \le n} \int _{|x|^2 +|y|^2 \le 1}| X^\alpha Y^\beta Tf(x,y)| \, dx_1 \dots dx_n dy_1\dots dy_n \end{aligned}$$

(3.4)

holds with a constant $C_n$ valid for all $f \in \mathcal {S}({\mathbb {C}}^n)$.

We first show that the derivatives $X_j Tf$ and $Y_j Tf$ are in $L^q({\mathbb {C}}^n),$ for $j=1,2,\dots , n$, whenever $f \in \mathcal {S}({\mathbb {C}}^n)$. Since T is linear and twisted translation invariant, we have

$$\begin{aligned} \frac{\tau _{tu_j} Tf(z) -Tf(z) }{t}= T\left( \frac{\tau _{tu_j} f(z) -f(z) }{t}\right) \end{aligned}$$

(3.5)

Note that for $f \in \mathcal {S}({\mathbb {C}}^n)$, we have $\frac{\tau _{tu_j} f(z) -f(z) }{t} \rightarrow X_j f$ in $\mathcal {S}({\mathbb {C}}^n)$, and hence in $L^p({\mathbb {C}}^n), 1 \le p \le \infty$ as $t \rightarrow 0$. Hence, by the boundedness of $T: L^p({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n)$, we see that the left hand side of (3.5) converges in $L^q({\mathbb {C}}^n)$ as $t \rightarrow 0$, and we get $X_j Tf= T(X_j f) \in L^q({\mathbb {C}}^n)$ for $f \in L^p({\mathbb {C}}^n), 1 \le p\le \infty$. Similarly $Y_j Tf= T(Y_j f) \in L^q({\mathbb {C}}^n)$, for $j=1,2,\dots ,n$.

Hence by boundedness of $T: L^p({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n)$, there exist a constant $C_N$ such that the following inequalities hold

$$\begin{aligned} \Vert X_j Tf \Vert _{L^q({\mathbb {C}}^n)} \le C_N \Vert X_j f \Vert _{L^p({\mathbb {C}}^n)},~ \Vert Y_j Tf \Vert _{L^q({\mathbb {C}}^n)}\le C_N \Vert Y_j f\Vert _{L^p({\mathbb {C}}^n)}. \end{aligned}$$

(3.6)

for $j=1,2,\dots , n, ~ 1\le p,q\le \infty$, and for all $~ f \in \mathcal {S}({\mathbb {C}}^n)$.

For $1\le p< \infty$, iterations of the above two inequalities yield

$$\begin{aligned} \Vert X^\alpha Y^\beta Tf \Vert _{L^q({\mathbb {C}}^n)}\le & {} \left( \int |(X^\alpha Y^\beta f) (w)|^p dw \right) ^{1/p} \nonumber \\\le & {} \tilde{C}_{N,\alpha ,\beta } \Vert f \Vert _{N+|\alpha |+|\beta |}. \end{aligned}$$

(3.7)

for all multi indices $\alpha , \beta$, for any integer $N>2n/p$. Here $\Vert f \Vert _M$ above denotes the Schwartz norm given by

$$\begin{aligned} \Vert f \Vert _M = \sup _{|\alpha | \le M }| (1+|z|)^M \partial _{x,y}^\alpha f(z)\Vert _{L^\infty ({\mathbb {C}}^n)} \end{aligned}$$

and (3.7), is valid with Schwartz norm $\Vert f \Vert _{N+|\alpha |+|\beta |}$.

When $p=\infty$ we have $q=\infty$ in (3.6), as T is twisted translation invariant. Thus by iteration we get

$$\begin{aligned} \Vert X^\alpha Y^\beta Tf \Vert _{L^\infty ({\mathbb {C}}^n)} \le C_N \Vert X^\alpha Y^\beta f \Vert _{L^\infty ({\mathbb {C}}^n)} \le C_N \Vert f\Vert _{|\alpha |+|\beta |}, \end{aligned}$$

(3.8)

which settles the case $p= \infty$.

