On the translation invariant operators in $\ell^p(\mathbb{Z}^d)$

TL;DR

This paper investigates the boundedness of translation invariant operators on discrete lattice spaces, providing a Mikhlin type multiplier theorem and analyzing the boundedness of a discrete wave equation.

Contribution

It introduces a Mikhlin type multiplier theorem for $ ext{ell}^p( ext{Z}^d)$ and studies $ ext{ell}^p- ext{ell}^q$ boundedness of a discrete wave equation, advancing discrete harmonic analysis.

Findings

Established a Mikhlin type multiplier theorem for discrete spaces.

Proved $ ext{ell}^p- ext{ell}^q$ boundedness for a discrete wave equation.

Identified conditions for boundedness of translation invariant operators.

Abstract

In this paper we study boundedness of translation invariant operators in the discrete space $\ell^p(\mathbb{Z}^d)$. In this context a Mikhlin type multiplier theorem is given, yielding boundedness for certain known operators . We also give $\ell^p-\ell^q$ boundedness of a discrete wave equation.

Equations

$${\mathcal{F}}_{{\mathbb{Z}}^d}(T(f))=m{\mathcal{F}}_{{\mathbb{Z}}^d}(f)$$

$${\int }_{{\mathbb{T}}^d}f(x)dx={\int }_{[0,1{)}^d}f(x)dx.$$

$$⟨f,g⟩={\int }_{{\mathbb{T}}^d}f(\xi )\overline{g(\xi )}d\xi .$$

$$x.y=x_1{y}_1+...+x_d{y}_d\text{and}|x|=(x.x{)}^{1/2}.$$

$$\Vert f{\Vert }_{{\ell }^p}={\left(\sum _{n\in {\mathbb{Z}}^d}|f(n){|}^p\right)}^{\frac{1}{p}}$$

$${\ell }^q\subset {\ell }^p;\text{and}\Vert f{\Vert }_{{\ell }^q}\le \Vert f{\Vert }_{{\ell }^p},1\le p\le q\le \infty .$$

$${\mathcal{F}}_{{\mathbb{Z}}^d}(f)(\xi )=\sum _{n\in {\mathbb{Z}}^d}f(n)e^{i2\pi n.\xi },\xi \in {\mathbb{R}}^d.$$

$${\mathcal{F}}_{{\mathbb{Z}}^d}^{-1}(u)(n)={\int }_{{\mathbb{T}}^d}u(\xi )e^{-i2\pi \xi .n}dx,u\in L^2({\mathbb{T}}^d).$$

$$\Vert {\mathcal{F}}_{{\mathbb{Z}}^d}(f){\Vert }_{L^{p^{\prime }}}\le \Vert f{\Vert }_{{\ell }^p},$$

$$\Vert {\mathcal{F}}_{{\mathbb{Z}}^d}^{-1}(u){\Vert }_{{\ell }^{p^{\prime }}}\le \Vert u{\Vert }_{L^p},$$

$$f{\ast }_{{\mathbb{Z}}^d}g(n)=g{\ast }_{{\mathbb{Z}}^d}f(n)=\sum _{k\in {\mathbb{Z}}^d}f(k)g(n-k);n\in {\mathbb{Z}}^d.$$

$${\mathcal{F}}_{{\mathbb{Z}}^d}(f{\ast }_{{\mathbb{Z}}^d}g)={\mathcal{F}}_{{\mathbb{Z}}^d}(f){\mathcal{F}}_{{\mathbb{Z}}^d}(g).$$

$$\Vert f{\ast }_{{\mathbb{Z}}^d}g{\Vert }_{{\ell }^r}\le \Vert f{\Vert }_{{\ell }^p}\Vert g{\Vert }_{{\ell }^q}(\text{Young’s Inequality}).$$

$$K=T(\text{mathds}{1}_{\{0\}}),$$

$$f=\text{mathds}{1}_{\{0\}}{\ast }_{{\mathbb{Z}}^d}f=\sum _{n\in {\mathbb{Z}}^d}f(n){\tau }_{-n}(\text{mathds}{1}_{\{0\}}).$$

