Notes on bilinear multipliers on Orlicz spaces

TL;DR

This paper studies bilinear multipliers on Orlicz spaces, characterizing their properties, providing examples, and deriving necessary conditions for specific multiplier forms, extending known results from Lebesgue spaces.

Contribution

It introduces the concept of bilinear multipliers on Orlicz spaces, explores their properties, and establishes necessary conditions for certain classes of multipliers, generalizing previous Lebesgue space results.

Findings

Characterization of bilinear multipliers on Orlicz spaces.

Examples of such multipliers under certain conditions.

Necessary conditions for multipliers of the form m(ξ,η)=M(ξ−η).

Abstract

Let $\Phi_1 , \Phi_2 $ and $ \Phi_3$ be Young functions and let $L^{\Phi_1}(\mathbb{R})$, $L^{\Phi_2}(\mathbb{R})$ and $L^{\Phi_3}(\mathbb{R})$ be the corresponding Orlicz spaces. We say that a function $m(\xi,\eta)$ defined on $\mathbb{R}\times \mathbb{R}$ is a bilinear multiplier of type $(\Phi_1,\Phi_2,\Phi_3)$ if \[ B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(\xi) \hat{g}(\eta)m(\xi,\eta)e^{2\pi i (\xi+\eta) x}d\xi d\eta \] defines a bounded bilinear operator from $L^{\Phi_1}(\mathbb{R}) \times L^{\Phi_2}(\mathbb{R})$ to $L^{\Phi_3}(\mathbb{R})$. We denote by $BM_{(\Phi_1,\Phi_2,\Phi_3)}(\mathbb{R})$ the space of all bilinear multipliers of type $(\Phi_1,\Phi_2,\Phi_3)$ and investigate some properties of such a class. Under some conditions on the triple $(\Phi_1,\Phi_2,\Phi_3)$ we give some examples of bilinear multipliers of type $(\Phi_1,\Phi_2,\Phi_3)$. We will focus on the case $m(\xi,\eta)=M(\xi-\eta) $ and get necessary conditions on $(\Phi_1,\Phi_2,\Phi_3)$ to get non-trivial multipliers in this class. In particular we recover some of the the known results for Lebesgue spaces.

Equations

$$B_m(f,g)(x)={\int }_{\mathbb{R}}{\int }_{\mathbb{R}}\hat{f}(\xi )\hat{g}(\eta )m(\xi ,\eta )e^{2\pi i(\xi +\eta )x}d\xi d\eta$$

$$\widehat{({\tau }_{y}f)}(\xi )=M_{-y}\hat{f}(\xi ),\widehat{(M_{x}f)}(\xi )={\tau }_x\hat{f}(\xi ),\widehat{(D_{\lambda }f)}(\xi )={\hat{f}}_{\lambda }(\xi ).$$

$$N_{\Phi }(f)=\inf \{\lambda >0:{\int }_{\mathbb{R}}\Phi (|f(x)|/\lambda )dx\le 1\}$$

$$T_m(f)(x)={\int }_{\mathbb{R}}\hat{f}(\xi )m(\xi )e^{2\pi i\xi x}d\xi$$

$$B_m(f,g)(x)={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\hat{f}(\xi )\hat{g}(\eta )m(\xi ,\eta )e^{2\pi i⟨\xi +\eta ,x⟩}d\xi d\eta$$

$$N_{{\Phi }_3}(B_m(f,g))\le C{N}_{{\Phi }_1}(f)N_{{\Phi }_2}(g)$$

$$B_M(f,g)(x)={\int }_{{\mathbb{R}}^n}{\int }_{{\mathbb{R}}^n}\hat{f}(\xi )\hat{g}(\eta )M(\xi -\eta )e^{2\pi i⟨\xi +\eta ,x⟩}d\xi d\eta$$

$${\Phi }^{-1}(y)=\inf \{x>0:\Phi (x)>y\}$$

$$\Phi ({\Phi }^{-1}(x))\le x\le {\Phi }^{-1}(\Phi (x)),x\ge 0.$$

$$\Psi (y)=\sup \{x|y|-\Phi (x):x\ge 0\}$$

$$|x|\le {\Phi }^{-1}(x){\Psi }^{-1}(x)\le 2|x|,x\in \mathbb{R},$$

$$|xy|\le \Phi (x)+\Psi (y),x,y\in \mathbb{R}.$$

$${\Phi }_1^{-1}(x){\Phi }_2^{-1}(x)\le {\Phi }_3^{-1}(x),x\ge 0.$$

$$N_{{\Phi }_3}(fg)\le 2N_{{\Phi }_1}(f)N_{{\Phi }_2}(g).$$

$${\Phi }_1^{-1}(x){\Phi }_2^{-1}(x)\le x{\Phi }_3^{-1}(x),x\ge 0.$$

$$N_{{\Phi }_3}(f\star g)\le 2N_{{\Phi }_1}(f)N_{{\Phi }_2}(g).$$

$$N_{\Phi ,\gamma }(f)=\inf \{k>0:{\int }_{\mathbb{R}}\Phi (\frac{|f(x)|}{k})dx\le \gamma \}.$$

$$\frac{{\gamma }_1}{{\gamma }_2}{N}_{\Phi ,{\gamma }_1}(f)\le N_{\Phi ,{\gamma }_2}(f)\le N_{\Phi ,{\gamma }_1}(f).$$

$$C_{\Phi }(\lambda )=\Vert D_{\lambda }{\Vert }_{L^{\Phi }(\mathbb{R})\to L^{\Phi }(\mathbb{R})}.$$

$$\frac{1}{\max \{1,\lambda \}}\le C_{\Phi }(\lambda )\le \frac{1}{\min \{1,\lambda \}}.$$