In view of (3.7) and (3.8), an application of Holder’s inequality on the right hand side of (3.4) yields

$$\begin{aligned} ~|v(0)| \le \sum _{|\alpha | \le n} \sum _{|\beta | \le n} \tilde{C}_{N, p, \alpha , \beta } \,\, \Vert f \Vert _{N+|\alpha |+|\beta |} \le C_{N,p, \alpha ,\beta } \, \Vert f \Vert _{4n+1} \end{aligned}$$

as $\Vert f \Vert _{N+|\alpha |+|\beta |}\le \Vert f \Vert _{N+2n} \le \Vert f \Vert _{4n+1}$ since $N> 2n/p \in (0,2n+1)$. This proves the continuity of the functional $f \rightarrow Tf(0)$ on $\mathcal {S}({\mathbb {C}}^n)$ and completes the proof. $\square$

Theorem 3.6

Let $M \in B(L^2({\mathbb {R}}^n))$ be a Weyl multiplier for the Lebesgue pair $(L^p({\mathbb {C}}^n), L^q({\mathbb {C}}^n)),$ $1 \le p, q \le \infty$ and let T be the associated operator given by (1.8). Then there exists a $K \in \mathcal {S}'({\mathbb {C}}^n)$ such that $Tf = K \times f$ for all $f \in \mathcal {S}'({\mathbb {C}}^n)$.

Proof

By Proposition 3.3, the associated operator $T: L^p({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n)$ given by (1.8) is twisted translation invariant. Hence T has the required form by Theorem 3.5. $\square$

4 Weyl multipliers for $(L^p, L^q )$

Recall that a bounded operator M on $L^2(\mathbb {R}^n)$ is called a left Weyl multiplier for the pair $(L^p, L^q)$, if the the operator T defined by $W(Tf)=MW(f),$ for $f \in L^2({\mathbb {C}}^n) \cap L^p({\mathbb {C}}^n)$ satisfies the inequality $\Vert Tf \Vert _{L^q({\mathbb {C}}^n)} \le C \Vert f\Vert _{L^p({\mathbb {C}}^n)}$ with a constant C valid for all $f \in L^2({\mathbb {C}}^n) \cap L^p({\mathbb {C}}^n)$. We start with the following observation analogous to the Fourier multipliers.

Proposition 4.1

Let $0 \not =M$ be a Weyl multiplier for $(L^p, L^q), 1 \le p,q \le \infty$. Then $p \le q$, and $M=W(K)$ for some $K \in \mathcal {S}'({\mathbb {C}}^n)$.

Proof

By Proposition 3.3, the associated operator $T=T_M$ given by (1.8) is twisted translation invariant. The argument of Hörmander ([4], Theorem 1.1), for translation invariant operator applies for twisted translation invariant operators as well, see [5]. Hence the condition $p\le q$ follows for all p, including the case $p=\infty$, in view of Definition 3.1.

By Theorem 3.6, there exists a $K\in \mathcal {S}'({\mathbb {C}}^n)$ such that $T_M f = K \times f$ for all $f \in \mathcal {S}({\mathbb {C}}^n)$. Since $W(K \times f ) = W(K) W(f)$ by (2.5), we see that $M W(f) =W(K) W(f)$ for all $f \in \mathcal {S}({\mathbb {C}}^n)$. In particular, taking $f =(2\pi )^{-n} \, \varphi _k^{n-1}$, where $\varphi _k^{n-1}$ are the Laguerre functions given by $\varphi _k^{n-1}(z) = L_k^{n-1}(|z|^2 /2) e^{-|z|^2 /4}$, we see that $M P_k =W(K) P_k$ for all $k \in \mathbb {N}_0$ as $W(\varphi _k) =P_k$, the special Hermite projections on $L^2({\mathbb {C}}^n)$, see [2], page 21. Since the above identity is valid for all $k \in \mathbb {N}_0$, we see that $M-W(K)=0$ in $B(L^2({\mathbb {R}}^n))$, as asserted. $\square$

By the above result, any Weyl multiplier for the pair $(L^p, L^q )$ is of the form W(K) for some tempered distribution K on ${\mathbb {C}}^n$. When $K\in L^{p_0}({\mathbb {C}}^n)$, we give a description of the pairs $(L^p({\mathbb {C}}^n), L^q({\mathbb {C}}^n))$ for which M is a Weyl multiplier.