$$T(f)=T(\text{mathds}{1}_{\{0\}}{\ast }_{{\mathbb{Z}}^d}f)=\sum _{n\in {\mathbb{Z}}^d}f(n)T({\tau }_{-n}(\text{mathds}{1}_{\{0\}}))=\sum _{n\in {\mathbb{Z}}^d}f(n){\tau }_{-n}T(\text{mathds}{1}_{\{0\}}))=K{\ast }_{{\mathbb{Z}}^d}f.$$

$$f_j=\sum _{|n|\le j}f(n)\text{mathds}{1}_{\{n\}},j\ge 0$$

$$\Vert T(f_j)-K{\ast }_{{\mathbb{Z}}^d}f{\Vert }_{{\ell }^q}=\Vert K{\ast }_{{\mathbb{Z}}^d}(f_j-f){\Vert }_{{\ell }^q}\le \Vert K{\Vert }_{{\ell }^q}\Vert f_j-f{\Vert }_{{\ell }^1}$$

$$T(f)=K{\ast }_{{\mathbb{Z}}^d}f,f\in {\ell }^1({\mathbb{Z}}^d).$$

$${\mathcal{F}}_{{\mathbb{Z}}^d}(T_m(f))=m{\mathcal{F}}_{{\mathbb{Z}}^d}(f)$$

$${\int }_{\{\xi \in (0,1{)}^d,|m(\xi )|\ge s\}}dx\le \frac{c}{s^r},s>0.$$

$$|m(\xi )|\le c|\xi {|}^{-r};\xi \in (0,1{)}^d,\xi \ne 0,$$

$$1<p\le 2\le q<\infty ,\frac{1}{p}-\frac{1}{q}=\frac{1}{\alpha }.$$

$${\int }_{\{\xi \in (0,1{)}^d,\phi (\xi )\ge s\}}dx\le \frac{c}{s},s>0.$$

$$\Big{(}\int_{(0,1)^{d}}|\mathcal{F}_{\mathbb{Z}^{d}}(f)(\xi)|^{p}|\varphi(\xi)|^{2-p}d\xi\Big{)}^{\frac{1}{p}}\leq c_{p}\;\|f\|_{\ell^{p}};\quad f\in\ell^{p}(\mathbb{Z}^{d}).$$

$$T(f)=\frac{{\mathcal{F}}_{{\mathbb{Z}}^d}(f)}{\phi }.$$

$$\mu (\{\xi \in (0,1{)}^d,\phi (\xi )\le s\})$$

$$\mu (\{\xi \in (0,1{)}^d,|T(f)(\xi )|\ge s\})$$

$$\mu (\{\xi \in (0,1{)}^d,|T(f)(\xi )|\ge s\})\le \frac{\Vert f{\Vert }_{{\ell }^2}^2}{s^2},$$

$$\Big{(}\int_{(0,1)^{d}}\Big{|}\mathcal{F}_{\mathbb{Z}}(f)(\xi)(\varphi(\xi))^{(1/r-1/p^{{}^{\prime}})}\Big{|}^{r}d\xi\Big{)}^{1/r}\leq C_{p}\|f\|_{\ell^{p}};\qquad f\in\ell^{p}(\mathbb{Z}^{d}).$$

Full text

** Keywords**. Discrete Fourier transforms, Discrete Laplacian, Calderón-Zygmund operators.

2010 Mathematics Subject Classification. Primary 39A12; Secondary 47B38, 35L05.

∗Taibah University, College of Sciences, Department of Mathematics, P. O. BOX 30002, Al Madinah AL Munawarah, Saudi Arabia.

e-mail: bechiramri69@gmail.com

*∗∗*Université Tunis El Manar, Faculté des sciences de Tunis,

Laboratoire d’Analyse Mathématique et Applications,

LR11ES11, 2092 El Manar I, Tunisie.

e-mail: kh.karfaf@gmail.com

1 Introduction

It is well known that translation invariant operator from $L^{p}$ into $L^{q}$ may be represented by convolution with a tempered distribution, or equivalently by Fourier multiplier transformation. This was originally proved in the classical article of Hörmander [4]. Through many aspects of harmonic analysis, many studies have been devoted to the topic of the $L^{p}$-bounded of translation invariant operator. The most famous are the works of Calderón and Zygmund on the singular integral operators, with a large number of generalizations.