$$N_{\Phi }(D_{\lambda }f)=N_{\varphi ,\lambda }(f).$$

$$N_{\Phi ,\lambda }(f)\le N_{\Phi }(f)\le \lambda N_{\Phi ,\lambda }(f),\lambda \ge 1$$

$$\lambda N_{\Phi ,\lambda }(f)\le N_{\Phi }(f)\le N_{\Phi ,\lambda }(f),0<\lambda \le 1$$

$$\frac{\Vert f{\Vert }_1}{|A|{\Phi }^{-1}(|A{|}^{-1})}\le N_{\Phi }(f)\le \frac{\Vert f{\Vert }_{\infty }}{{\Phi }^{-1}(|A{|}^{-1})}$$

$$N_{\Phi }(f)$$

$$N_{\Phi }(D_{\lambda }{\chi }_{[0,a]})=N_{\Phi }({\chi }_{[0,\frac{a}{\lambda }]})=\frac{1}{{\Phi }^{-1}(\frac{\lambda }{a})},N_{\Phi }(f)=\frac{1}{{\Phi }^{-1}(\frac{1}{a})}.$$

$$C_{\Phi }(\lambda )\ge \underset{a>0}{\sup }\frac{N_{\Phi }(D_{\lambda }{\chi }_{[0,a]})}{N_{\Phi }({\chi }_{[0,a]})}=\underset{\mu >0}{\sup }\frac{{\Phi }^{-1}(\mu )}{{\Phi }^{-1}(\lambda \mu )}.$$

$${\Phi }_1(s){\int }_{\mathbb{R}}\Phi (\frac{|D_{\lambda }f(x)|}{k})dx=\frac{{\Phi }_1(s)}{\lambda }{\int }_{\mathbb{R}}\Phi (\frac{|f(x)|}{k})dx\le \frac{1}{\lambda }{\int }_{\mathbb{R}}\Phi (s\frac{|f(x)|}{k})dx.$$

Full text

[TABLE]

defines a bounded bilinear operator from $L^{\Phi_{1}}(\mathbb{R})\times L^{\Phi_{2}}(\mathbb{R})$ to $L^{\Phi_{3}}(\mathbb{R})$. We denote by $BM_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ the space of all bilinear multipliers of type $(\Phi_{1},\Phi_{2},\Phi_{3})$ and investigate some properties of such a class. Under some conditions on the triple $(\Phi_{1},\Phi_{2},\Phi_{3})$ we give some examples of bilinear multipliers of type $(\Phi_{1},\Phi_{2},\Phi_{3})$. We will focus on the case $m(\xi,\eta)=M(\xi-\eta)$ and get necessary conditions on $(\Phi_{1},\Phi_{2},\Phi_{3})$ to get non-trivial multipliers in this class. In particular we recover some of the the known results for Lebesgue spaces.

The first author is partially supported by Proyecto MTM2014-53009-P(MINECO Spain) and the second author is supported by ”The Scientific and Technological Research Council of Turkey” TUBITAK-BIDEB grant no 1059B191600535

1. Introduction.

The theory of bilinear multipliers was originated in the work by R. Coiffman and C. Meyer ([10]) in the eighties of the last century and continued by L. Grafakos and R. Torres ([15]) and many others. A renewed interest appeared in the nineties after the celebrated result by M. Lacey and C. Thiele ([19, 20]), solving the old standing conjecture of Calderón on the boundedness of the bilinear Hilbert transform. Let us recall that for a couple of functions $f,g:\mathbb{R}\to\mathbb{C}$ such that $\hat{f}$ and $\hat{g}$ are compactly supported and for any locally integrable function $m(\xi,\eta)$ defined on $\mathbb{R}\times\mathbb{R}$ one can consider the mapping

[TABLE]

and ask himself about its boundedness on certain function spaces. In such a way the bilinear versions of several classical operators appearing in Harmonic Analysis, such as the Hilbert transform, the fractional integrals, the Hardy-Littlewood maximal function and many others have been considered in the last decades and their boundedness on several spaces have been addressed.

The study of bilinear multipliers for smooth symbols (where $m(\xi,\eta)$ is a “nice” regular function with at most a single point singularity) goes back to the work by R.R. Coifman and Y. Meyer in [10]. A particularly interesting case is $m(\xi,\eta)=M(\xi-\eta)$ for a measurable function $M$ where, for instance the case $M(x)=\frac{1}{|x|^{1-\alpha}}$, corresponding to the bilinear fractional transform, was shown to define a bilinear multiplier mapping $L^{p_{1}}(\mathbb{R})\times L^{p_{2}}(\mathbb{R})$ into $L^{p_{3}}(\mathbb{R})$ for $1/p_{3}=1/p_{1}+1/p_{2}-\alpha$ for $1<p_{1},p_{2}<\infty$ and $0<\alpha<1/p_{1}+1/p_{2}$ (see [16, 14]) or the celebrated result of the bilinear Hilbert transform, given by the case $M(x)=sign(x)$, was shown to define a bilinear multiplier of type $(p_{1},p_{2},p_{3})$ for $1/p_{3}=1/p_{1}+1/p_{2}$ for $1<p_{1},p_{2}<\infty$ and $p_{3}>2/3$ ([19, 20, 21]). The case of more general non-smooth symbols was later analyzed by J. Gilbert and A. Namod (see [12, 13]).

Bilinear multipliers acting on other groups such as torus $\mathbb{T}$ or integers $\mathbb{Z}$ have also been studied. Their corresponding analogues have been achieved using transference properties first by D. Fan and S. Sato [11] and later by the results in several papers by E. Berkson, O.Blasco, M.J. Carro and A.Gillespie (see [5, 8, 3, 4]). More recently several results on bilinear multipliers defined on locally compact abelian groups and acting on rearrangement invariant quasi-Banach spaces have been obtained by S. Rodriguez-López [24]. Other function spaces such as Lorentz spaces have been studied mainly by O. Blasco and F. Villarroya (see [9, 26]) and for also for weighted Lebesgue spaces or Lebesgue spaces with variable exponent by T. Gürkanli and O. Kulak [14]. Our objective will be to deal with bilinear multipliers on $\mathbb{R}$ (although similar results can be presented in $\mathbb{R}^{n}$) acting on Orlicz spaces.