Theorem 4.2

Let M be a bounded operator on $L^2({\mathbb {R}}^n)$ of the form $M=W(K)$ with $K \in L^{p_0}({\mathbb {C}}^n)$ for some $p_0 \in [1, 2]$. Then M is a Weyl multiplier for the pair $(L^p({\mathbb {C}}^n), L^q({\mathbb {C}}^n))$ and also for $(L^{q'}({\mathbb {C}}^n), L^{p'}({\mathbb {C}}^n))$ for all (p, q) such that $0 \le \frac{1}{p_0}+ \frac{1}{p}-1 \le \frac{1}{q} \le \text {min}\{ \frac{1}{p}, \frac{1}{p_0}\}$.

Proof

In view of Proposition 3.3 and Theorem 3.5, we see that if M is a Weyl multiplier for $(L^p, L^q )$, then the corresponding operator T is given by twisted convolution with some $K \in \mathcal {S}'({\mathbb {C}}^n)$. More over if $K \in L^p(C^n)$, $1 \le p \le 2$, then $W(K) \in B(L^2({\mathbb {R}}^n))$. This follows since $L^p(C^n) \subset L^1(C^n) + L^2(C^n)$ for $1 \le p \le 2$,

Now Theorem A stated at the end of Sect. 2 shows that if $K\in L^{p_0}({\mathbb {C}}^n)$, then the linear map $T: L^p ({\mathbb {C}}^n) \rightarrow L^q ({\mathbb {C}}^n)$ defined by $T f = K \times f$ is bounded for all (p, q) such that $0 \le \frac{1}{p}+ \frac{1}{p_0}-1 \le \frac{1}{q} \le \text {min}\{ \frac{1}{p},\frac{1}{p_0}\}$. Thus clearly M defines a Weyl multiplier for the pair $(L^p({\mathbb {C}}^n), L^q({\mathbb {C}}^n))$ for the above range of (p, q).

On the other hand since $K^\sharp$ is also in $L^{p_0}({\mathbb {C}}^n)$, where $K^\sharp = \bar{\tilde{K}}$, we see again by Theorem A that $T^\sharp : f \rightarrow K^\sharp \times f$ is a bounded linear map from $L^p ({\mathbb {C}}^n) \rightarrow L^q ({\mathbb {C}}^n)$ for all (p, q) such that $0 \le \frac{1}{p}+ \frac{1}{p_0}-1 \le \frac{1}{q} \le \text {min}\{ \frac{1}{p},\frac{1}{p_0}\}$.

Let S denote the space of simple functions, i.e., the linear span of the characteristic functions of measurable sets with finite measure on ${\mathbb {C}}^n$. For $f, g \in S$,

$$\begin{aligned} \langle K^\sharp \times f, \bar{g} \rangle= & {} \int _{{\mathbb {C}}^n} K^\sharp \times f (w) \bar{g}(w) \, dw = ( K^\sharp \times f) \times g^\sharp (0) \\= & {} g^\sharp \times (K^\sharp \times f) (0) = ( g^\sharp \times K^\sharp ) \times f (0) = \int _{{\mathbb {C}}^n} (K \times g)^\sharp (w) \tilde{f}(w) \, dw \end{aligned}$$

as $(K \times g)^\sharp =g^\sharp \times K^\sharp$. Thus using the identity $\int f_1(z) f_2(z) dz =\overline{ \int f_1^\sharp (z) f_2^\sharp (z) dz }$ and the fact that $\tilde{f} = \bar{f}^\sharp$, we see that

$$\begin{aligned} | \langle K^\sharp \times f, \bar{g} \rangle | = \left| \int _{{\mathbb {C}}^n}(K \times g)(w) \bar{f}(w) \, dw \right| = | \langle {K \times g}, \bar{f} \rangle |. \end{aligned}$$