In this paper we consider translation invariant operator $T$ on $\mathbb{Z}^{d}$. The problem is essentially the multiplier problem,

[TABLE]

where the function $m$ is defined on the Torus $\mathbb{R}^{d}/\mathbb{Z}^{d}$. In this setting a Hörmander’s type theorem for $\ell^{p}-\ell^{q}$ boundedness of $T$ and an $\ell^{p}$-theorem of Mikhlin- type are given. We apply our results to get $L^{p}$-estimate for the discrete wave equation.

We begin by introducing the following notations. Let $\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$ be the $d$-dimensional torus. Functions on $\mathbb{T}^{d}$ are functions $f$ on $\mathbb{R}^{d}$ that satisfy $f(x+n)=f(x)$ for all $x\in\mathbb{R}^{d}$ and $n\in\mathbb{Z}^{d}$. Such functions are called $1$-periodic in every coordinate. Haar measure on $\mathbb{T}^{d}$ is the restriction of $d$-dimensional Lebesgue measure to the set $[0,1)^{d}$. This measure is still denoted by $dx$ and given by

[TABLE]

We denote by $L^{p}(\mathbb{T}^{d})$, $1\leq p\leq\infty$ the Lebesgue space $L^{p}([0,1)^{d},dx)$. The inner product of the Hilbert space $L^{2}(\mathbb{T}^{d})$ is given by

[TABLE]

The functions $\psi_{n}:\xi\rightarrow e^{2i\pi\xi.n}$, indexed by $n\in\mathbb{Z}^{d}$, form a complete orthonormal system of $L^{2}(\mathbb{T}^{d})$ where for $x=(x_{1},...,x_{d})$ and $y=(y_{1},...,y_{d})$ in $\mathbb{R}^{d}$

[TABLE]

By $\ell^{p}=\ell^{p}(\mathbb{Z}^{d})$, $1\leq p<\infty$ , we denote the usual Banach space of p-summable complex-valued function $f=(f(n))_{n\in\mathbb{Z}^{n}}$ equipped with the norm

[TABLE]

and $\ell^{\infty}$ the space of bounded function on $\mathbb{Z}^{d}$ with $\|f\|_{\ell^{\infty}}=\sup_{n\in\mathbb{Z}^{d}}|f(n)|$. We note the following elementary embedding relations

[TABLE]

For $f\in\ell^{2}(\mathbb{Z}^{d})$ its Fourier transform is given by

[TABLE]

The Fourier transform $\mathcal{F}_{\mathbb{Z}^{d}}$ is an isometry from $\ell^{2}(\mathbb{Z}^{d})$ into $L^{2}(\mathbb{T}^{d})$ and its inverse $\mathcal{F}_{\mathbb{Z}^{d}}^{-1}$ is given by

[TABLE]

By Riesz-Thorin convexity theorem, the Fourier transform $\mathcal{F}_{\mathbb{Z}^{d}}$ and its inverse $\mathcal{F}_{\mathbb{Z}^{d}}^{-1}$ satisfy the Hausdorff-Young inequalities

[TABLE]

and

[TABLE]

for , $1<p\leq 2$ and $1/p+1/p^{\prime}=1$.

Convolution product of two functions $f$ and $g$ of $\ell^{2}(\mathbb{Z}^{d})$ is defined by

[TABLE]

If $f,\;g\in\ell^{1}(\mathbb{Z}^{d})$ then $f*_{\mathbb{Z}^{d}}g\in\ell^{1}(\mathbb{Z}^{d})$ and

[TABLE]

Suppose $f\in\ell^{p}(\mathbb{Z}^{d})$ and $g\in\ell^{q}(\mathbb{Z}^{d})$ with $1\leq p,q,r\leq\infty$, $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}+1.$ Then

[TABLE]

2 Translation invariant operators

In this section we shall be concerned with the space of bounded operators $T$ from $\ell^{p}$ to $\ell^{q}$ for $p\leq q$, which commute with translations; that is, $\tau_{n}T=T\tau_{n}$ for all $n\in\mathbb{Z}^{d}$, where $\tau_{n}(f)(k)=f(n+k)$. It is not difficult to see that $T$ is a convolution operator. Indeed, let

[TABLE]

where $\mathds{1}_{A}$ is the characteristic function of a set $A$. Consider first, functions $f$ with compact support and write

[TABLE]

Since $T$ is translation invariant operator then

[TABLE]