Throughout the paper ${\mathcal{P}}(\mathbb{R})$ stands for the set of functions such that $supp\hat{f}$ is compact and $S(\mathbb{R})$ for the Schwartz class on $\mathbb{R}$, i.e. $f:\mathbb{R}\to\mathbb{C}$ such that $f\in C^{\infty}(\mathbb{R})$ and $x^{k}f^{(n)}(x)$ is bounded for any $k$ and $n$. We write the Fourier transform by $\hat{f}(\xi)=\int_{\mathbb{R}}f(x)e^{-2\pi ix\xi}dx$ and we denote the translation by $\tau_{y}f(x)=f(x-y)$, the modulation by $M_{x}f(y)=f(y)e^{2\pi\textit{i}xy}$ and the dilation by $D_{\lambda}f(x)=f(\lambda x)$ for $x,y\in\mathbb{R}$ and $\lambda>0$. As usual for $g$ defined in $\mathbb{R}^{n}$ we write $g_{t}(x)=\frac{1}{t^{n}}g(\frac{x}{t})=\frac{1}{t^{n}}D_{1/t}g(x)$ for $x\in\mathbb{R}^{n}$ and $t>0$. Clearly one has for each $f\in L^{1}(\mathbb{R})$, $y\in\mathbb{R}$ and $\lambda>0$

[TABLE]

Given a Young function $\Phi$, the Orlicz space $L^{\Phi}(\mathbb{R})$ consists of the set of all measurable functions $f:\mathbb{R}\to\mathbb{C}$ such that $\int_{\mathbb{R}}\Phi(|f(x)|/\lambda)dx<\infty$ for some $\lambda>0$, which equipped with the so called Luxemburg norm

[TABLE]

becomes a Banach space.

It is known that if a Young function $\Phi$ satisfies the $\Delta_{2}$-condition (i.e. there exists a constant $k>0$ such that $\Phi(2x)\leq k\Phi(x)$ for all $x\geq 0$), then the space of compactly supported functions in $C^{\infty}(\mathbb{R})$ is dense in $L^{\Phi}(\mathbb{R})$ with respect to the norm $N_{\Phi}(\cdot)$. Hence, in this case $S(\mathbb{R})$ and $\mathcal{P}(\mathbb{R})$ are also dense in $L^{\Phi}(\mathbb{R})$.

Given two Young functions $\Phi_{1}$ and $\Phi_{2}$ the space $\mathcal{M}_{\Phi_{1},\Phi_{2}}(\mathbb{R})$ stands for the space of bounded functions $m$ defined on $\mathbb{R}$ such that

[TABLE]

defines a bounded operator from $L^{\Phi_{1}}(\mathbb{R})$ to $L^{\Phi_{2}}(\mathbb{R})$. We endow the space with the “norm” of the operator $T_{m}$, that is $\|m\|_{\Phi_{1},\Phi_{2}}=\|T_{m}\|$. We refer the reader to [2, 25] for the case $\Phi_{1}(x)=|x|^{p}$ and $\Phi_{2}(x)=|x|^{q}$, to be denoted $\mathcal{M}_{p,q}(\mathbb{R})$.

Definition 1.1**.**

Given three Young functions $\Phi_{i}$ for $i=1,2,3$, a locally integrable function $m$ defined on $\mathbb{R}^{2}$ is said to be a bilinear multiplier of type $(\Phi_{1},\Phi_{2};\Phi_{3})$ if there exists a constant $C>0$ such that

[TABLE]

satisfies

[TABLE]

for any $f,g\in\mathcal{P}(\mathbb{R})$.

We write $\mathcal{BM}_{(\Phi_{1},\Phi_{2};\Phi_{3})}(\mathbb{R})$ for the space of bilinear multipliers of type $(\Phi_{1},\Phi_{2};\Phi_{3})$ and $\|m\|_{(\Phi_{1},\Phi_{2};\Phi_{3})}=\|B_{m}\|$.

We denote by $\tilde{\mathcal{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ the space of locally integrable functions $M$ defined on $\mathbb{R}$ such that $m(\xi,\eta)=M(\xi-\eta)\in\mathcal{BM}_{(\Phi_{1},\Phi_{2};\Phi_{3})}(\mathbb{R})$.

Note that in the case that $\Phi_{1}$ and $\Phi_{2}$ satisfy $\Delta_{2}$-condition then $M\in\mathcal{BM}_{(\Phi_{1},\Phi_{2};\Phi_{3})}(\mathbb{R})$ means that

[TABLE]

extends to a bounded bilinear map from $L^{\Phi_{1}}(\mathbb{R})\times L^{\Phi_{2}}(\mathbb{R})$ into $L^{\Phi_{3}}(\mathbb{R})$. We keep the notation $\|M\|_{(\Phi_{1},\Phi_{2};\Phi_{3})}=\|B_{M}\|.$ This generalize the case $\Phi_{i}(x)=x^{p_{i}}$ considered in [6] and denoted $\mathcal{BM}_{(p_{1},p_{2},p_{3})}(\mathbb{R})$ and $\tilde{\mathcal{M}}_{(p_{1},p_{2},p_{3})}(\mathbb{R})$ respectively.