It follows that

$$\begin{aligned} \Vert T\Vert _{L^{q'}({\mathbb {C}}^n) \rightarrow L^{p'}({\mathbb {C}}^n)}= & {} \sup _{ \Vert g \Vert _{L^{q'}({\mathbb {C}}^n)} = 1= \Vert f\Vert _{L^p({\mathbb {C}}^n)} } | \langle {K \times g}, \bar{f} \rangle | \\= & {} \sup _{ \Vert g \Vert _{L^{q'}({\mathbb {C}}^n)} = 1= \Vert f\Vert _{L^p({\mathbb {C}}^n)} } | \langle {K^\sharp \times f, \bar{g}} \rangle | = \Vert T^\sharp \Vert _{L^p({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n)} \end{aligned}$$

where the above supremum is taken over $f,g \in S$. Since $\Vert T^\sharp \Vert _{L^p({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n)}< \infty$ for all (p, q) such that $0 \le \frac{1}{p}+ \frac{1}{p_0}-1 \le \frac{1}{q} \le \text {min}\{ \frac{1}{p},\frac{1}{p_0}\}$ as observed above, the proof is complete. $\square$

We conclude this section with a composition formula for the Weyl multipliers. Recall that by Proposition 4.1, we have $\mathcal {M}_p^r$ is non trivial only if $r \ge p$.

Proposition 4.3

Let $M_1, M_2 \in B(L^2({\mathbb {R}}^n))$ be of the form $M_i = W(K_i)$ with $K_i \in L^{p_i}({\mathbb {C}}^n)$ for $1\le p_i \le 2,$ $i=1,2$. Then $M_2M_1 \in \mathcal {M}_p^q$ for all (p, q) such that $0 \le \frac{1}{p} + \frac{1}{p_1} + \frac{1}{p_2} -2 \le \frac{1}{q} \le \text {min} \{ \frac{1}{p}, \frac{1}{p_1}, \frac{1}{p_2} \}$.

Proof

Since $K_1 \in L^{p_1}({\mathbb {C}}^n)$, the map $T_{M_1}: f \rightarrow K_1 \times f$ is bounded from $L^{p}({\mathbb {C}}^n) \rightarrow L^{q_1}({\mathbb {C}}^n)$ by Theorem A, for all $(p,q_1)$ such that

$$\begin{aligned} 0 \le \frac{1}{p} + \frac{1}{p_1} -1 \le \frac{1}{q_1} \le \text {min} \left\{ \frac{1}{p}, \frac{1}{p_1} \right\} . \end{aligned}$$

Similarly the map $T_{M_2}: f \rightarrow K_2 \times f$ is bounded from $L^{q_1}({\mathbb {C}}^n) \rightarrow L^{q}({\mathbb {C}}^n)$ for all $(q_1,q)$ such that

$$\begin{aligned} 0 \le \frac{1}{p_2} + \frac{1}{q_1} -1 \le \frac{1}{q} \le \text {min} \left\{ \frac{1}{p_2}, \frac{1}{q_1} \right\} . \end{aligned}$$