Now for function $f\in\ell^{1}(\mathbb{Z}^{d})$ we let

[TABLE]

Clearly the sequence $(f_{j})_{j}$ converges to $f$ in $\ell^{r}(\mathbb{Z}^{d})$ for all $1\leq r<\infty$ and

[TABLE]

which implies that $(T(f_{j}))_{j}$ converges to $K*_{\mathbb{Z}^{d}}f$ in $\ell^{q}(\mathbb{Z}^{d})$. But $T:\ell^{p}(\mathbb{Z}^{d})\rightarrow\ell^{q}(\mathbb{Z}^{d})$ is bounded and $(f_{j})_{j}$ converges to $f$ in $\ell^{p}(\mathbb{Z}^{d})$, thus by uniqueness of the limit we may have $T(f)=K*_{\mathbb{Z}^{d}}f$. Notice that the $\ell^{p}-\ell^{q}$ boundedness of $T$ implies that $K\in\ell^{q}(\mathbb{Z}^{d})$. We state the following

Theorem 2.1**.**

If $T$ is a bounded translation invariant operator from $\ell^{p}(\mathbb{Z}^{d})$ to $\ell^{q}(\mathbb{Z}^{d})$, $p\leq q$, then there is exists a function $K\in\ell^{q}(\mathbb{Z}^{d})$ such that

[TABLE]

Translation invariant operator can also be described as Fourier multiplier transformation $T_{m}$ defined by

[TABLE]

where $m$ is a bounded measurable function $m$ on $\mathbb{T}^{d}$. An important class of $T_{m}$ is given by $L^{r,\infty}(\mathbb{T}^{d})$ for $r>1$, that is the space of measurable functions $m$ such that for some constant $c>0$,

[TABLE]

In particular if $m$ satisfies the estimate

[TABLE]

for $0<\alpha<d$ then $m\in L^{d/r,\infty}(\mathbb{T}^{d})$.

Theorem 2.2**.**

If $m\in L^{\alpha,\infty}(\mathbb{T}^{d})$ with $\alpha>1$ then $T_{m}$ is a bounded operator from $\ell^{p}(\mathbb{Z}^{d})$ into $\ell^{q}(\mathbb{Z}^{d})$, provided that

[TABLE]

This result is originally proved in [4] for translation invariant operator on $\mathbb{R}^{d}$ and in [stein] for translation invariant operator on $\mathbb{Z}$, for completeness we extended this result to $\mathbb{Z}^{d}$. The proof of Theorem 2.2 follows closely the argument of [4].

Lemma 2.3**.**

Let $\varphi\geq 0$ be a measurable function such that for some constant $c>0$

[TABLE]

Then for all $1<p\leq 2$ there exists a constant $c_{p}>0$ such that

[TABLE]

Proof.

Put $d\mu(\xi)=\varphi^{2}(\xi)\;d\xi$ and let $T$ be the operator defined on $\ell^{1}(\mathbb{Z}^{d})$ by

[TABLE]

Noting that $T(f)$ is well defined $\mu$-almost everywhere on $(0,1)^{d}$, since we have that $\mu(\{\xi\in(0,1)^{d},\;\;\varphi(\xi)=0\})=0$. In fact, for $s>0$ we have

[TABLE]

which implies that $\mu(\{\xi\in(0,1)^{d},\;\;\varphi(\xi)=0\})\leq 2cs$, for all $s>0$.

Now for $s>0$ and $f\in\ell^{1}(\mathbb{Z}^{d})$ we have

[TABLE]

Hence $T$ is of weak type $(1.1)$. In addition, from Plancherel Theorem

[TABLE]

which mean that $T$ is of weak type (2,2). We thus obtain Lemma 2.3 by using Marcinkiewicz interpolation Theorem. ∎

Lemma 2.4**.**

If $\varphi$ satisfies (2.3) and $1<p<r<p^{{}^{\prime}}<\infty$, then we have

[TABLE]

Proof.

Put $a=(p^{{}^{\prime}}-p)/(p^{{}^{\prime}}-r)$ and $a^{{}^{\prime}}$ it’s conjugate. We note the following

[TABLE]

Then using Holder’s inegality,(2.5) and the Hausdorff-Young inegality (1.2)

[TABLE]

which is the desired statement. ∎

Proof of Theorem 2.2.