In this paper, we shall investigate some properties of the spaces $\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ and $\tilde{\mathcal{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$. The paper is divided into five sections. The first section is devoted to recall some notions on Orlicz spaces to be used in the sequel. In particular we shall analyze the norm of the dilation operator $D_{\lambda}$ acting on Orlicz spaces. In Section 3 we shall give elementary examples of bilinear multipliers and procedures to generate them. In Section 4 we mainly focus on the case $m(\xi,\eta)=M(\xi-\eta)$ and give some sufficient conditions to define a bilinear multiplier on Orlicz spaces. Finally we use the last section to investigate some necessary conditions to get a non-zero bilinear multipliers in the class $\tilde{\mathcal{M}}_{(\Phi_{1},\Phi_{2};\Phi_{3})}(\mathbb{R})$, generalizing the known results for Lebesgue spaces.

2. Orlicz spaces

A non-zero function $\Phi:\mathbb{R}\to[0,\infty]$ is called a Young function if $\Phi$ is convex, even and $\Phi(0)=0$. If $\Phi$ is a Young function then $\Phi^{-1}$ is defined for $0\leq y$ by

[TABLE]

where $\inf\emptyset=\infty$ and it is easy to see [22] that

[TABLE]

Given a Young function $\Phi$, its complementary function is defined by

[TABLE]

for $y\in\mathbb{R}$. It can be seen that $\Psi$ is still a Young function in the sense of above definition. Then $(\Phi,\Psi)$ is called a complementary pair of Young functions and they satisfy

[TABLE]

and the Young inequality

[TABLE]

There are several inequalities to be used throughout the paper when dealing with Orlicz spaces: One deals with the generalization of Hölder’s inequality (see [22],[23, page 64]): Let $\Phi_{i}$, $i=1,2,3$ be Young’s functions satisfying

[TABLE]

If $f\in L^{\Phi_{1}}(\mathbb{R})$ and $g\in L^{\Phi_{2}}(\mathbb{R})$ then $fg\in L^{\Phi_{3}}(\mathbb{R})$ and

[TABLE]

The other one refers to Young’s inequality for convolutions (see [22],[23, page 64]): Let $\Phi_{i}$, $i=1,2,3$ be Young functions satisfying

[TABLE]

If $f\in L^{\Phi_{1}}(\mathbb{R})$ and $g\in L^{\Phi_{2}}(\mathbb{R})$ then the convolution $f\star g\in L^{\Phi_{3}}(\mathbb{R})$ and

[TABLE]

The reader is referred to [23] for the proofs of these results and for further information about Orlicz spaces.

In this section, we shall give some estimates to the norms of the dilation operator on Orlicz spaces which will be useful in the sequel.

Given $\gamma>0$ one can define

[TABLE]

Of course $N_{\Phi,1}=N_{\Phi}$. Let us observe that these quantities give equivalent norms in $L^{\Phi}(\mathbb{R})$. In fact, by convexity, we can easily see the following property of these norms: If $0<\gamma_{1}<\gamma_{2}$ and $f$ is a measurable function then

[TABLE]

Throughout the paper

[TABLE]

Of course $C_{\Phi}(\lambda)$ is non-increasing, submultiplicative and $C_{\Phi}(1)=1$.

Proposition 2.1**.**

Let $\lambda>0$ and $\Phi$ a Young function. Then

[TABLE]

Proof.

It is straightforward that for $f\in L^{\Phi}(\mathbb{R})$ and $\lambda>0$ one has

[TABLE]

Using now (10) we have

[TABLE]

and

[TABLE]

The result now follows from (11). $\blacksquare$

Let us now get better estimates for $C_{\Phi}(\lambda)$ using the following lemma.

Lemma 2.2**.**

Let $\Phi$ be a Young function and $A\subset\mathbb{R}$ be measurable with $0<|A|<\infty$. If $f$ be a bounded function supported on $A$ then

[TABLE]

where $|A|$ stands for the Lebesgue measure of $A$.

In particular if $|f(x)|=1$ for $x\in A$ then $N_{\Phi}(f)=\frac{1}{\Phi^{-1}(|A|^{-1})}$.

Proof.

From (3) one sees that $\{x>0:\Phi(x)\leq a\}=\{x>0:x\leq\Phi^{-1}(a)\}$ for $a>0$. Therefore since $|f(x)|\leq\|f\|_{\infty}\chi_{A}(x)$ we have

[TABLE]

For the other inequality we use Jensen inequality for convex functions. Indeed

[TABLE]

$\blacksquare$

Proposition 2.3**.**

Let $\Phi$ be a Young function. Then $C_{\Phi}(\lambda)\geq\sup_{\mu>0}\frac{\Phi^{-1}(\mu)}{\Phi^{-1}(\lambda\mu)}.$

Proof.

Taking $A=[0,a]$ and $f=\chi_{A}$ in Lemma 2.2, since $D_{\lambda}(\chi_{[0,a]})=\chi_{[0,\frac{a}{\lambda}]}$ one obtains

[TABLE]

Hence

[TABLE]

$\blacksquare$

Theorem 2.4**.**

Let $\Phi$ be a Young function.

(i) If $\Phi(st)\geq\Phi_{1}(s)\Phi(t)$ for all $s,t\geq 0$ for some $\Phi_{1}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ non-decreasing and left continuous then $C_{\Phi}(\lambda)\leq\Phi_{1}^{-1}(\frac{1}{\lambda}).$

(ii)If $\Phi(st)\leq\Phi_{2}(s)\Phi(t)$ for all $s,t\geq 0$ for some $\Phi_{2}:\mathbb{R}^{+}\to\mathbb{R}^{+}$ non-decreasing and left continuous then $C_{\Phi}(\lambda)\leq\frac{1}{\Phi_{2}^{-1}(\lambda)}.$

Proof.