Eliminating $q_1$ from these two conditions, we see that $T_{M_2}T_{M_1}: L^p({\mathbb {C}}^n) \rightarrow L^q({\mathbb {C}}^n)$ is bounded for all (p, q) such that

$$\begin{aligned} 0 \le \frac{1}{p} + \frac{1}{p_1} + \frac{1}{p_2} -2 \le \frac{1}{q} \le \text {min} \left\{ \frac{1}{p}, \frac{1}{p_1}, \frac{1}{p_2} \right\} \end{aligned}$$

as $\text {min} \left\{ \frac{1}{p_2}, \text {min} \left\{ \frac{1}{p}, \frac{1}{p_1} \right\} \right\} = \text {min} \left\{ \frac{1}{p_1}, \frac{1}{p_2}, \frac{1}{p} \right\}$. $\square$

5 Weyl multipliers for Lorentz spaces

In this section we discuss the Weyl multipliers for pairs of Lorentz spaces $( L^{p_1,s_1}({\mathbb {C}}^n), L^{p_2,s_2}({\mathbb {C}}^n) )$ and prove the analogues of Theorem 4.2 and Proposition 4.3 in this setting.

The Lorentz spaces $L^{p,q}({\mathbb {R}}^n)$ are defined in terms of the distribution functions. Recall that the distribution function of a Lebesgue measurable function f on $\mathbb {R}^n$ is a function on ${\mathbb {R}}_+= [0,\infty )$ given by $\lambda _f (\alpha )= | \{x\in {\mathbb {R}}^n: |f(x) | > \alpha \}|$, where |E| denote the Lebesgue measure of the measurable set $E\subset {\mathbb {R}}^n$. The non increasing rearrangement of f is defined by $f^* (t) = \inf \{s \ge 0: \lambda (s) \le t \}$. For $0 < p, q \le \infty$, define

$$\begin{aligned} \Vert f \Vert _{p,q} = \left\{ \begin{array}{ll} \left( \int _0 ^\infty \left[ t^{\frac{1}{p}} f^*(t) \right] ^q \frac{dt}{t} \right) ^\frac{1}{q}, &{} \quad \text {if } 0\le q< \infty \\ \sup _{t>0}\, t^\frac{1}{p}f^*(t) &{} \quad \text {if } q= \infty \end{array} \right. \end{aligned}$$

(5.1)

The Lorentz space $L^{p,q}(\mathbb {R}^n)$ is the space of all measurable functions f on ${\mathbb {R}}^n$ with $\Vert f\Vert _{p,q} <\infty$. $L^{p,q}(\mathbb {R}^n)$ is Banach space with respect to the topology given by the quasi norm $\Vert \cdot \Vert _{p,q}$, for $1\le p,q \le \infty$.

We have $L^{p,p} = L^p$ and $L^{p,q_1} \subset L^{p,q_2}$ for $q_1 \le q_2$, in particular $L^{p,q} \subset L^{p}$ for $1 \le q \le p \le 2$. Hence the Weyl transform of $f \in L^{p,q}$ are defined and $W(f) \in B(L^2({\mathbb {R}}^n))$ for this range of p, q. We refer to [7] and [8] for more on Lorentz spaces.

Now we proceed to define the Weyl multipliers for a pair of Lorentz spaces. Note that the space S of simple functions on ${\mathbb {C}}^n$ mentioned in the previous section, is dense in $L^{p,r}(\mathbb {C}^n)$ for $1<p<\infty$, $1<r \le \infty$.

Definition 5.1

A bounded operator M on $L^2({\mathbb {R}}^n)$ is said to be a Weyl multiplier for the pair $(L^{p, r}(\mathbb {C}^n), L^{q,s}(\mathbb {C}^n))$ if the operator $T:S \rightarrow \mathcal {S}'$ defined by $M W(h)=W(Th)$, satisfies the norm estimate

$$\begin{aligned} \Vert Th \Vert _{L^{q,s}(\mathbb {C}^n)} \le C \Vert h\Vert _{L^{p,r}(\mathbb {C}^n)} \end{aligned}$$

(5.2)

with a constant C valid for all $h \in S$.