Assume first that $p\leq q^{{}^{\prime}}$ and let $\varphi=|m|^{\alpha}$. Clearly from (2.1) the function $\varphi$ satisfies the condition (2.3). Hence using Lamma 2.4 with $r=q^{{}^{\prime}}$ and the fact that $1/p-1/q=1/q^{{}^{\prime}}-1/p^{{}^{\prime}}=1/\alpha,$ we obtain that

[TABLE]

Now the Hausdorff-Young inegality ( 1.3 ) implies

[TABLE]

When $q^{{}^{\prime}}<p=(p^{{}^{\prime}})^{{}^{\prime}}$, we can apply the similar argument to the adjoint operator $T^{*}_{m}=T_{\overline{m}}$, since $1<q^{{}^{\prime}}\leqslant 2\leqslant p^{{}^{\prime}}<\infty$ and $1/q^{{}^{\prime}}-1/p^{{}^{\prime}}=1/\alpha$. Hence by duality it follows that

[TABLE]

This finishes the proof of Theorem 2.2. ∎

Corollary 2.5**.**

If $m$ satifies (2.2) with $0<r<d$ then $T_{m}$ is bounded from $\ell^{p}(\mathbb{Z}^{d})$ into $\ell^{q}(\mathbb{Z}^{d})$, provided that

[TABLE]

Now observe that the inequality (2.1) can be restricted only to $s\geq 1$ which implies that $L^{\alpha,\infty}(\mathbb{T}^{d})\subset L^{\beta,\infty}(\mathbb{T}^{d})$ for all $1<\beta\leq\alpha$. Thus one can state

Corollary 2.6**.**

If $m\in L^{\alpha,\infty}(\mathbb{T}^{d})$ with $\alpha>1$ then the operator $T_{m}$ is bounded from $\ell^{p}(\mathbb{Z}^{d})$ into $\ell^{q}(\mathbb{Z}^{d})$, provided that

[TABLE]

We now study $L^{p}$-boundedness of the multiplier operator $T_{m}$. We begin by the following:

Theorem 2.7**.**

If $m$ is a $C^{d+1}$- function on $\mathbb{T}^{d}$ then $T_{m}$ is a bounded operator from $\ell^{p}(\mathbb{Z}^{d})$ into itself for all $1\leq p\leq\infty$.

Proof.

We note first that the kernel of $T_{m}$ is given by

[TABLE]

Using integrations by parts we have

[TABLE]

for all $\gamma_{1},...,\gamma_{d}\in\mathbb{N}$ with $\gamma_{1}+...+\gamma_{d}\leq d+1$ . It follows that

[TABLE]

for all $j=1,...,d$. By varying $j$ from $1$ to $d$ we deduce the following estimate

[TABLE]

which implies that the kernel $K$ is in $\ell^{1}(\mathbb{Z}^{d})$. This yields the result. ∎

Our main result is the following Hörmander-Mihlin type multiplier theorem where we may consider $T_{m}$ as a Calderón-Zygmund operator.

Theorem 2.8**.**

Let $m$ be a bounded function on the torus $\mathbb{T}^{d}$. We assume that $m$ is $C^{d+1}$-function on $\mathbb{R}^{d}\setminus\mathbb{Z}^{d}$ and satisfies the Mikhlin condition,

[TABLE]

for all $\alpha=(\alpha_{1},...,\alpha_{d})\in\mathbb{N}^{d}$ with $|\alpha|=\alpha_{1}+...+\alpha_{d}\leq d+1$. Then $T_{m}$ extended to a bounded operator from $\ell^{p}$ into itself for all $1<p<\infty$.

Proof.