(i) Assume that $\Phi(st)\geq\Phi_{1}(s)\Phi(t)$ for $s,t\geq 0$. Note that for any $s>0$ and $k>0$ we have

[TABLE]

In particular whenever $\Phi_{1}(s)>\frac{1}{\lambda}$ one obtains that

[TABLE]

Select a decreasing sequence $s_{n}$ converging to $\Phi_{1}^{-1}(\frac{1}{\lambda})$ and invoke the Lebesgue convergence theorem to get

[TABLE]

Therefore for $k_{\lambda}=\Phi_{1}^{-1}(\frac{1}{\lambda})N_{\Phi}(f)$ one gets $\int_{\mathbb{R}}\Phi(\frac{|D_{\lambda}f(x)|}{k_{\lambda}})dx\leq 1.$ This gives that $N_{\Phi}(D_{\lambda}f)\leq\Phi_{1}^{-1}(\frac{1}{\lambda})N_{\Phi}(f)$ and we obtain (i).

(ii) Assume now $\Phi(st)\leq\Phi_{2}(s)\Phi(t)$. As above for $s>0$

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Choosing $s=\Phi_{2}^{-1}(\lambda)$ one obtains from (3) that $\Phi_{2}(s)\leq\lambda$. Hence

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Now selecting $k=N_{\Phi}(f)$ we get $N_{\Phi}(D_{\lambda}f)\leq\frac{1}{\Phi_{2}^{-1}(\lambda)}N_{\Phi}(f)$. This finishes the proof of (ii). $\blacksquare$

Invoking Theorem 2.4 and Proposition 2.3 we obtain the following result.

Corollary 2.5**.**

Let $\Phi$ be a Young function satisfying $\Phi(st)\leq\Phi(s)\Phi(t)$ for all $s,t\geq 0$. Then

[TABLE]

Remark 2.6**.**

If $\Phi$ is sub-multiplicative and $\Phi(1)=1$ then $C_{\Phi}(\lambda)=\frac{1}{\Phi^{-1}(\lambda)}$. This is the case for $\Phi(x)=|x|^{p}$ where we obtain $C_{\Phi}(\lambda)=\lambda^{-1/p}$.

3. Bilinear multipliers: The basics

Let us start with some elementary properties of the bilinear multipliers acting on Orlicz spaces. We follow the arguments in [6] where the case of Lebesgue spaces was studied. Since the norm in Orlicz spaces is invariant under translations and modulations one can easily obtain the following results.

Proposition 3.1**.**

Let $\Phi_{i}$ for $i=1,2,3$ and $\tilde{\Phi}_{j}$ for $j=1,2$ be Young functions and let $m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$.

1. (a)

If $m_{1}\in\mathcal{M}_{\tilde{\Phi}_{1},\Phi_{1}}(\mathbb{R})$, $m_{2}\in\mathcal{M}_{\tilde{\Phi}_{2},\Phi_{2}}(\mathbb{R})$ and $\tilde{m}(\xi,\eta)=m_{1}(\xi)m(\xi,\eta)m_{2}(\eta)$ then $\tilde{m}\in\mathcal{BM}_{(\tilde{\Phi}_{1},\tilde{\Phi}_{2},\Phi_{3})}(\mathbb{R})$. Moreover

[TABLE] 2. (b)

If $\tau_{(\xi_{0},\eta_{0})}m(\xi,\eta)=m(\xi-\xi_{0},\eta-\eta_{0})$ then $\tau_{(\xi_{0},\eta_{0})}m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ for each $(\xi_{0},\eta_{0})\in\mathbb{R}^{2}$ and

[TABLE] 3. (c)

If $M_{(\xi_{0},\eta_{0})}m(\xi,\eta)=e^{2\pi i(\xi_{0}\xi+\eta_{0}\eta)}$ then $M_{(\xi_{0},\eta_{0})}m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ for each $(\xi_{0},\eta_{0})\in\mathbb{R}^{2}$ and

[TABLE]

Proof.

For each $f,g\in\mathcal{P}(\mathbb{R})$ the following formulae are straightforward

[TABLE]

[TABLE]

[TABLE]

The result now follows easily. $\blacksquare$

Proposition 3.2**.**

Let $\Phi_{i}$ for $i=1,2,3$ be Young functions. If $m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ and $t>0$ then $D_{t}m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$. Moreover and

[TABLE]

Proof.

We first observe that

[TABLE]

for each $f,g\in\mathcal{P}(\mathbb{R})$. Indeed,

[TABLE]

This gives

[TABLE]

This completes the proof. $\blacksquare$

Let us combine the previous results to get new bilinear multipliers from a given one.

Proposition 3.3**.**

Let $\Phi_{i}$ for $i=1,2,3$ be Young functions, $\varphi\in L^{1}(\mathbb{R}^{2})$ and $m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$. Then

1. (a)

$\varphi*m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$* and $\|\varphi*m\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}\leq\|\varphi\|_{1}\|m\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}.$* 2. (b)

$\hat{\varphi}m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$* and $\|\hat{\varphi}m\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}\leq\|\varphi\|_{1}\|m\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}.$* 3. (c)

Let $W(t)=C_{\Phi_{3}}({1/t})C_{\Phi_{1}}(t)C_{\Phi_{2}}(t)$ and $\psi\in L^{1}(\mathbb{R}^{+},W)$ and assume that $t\to m(t\xi,t\eta)\psi(t)$ is integrable in $\mathbb{R}^{+}$ for each $(\xi,\eta)\in\mathbb{R}^{2}$. Define $m_{\psi}(\xi,\eta)=\int_{0}^{\infty}m(t\xi,t\eta)\psi(t)dt.$ Then $m_{\psi}\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ and $\|m_{\psi}\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}\leq\|\psi\|_{L^{1}(\mathbb{R}^{+},W)}\|m\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}.$

Proof.

(a) Note that

[TABLE]

From the vector-valued Minkowski inequality and Proposition 3.1 part (b), we have

[TABLE]

(b) Observe that

[TABLE]

Argue as above, using now Proposition 3.1 part (c), to conclude the result.