By density of S in $L^{p,r}(\mathbb {C}^n)$ for $1<p< \infty , 1 \le r < \infty$, the operator T actually extends as a bounded operator from $L^{p,r}(\mathbb {C}^n) \rightarrow L^{q,s}(\mathbb {C}^n)$. Note that Theorem A has an extension to the Lorentz spaces, obtained as Theorem 4.2 in [9]. We restate it as Theorem B below, which is crucial in the deduction of the results for the Weyl multipliers for Lorentz pairs, from that for the Lebesgue pairs.

Theorem B. Let $\Delta = \{(p_1, p_2, p) \in [1,\infty )^3: 0< \frac{1}{p_1}+ \frac{1}{p_2} -1< \frac{1}{p} < \min \{ \frac{1}{p_1}, \frac{1}{p_2} \} \}$. Then the inequality

$$\begin{aligned} \Vert f \times g\Vert _{L^{p, r} (\mathbb {C}^n)} \le C \Vert f \Vert _{L^{p_1, r_1}(\mathbb {C}^n)} \Vert g\Vert _{L^{p_2, r_2}(\mathbb {C}^n)} \end{aligned}$$

holds with a constant C valid for all $f \in L^{p_1,r_1}(\mathbb {C}^n)$ and $g \in L^{p_2, r_2}(\mathbb {C}^n)$ for all $(p_1, p_2, p) \in \Delta$ and all r such that $\min \{ r_1, r_2\} \le r \le \infty$.

The above theorem leads to the following condition on the Lorentz pairs $(L^{p_1, s_1}(\mathbb {C}^n), L^{p_2, s_2}(\mathbb {C}^n) )$ for which $M= W(K)$ with $K \in L^{p_0, r_0}(\mathbb {C}^n), \, 1 \le r_0 \le p_0 \le 2$ is a Weyl multiplier.

Theorem 5.2

Let $M \in B(L^2({\mathbb {R}}^n))$ be of the form $M=W(K)$ with $K\in L^{p_0, r_0}, 1 \le r_0 \le p_0 \le 2$. Then M defines a Weyl multiplier for all Lorentz pairs $(L^{p, r}(\mathbb {C}^n), L^{q, s}(\mathbb {C}^n) )$ such that $(p_0,p,q) \in \Delta$ and $s \ge \min \{ r_0, r\}$, where $1< r < \infty$.

Proof

Note that since $L^{p_0,r_0}({\mathbb {C}}^n) \subset L^{p_0}({\mathbb {C}}^n)$ for $r_0\le p_0$, and since $p_0 \le 2$, we have $K \in L^1({\mathbb {C}}^n) + L^2({\mathbb {C}}^n)$, hence $W(K) \in B(L^2({\mathbb {R}}^n))$ as observed before. Hence the corresponding operator T is of the form $f \rightarrow K \times f$. Now by Theorem B, the operator $T:L^{p, r}(\mathbb {C}^n) \rightarrow L^{q, s}(\mathbb {C}^n)$ is bounded for $1< p,q, r,s < \infty$, whenever $(p_0,p,q) \in \Delta$ and $s \ge \min \{ r_0, r\}$. $\square$

We also have the following result for composition of two Weyl multipliers in the Lorentz space setting, whose proof follows by similar arguments as in the proof of Proposition 4.3.

Proposition 5.3

Let $M_1, M_2 \in B({\mathbb {R}}^n)$ are of the form $M_i = W(K_i)$ with $K_i \in L^{p_i, r_i}({\mathbb {C}}^n)$ for $1\le r_i \le p_i \le 2$, $i=1,2$. Then $M_2M_1 \in \mathcal {M}_{p,r}^{q,s}$ for all (p, q) and (r, s) such that $0< \frac{1}{p} + \frac{1}{p_1} + \frac{1}{p_2} -2< \frac{1}{q} < \text {min} \{ \frac{1}{p}, \frac{1}{p_1}, \frac{1}{p_2} \}$, and $s \ge \text { min} \{ r, r_1, r_2\}$, $1 \le r_1, r_2 <\infty$.