Taking $\psi$ a $C^{\infty}$-function on $\mathbb{R}^{d}$ such that $\psi(\xi)=1$ for $|\xi|\leq 1/16$ and $\psi(\xi)=0$ for $|\xi|\geq 1/8$. In $(-1/2,1/2)^{d}$ we split $m$ into

[TABLE]

Since $m_{1}=m$ near the sides $|\xi_{j}|=1/2$, then $m_{1}$ can be extended to $C^{d+1}$-function on $\mathbb{T}^{d}$ and by Theorem 2.7 the operator $T_{m_{1}}$ is bounded on $\ell^{p}$ for all $1\leq p\leq\infty$. It is therefore enough to prove boundedness of $T_{m_{2}}$. Introduce the function $\mathcal{K}$ by

[TABLE]

and $K$ its restriction to $\mathbb{Z}^{d}$. One can write

[TABLE]

Our aim is to prove that $T_{m_{2}}$ is a Calderon-Zygmund operator. We consider here $\mathbb{Z}^{d}$ as a space of homogeneous type in the sense of Coifman and Weiss [CW], equipped with the Euclidean metric $(r,s)\rightarrow|r-s|$ and the counting measure. Precisely, we will prove that the kernel $K$ satisfies the integral Hörmander condition: there exists constant $c>0$ such that for all $s\in\mathbb{Z}^{d}$,

[TABLE]

To begin, let $\phi$ be a $C^{\infty}$- function on $\mathbb{R}$, such that $supp(\phi)\subset\{t\in\mathbb{R};\>\;1/2\leq|t|\leq 2\}$ and

[TABLE]

So we have

[TABLE]

and we may write

[TABLE]

Notice that

[TABLE]

for some constant $c>0$, since this sum contains at most three non-null terms. We now set

[TABLE]

By Fubini’s Theorem and (2.8) the sum $\sum_{j}\mathcal{K}_{j}$ converges and

[TABLE]

Next we shall give estimates of the kernels $\mathcal{K}$. Observe first that $m_{2}$ satisfies the condition $(\ref{mik})$ and from this estimate and the compactness of $supp(\varphi)$ one can obtain the following

[TABLE]

for $i=1,...,d$ and for $|\alpha|\leq d+1$. It follows that

[TABLE]

and

[TABLE]

Using (2.9) with $s=0$ and $s=d+1$ we get

[TABLE]

Similarly by (2.10) with $s=0$ and $s=d+2$

[TABLE]

Therefore we obtain the following estimates

[TABLE]

Note that $\mathcal{K}$ is a $C^{\infty}$-function and all of its derivatives are bounded.

Now we come to the proof of (2.7). Let $r,s\in\mathbb{Z}^{d}$ with $|r|\geq 2|s|$. By mean value Theorem we have

[TABLE]

Using (2.11) and the fact that

[TABLE]

we obtain the following

[TABLE]

Now use that

[TABLE]

we have

[TABLE]

It is not hard to see that

[TABLE]

and from which

[TABLE]

Combine (2.13) with (2.12), yield that

[TABLE]

which proves (2.7). The proof of Theorem 2.8 follows. ∎

In the next we shall be concerned with Mikhlin type multiplier on $\mathbb{Z}$. Thus one can read Theorem 2.8 as follows

Theorem 2.9**.**

If $m$ is a bounded $C^{2}-$function on $(0,1)$ such that

[TABLE]

then $T_{m}$ is a bounded operator from $\ell^{p}(\mathbb{Z})$ into $\ell^{p}(\mathbb{Z})$ for $1<p<\infty$.

Clearly condition (2.14) is exactly (2.6) when extending $m$ to a periodic function. Now we replace the interval $(0,1)$ by a bounded interval $(a,b)$. For $f\in\ell^{2}(\mathbb{Z})$ we define its Fourier transform by

[TABLE]

and its inverse

[TABLE]

For a bounded function $m$ on $(a,b)$ we define on $\ell^{2}(\mathbb{Z})$ the operator $T_{m}^{a,b}$ by

[TABLE]

for $n\in\mathbb{Z}$. According to Theorem 2.9 we have the following

Corollary 2.10**.**

If $m$ is a bounded $C^{2}$-function on a bounded interval $(a,b)$, such that for some constant $c>0$

[TABLE]

then $T_{m}^{a,b}$ is a bounded operator from $\ell^{p}(\mathbb{Z})$ into $\ell^{p}(\mathbb{Z})$ for $1<p<\infty$.

As a typical example we have $m=\chi_{(a,b)}$ the characteristic function of $(a,b)$. Theorem 2.9 can be generalized as follows.

Theorem 2.11**.**

Let $(a_{j})_{0\leq j\leq s}$ be a subdivision of $[0,1]$. If $m$ is a bounded $C^{2}$-function on $(0,1)$ such that, for some constant $c>0$,

[TABLE]

then $T_{m}$ is a bounded operator from $\ell^{p}(\mathbb{Z})$ into $\ell^{p}(\mathbb{Z})$ for $1<p<\infty$.