(c) Use the formula

[TABLE]

and Proposition 3.2 to finish the proof. $\blacksquare$

Let us now present an elementary example of bilinear multipliers. If $\mu$ is a Borel regular measure in $\mathbb{R}$ we denote $\hat{\mu}(\xi)=\int_{\mathbb{R}}e^{-2\pi ix\xi}d\mu(x)$ its Fourier transform.

Proposition 3.4**.**

Let $\Phi_{1}$, $\Phi_{2}$ and $\Phi_{3}$ be Young functions such that

[TABLE]

If $(\alpha,\beta)\in\mathbb{R}^{2}$ and $m(\xi,\eta)=\hat{\mu}(\alpha\xi+\beta\eta)$ where $\mu$ is a regular Borel measure on $\mathbb{R}$ then $m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ and $\|m\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}\leq 2\|\mu\|_{1}$.

Proof.

Let us first rewrite the value $B_{m}(f,g)$ for each $f,g\in\mathcal{S}(\mathbb{R})$ as follows:

[TABLE]

Hence, using Minkowski’s inequality, (7) and invariance under traslations one gets

[TABLE]

This gives the result. $\blacksquare$

This basic example combined with the procedures exhibited in Proposition 3.3 produces a number of multipliers in this setting.

Also, if we consider a complementary pair of Young functions, then we can give the following result as a corollary of Proposition 3.4.

Corollary 3.5**.**

Let $(\Phi,\Psi)$ be a complementary pair of Young functions. If $(\alpha,\beta)\in\mathbb{R}^{2}$ and $m(\xi,\eta)=\hat{\mu}(\alpha\xi+\beta\eta)$ where $\mu$ is a regular Borel measure on $\mathbb{R}$ then $m\in\mathcal{BM}_{(\Phi,\Psi,1)}$ and $\|m\|_{(\Phi,\Psi,1)}\leq 4\|\mu\|_{1}$.

Proof.

It is enough to take $\Phi_{1}=\Phi$, $\Phi_{2}=\Psi$ and $\Phi_{3}(x)=\frac{1}{2}|x|$, $x\in\mathbb{R}$ in Proposition 3.4, since $\Phi$ and $\Psi$ satisfy the inequality (4), noticing that $L^{\Phi_{3}}(\mathbb{R})=L^{1}(\mathbb{R})$ and $N_{\Phi_{3}}(f)=\frac{1}{2}\|f\|_{1}$ for any $f\in L^{1}(\mathbb{R})$. $\blacksquare$

Let us now give a necessary condition for multipliers homogeneous of degree [math]. This will depend upon the Boyd indices of the spaces. Recall that for a rearrangement invariant Banach space $X$ one defines

[TABLE]

where $\tilde{X}$ is the r.i. space defined on $(0,\infty)$ with the same distribution function. The Boyd indices (see [1, page 149]) are given by

[TABLE]

We denote by $\underline{\alpha}_{\Phi}$ and $\overline{\alpha}_{\Phi}$ the case $X=L^{\Phi}(\mathbb{R})$.

Proposition 3.6**.**

Let $m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ a non zero multiplier such that $m(t\xi,t\eta)=m(\xi,\eta)$ for any $t>0$. Then

[TABLE]

and

[TABLE]

Proof.

From assumption $D_{t}m=m$ for $t>0$. Using now Proposition 3.2 we can write

[TABLE]

It is elementary to show that $C_{\Phi}(t)=h_{L^{\Phi}}(1/t)$. Hence, denoting by $h_{\Phi}=h_{L^{\Phi}}$, we have

[TABLE]

Therefore

[TABLE]

This shows that

[TABLE]

[TABLE]

Hence making limits as $t\to\infty$ and $t\to 0$ one obtains (16) and (17) respectively. $\blacksquare$

Remark 3.7**.**

Let $m\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ and $m(t\xi,t\eta)=m(\xi,\eta)$ for any $t>0$. In the case $\underline{\alpha}_{\Phi_{3}}=\overline{\alpha}_{\Phi_{3}}$ one has

[TABLE]

In the case $\underline{\alpha}_{\Phi_{i}}=\overline{\alpha}_{\Phi_{i}}$ for $i=1,2$ one has,

[TABLE]

For Orlicz spaces where $\underline{\alpha}_{\Phi_{i}}=\overline{\alpha}_{\Phi_{i}}$ for $i=1,2,3$ the Bilinear Hilbert transform $m(\xi,\eta)=sign(\xi-\eta)$ can only belong to $\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ whenever $\alpha_{\Phi_{3}}=\alpha_{\Phi_{1}}+\alpha_{\Phi_{2}}$.

4. Bilinear multipliers when $m(\xi,\eta)=M(\xi-\eta)$

Let us restrict ourselves to a class of multipliers where $m(\xi,\eta)=M(\xi-\eta)$ for some function $M$ defined in $\mathbb{R}$. As in the introduction we use the notation $\tilde{\mathcal{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ for the space of locally integrable functions $M:\mathbb{R}\to\mathbb{C}$ such that $m(\xi,\eta)=M(\xi-\eta)\in\mathcal{BM}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R}),$ We keep the notation $\|M\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}=\|B_{m}\|.$

We recall several formulations for $B_{M}$ (see [7, Proposition 3.3]): Let $M\in L^{1}_{loc}(\mathbb{R})$, $f,g\in{\mathcal{P}}(\mathbb{R})$. Then

[TABLE]

[TABLE]

A basic characterization for integrable symbols is the following (see [7, Proposition 3.4]): If $M\in L^{1}(\mathbb{R})$ and $K=\check{M}$, where $\check{M}(\xi)=\hat{M}(-\xi)$, and $f,g\in\mathcal{P}(\mathbb{R})$ then

[TABLE]

A first elementary example of multiplier in $\tilde{\mathcal{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ is giving selecting $\alpha=1$ and $\beta=-1$ in Proposition 3.4 obtaining the following result (which follows from (7):

Theorem 4.1**.**

Let $\Phi_{1}$, $\Phi_{2}$ and $\Phi_{3}$ be Young functions such that

[TABLE]

If $\mu\in M(\mathbb{R})$ and $M(\xi)=\hat{\mu}(\xi)$ then $M\in\tilde{\mathcal{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$. Moreover

[TABLE]

Another elementary case is the following one.

Theorem 4.2**.**

Let $\Phi_{1}$, $\Phi_{2}$ and $\Phi_{3}$ be Young functions such that

[TABLE]

If $M\in L^{1}(\mathbb{R})$ then $M\in\tilde{\mathcal{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$. Moreover

[TABLE]

Proof.

Making the change of variable $\beta=\xi-\eta$ and $\gamma=\xi$

[TABLE]

Then by taking norm of this expression in $L^{\Phi_{3}}(\mathbb{R})$, and using (9), we obtain

[TABLE]

The proof is then complete. $\blacksquare$

Remark that, if we consider the complementary pair of Young functions $(\Phi,\Psi)$, then we could also obtain the following new result as a corollary of Theorem 4.2.

Corollary 4.3**.**

Let $(\Phi,\Psi)$ be a complementary pair of Young function. If $M\in L^{1}(\mathbb{R})$ then $M\in\tilde{\mathcal{M}}_{(\Phi,\Psi,\infty)}(\mathbb{R})$. Moreover $\|M\|_{(\Phi,\Psi,\infty)}\leq 2\|M\|_{1}.$

Proof.

We take in Theorem 4.2 the functions $\Phi_{1}=\Phi$, $\Phi_{2}=\Psi$ and $\Phi_{3}$ is such a way that $\Phi_{3}^{-1}=2$, that is to say $\Phi_{3}(x)=0$ for $|x|\leq 2$ and $\Phi_{3}(x)=\infty$ for $|x|>2$. Then the proof is complete since $L^{\Phi_{3}}(\mathbb{R})=L^{\infty}(\mathbb{R})$ and the complementary pair of Young functions satisfy the inequality (4). $\blacksquare$

As in the previous section we can generate new multipliers in $\mathcal{\tilde{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ using the following methods and the previous examples. The proof follows the same ideas as in [7] and Proposition 3.3 and it is left to the reader.

Proposition 4.4**.**

Let $\phi\in L^{1}(\mathbb{R})$ and $M\in\mathcal{\tilde{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$. Then

1. (a)

$\phi*M\in\mathcal{\tilde{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$* and $\|\phi*M\|_{p_{1},p_{2},p_{3}}\leq\|\phi\|_{1}\|M\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}.$* 2. (b)

$\hat{\phi}M\in\mathcal{\tilde{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$* and $\|\hat{\phi}M\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}\leq\|\phi\|_{1}\|M\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}.$* 3. (c)

If $\psi\in L^{1}(\mathbb{R}^{+},W)$ then $M_{\psi}(\xi)=\int_{0}^{\infty}M(t\xi)\psi(t)dt\in\mathcal{\tilde{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$. Moreover $\|M_{\psi}\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}\leq\|\psi\|_{L^{1}(\mathbb{R}^{+},W)}\|M\|_{(\Phi_{1},\Phi_{2},\Phi_{3})}.$

5. On necessary conditions for $\mathcal{\tilde{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})\neq\{0\}$

Let us show that the classes $\mathcal{\tilde{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ are reduced to $\{0\}$ in certain cases. We shall use arguments from [7, Theorem 3.7, Theorem 3.9] and [24, Theorem 5.10].

We need the following lemma to give a result about the bilinear multipliers in the class $M_{(\Phi_{1},\Phi_{2},\Phi_{3})}$.

Lemma 5.1**.**

Let $g$ be a continuous function in $\mathbb{R}$ with $supp(g)\subset[0,a]$ for some $a>0$ and let $\Phi$ be a Young function. Then

[TABLE]

where $\epsilon_{k}\in\{\pm 1\}$.

Proof.

Note that $supp(\tau_{y}g)\subset[y,y+a]$ and then $\tau_{[a+1]k}g$ are disjointly supported. Hence if $h=\Sigma_{k=0}^{N}\epsilon_{k}\tau_{[a+1]k}g$, $I_{0}=[0,a]$ and $I_{k}=[[a+1]k,[a+1]k+a]$ then, using Jensen’s inequality, one has

[TABLE]

where the last equality follows same argument as in the proof of Lemma 2.2. $\blacksquare$

Theorem 5.2**.**

Let $\Phi_{1},\Phi_{2},\Phi_{3}$ be Young functions.

(i) If

[TABLE]

then $\mathcal{\tilde{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})\neq\{0\}$.

(ii) If $\tilde{\mathcal{M}}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})\neq\{0\}$ then for all $a>0$ one has

[TABLE]

Proof.

(i) follows from Theorem 4.1.

(ii) Let $0\neq M\in\tilde{M}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$. Using Proposition 4.4 we may assume that there exists $0\neq M\in L^{1}(\mathbb{R})\cap\tilde{M}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$. Hence, from (20) one has that

[TABLE]

for any $f$ and $g$ continuous functions compactly supported in $[0,a]$. Consider the Rademacher system in $[0,1]$ and observe that for each $N$ and $y\in\mathbb{R}$, the orthonormality of the system gives

[TABLE]

Therefore, since $B_{M}(\tau_{ky}f,\tau_{ky}g)=\tau_{ky}B_{M}(f,g)$, we have

[TABLE]

for any $f,g$ compactly supported in $[0,a]$. Now, let us consider the functions $f=g=\chi_{[0,a]}$, where $a>0$ is arbitrary constant, $y=[a+1]$ where $[\cdot]$ is the integer part. For each $N\in\mathbb{N}$ and $t\in[0,1]$ we denote

[TABLE]

Then for the functions $f$ and $f_{N}$, by using (21) we have

[TABLE]

where $suppB_{M}(f,f)\subset[0,2a]$.

By taking norm of the right hand side of this equality in $L^{\Phi_{3}}(\mathbb{R})$ and using the Lemma 5.1 we observe that

[TABLE]

On the other hand, by using Minkowski’s inequality and Lemma 2.2 we have

[TABLE]

which combining with (22) gives, for each $a>0$ and for all $N\in\mathbb{N}$,

[TABLE]

This implies that for any $a>0$ there exists $C=C_{a}>0$ such that

[TABLE]

where $C_{a}=\frac{a\|B_{m}\|}{\|B_{M}(\chi_{[0,a]},\chi_{[0,a]})\|_{1}}>0.$ This completes the proof. $\blacksquare$

Note that, if we take $\Phi_{i}(x)=|x|^{p_{i}}$ for $i=1,2,3$, then $L^{\Phi_{i}}(\mathbb{R})=L^{p_{i}}(\mathbb{R})$ and $\Phi_{i}^{-1}(x)=|x|^{1/p_{i}}$. Theorem 5.2 becomes now $sup_{N\geq 1}(\frac{1}{Na})^{\frac{1}{p_{1}}+\frac{1}{p_{2}}-\frac{1}{p_{3}}}<\infty$. This gives the following corollary.

Corollary 5.3** ([7]).**

Let $p_{1},p_{2},p_{3}\geq 1$ such that $\frac{1}{p_{1}}+\frac{1}{p_{2}}<\frac{1}{p_{3}}$. Then $\tilde{M}_{(p_{1},p_{2},p_{3})}(\mathbb{R})=\{0\}$.

Let us now use another approach following [7] to get other necessary conditions on multipliers.

Lemma 5.4**.**

Let $M\in\tilde{M}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ such that $F_{M}(\lambda)=|\int_{\mathbb{R}}e^{-\lambda^{2}v^{2}}M(v)dv|<\infty$ for all $\lambda>0$. Then there exists a constant $A>0$ such that

[TABLE]

Proof.

Let $\lambda>0$ and recall that $G_{\lambda}(x)=\frac{1}{\lambda}G(\frac{x}{\lambda})$ with $\widehat{G_{\lambda}}=D_{\lambda}\hat{G}$. Take $G$ such that $\hat{G}(\xi)=e^{-2\xi^{2}}$. Using formula (18) one has

[TABLE]

Since $M\in\tilde{M}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ we have

[TABLE]

Since

[TABLE]

and, using $G=D_{\lambda/\sqrt{2}}D_{\sqrt{2}/\lambda}G$, also

[TABLE]

We can write

[TABLE]

Hence we have

[TABLE]

for some constant $A>0$. $\blacksquare$

Theorem 5.5**.**

If there exists a non-zero continuous and integrable function $M\in\hat{M}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})$ then

[TABLE]

and

[TABLE]

Proof.

Let $y\in\mathbb{R}$ such that $M(y)\neq 0$. By using Lemma 5.4 to the function $M(y-\cdot)$ we obtain

[TABLE]

Therefore, using that $M\in C_{0}(\mathbb{R})$, the convolution with approximation of the identity and taking limits as $\lambda\to\infty$ one gets

[TABLE]

This gives (26).

Since $\hat{M}\neq 0$ there exists $y\in\mathbb{R}$ such that $\hat{M}(y)\neq 0$. Using again Lemma 5.4, applied to $M_{-y}M$ we obtain

[TABLE]

Therefore, taking limits as $\lambda\to 0$ we get

[TABLE]

Hence we get (27). $\blacksquare$

Corollary 5.6**.**

Let $\Phi_{1},\Phi_{2},\Phi_{3}$ be Young functions and let

[TABLE]

and

[TABLE]

If $\alpha=0$ or $\beta=0$ then $\tilde{M}_{(\Phi_{1},\Phi_{2},\Phi_{3})}(\mathbb{R})=\{0\}$.

Corollary 5.7**.**

(see [7, 26]) Let $1\leq p_{i}<\infty$ for $i=1,2,3$. If $\tilde{\mathcal{M}}_{(p_{1},p_{2},p_{3})}(\mathbb{R})\neq\{0\}$ then

[TABLE]

Proof.

For $\Phi_{i}(x)=|x|^{p_{i}}$ for $i=1,2,3$ the dilation operator $D_{\lambda}$ has norm $C_{p_{i}}(\lambda)=\lambda^{-1/p_{i}}$ for $i=1,2,3$. In this case the constants $\alpha$ and $\beta$ in the Corollary 5.6 become

[TABLE]

and

[TABLE]

Hence $\alpha=0$ and $\beta=0$ correspond to $\frac{1}{p_{3}}>\frac{1}{p_{1}}+\frac{1}{p_{2}}$ and $1+\frac{1}{p_{3}}<\frac{1}{p_{1}}+\frac{1}{p_{2}}$ respectively. The result now follows from Corollary 5.6. $\blacksquare$

Remark 5.8**.**

The reader is also referred to the work of S. Rodriguez [24] where the existence of a non-zero bilinear multiplier on r.i Banach spaces (in particular to Orlicz spaces) is related to Boyd indices of the spaces.

Acknowledgment: The second named author would like to express his gratitude to the Department of Mathematical Analysis of Valencia University for their hospitality during his stay in Valencia as a visiting researcher and to TÜBÍTAK for their support to make this work. We both would like to thank M.J Carro for calling to our attention [24, Theorem 5.10].