Proof.

Assume first that $s=2$ and let $0=a_{0}<a_{1}<a_{2}=1$. Let $\varepsilon>0$ and $\varphi$ a be $C^{\infty}$ function on $\mathbb{R}$ such that $]a_{1}-2\varepsilon,a_{1}+2\varepsilon[\subset]0,1[$ and $\varphi(\xi)=0$ for $\xi\in]a_{1}-\varepsilon,a_{1}+\varepsilon[$ and $\varphi(\xi)=1$ for all $\xi\notin]a_{1}-2\varepsilon,a_{1}+2\varepsilon[$. Put

[TABLE]

and

[TABLE]

Clearly the boundedness of $T_{m_{1}}$ is a consequence of Theorem 2.9. However, if we consider $\widetilde{m_{2}}$ the $1$- periodic function such that $\widetilde{m_{2}}(\xi)=m_{2}(\xi)$ for $\xi\in[0,1]$, then one can write

[TABLE]

where $\widetilde{f}$ is the function given by $\widetilde{f}(n)=e^{i2\pi na_{1}}f(n)$, $n\in\mathbb{Z}$. Now obseve that $\widetilde{m_{2}}$ satisfies the hypothesis of Corollary 2.10 on $(a_{1},a_{1}+1)$, then the boundedness of $T_{m_{2}}$ follows.

Now for $s\geq 3$ we proceed as follows: choose $\varepsilon>0$ such that the intervals $]a_{j}-2\varepsilon,a_{j}+2\varepsilon[$ are disjoint for all $1\leq j\leq s-1$ and $C^{\infty}$ functions $\varphi_{j}$ with $\varphi_{j}(\xi)=0$ for $\xi\in]a_{j}-\varepsilon,a_{j}+\varepsilon[$ and $\varphi(\xi)=1$ for $\xi\notin]a_{j}-2\varepsilon,a_{j}+2\varepsilon[$. Put

[TABLE]

and write

[TABLE]

and

[TABLE]

Therefore from the above argument all the operators $T_{m_{j}}$ are bounded on $\ell^{p}(\mathbb{Z})$, $1<p<\infty$. ∎

3 Applications

3.1 Discrete Riesz Transforms

For a complex-valued function $f$ on $\mathbb{Z}^{d}$ its discrete Laplacian is given by

[TABLE]

where $\partial_{j}f(n)=f(n+e_{j})-f(n)$ and $\partial^{*}_{j}f(n)=f(n)-f(n-e_{j})$. We have

[TABLE]

The discrete Laplacian $\Delta_{d}$ is a bounded self-adjoint operator on $\ell^{2}(\mathbb{Z}^{d})$ and one has

[TABLE]

The discrete Riesz transforms $R_{j}$ , $j=1,...,d$, associated with $\Delta_{d}$ are defined on $\ell^{2}(\mathbb{Z}^{d})$ as the multiplier operators

[TABLE]

its can be interpret as $R_{j}(f)=\partial_{j}\Delta_{d}^{-1/2}$. Let us set

[TABLE]

and prove that $\psi_{j}$ satisfies the Mikhlin condition (2.6). This can be seen by using the fact that $\gamma\in\mathbb{N}^{d}$, $\partial^{\gamma}\psi_{j}$ is a linear combination of the following functions

[TABLE]

where $\sum_{k=1}^{d}\alpha_{k}\leq|\gamma|+1$ and $\sum_{k=1}^{d}\beta_{k}\leq|\gamma|$. Hence using the fact that for $k=1,...,d$, $\xi_{k}\in(-1/2,1/2)$ and $2|\xi_{k}|\leq|\sin\pi\xi_{k}|\leq\pi|\xi_{k}|$ we obtain

[TABLE]

Therefore we can apply Theorem 2.8 to assert that $R_{j}$ is bounded on $\ell^{p}(\mathbb{Z}^{d})$ for $1<p<\infty$.

3.2 Imaginary powers of the discrete Laplace operator $\Delta_{d}$

Theorem 2.8 also applies to imaginary powers of the discrete Laplacian: $(-\Delta_{d})^{it}$ for $t\in\mathbb{R}$, it is the multiplier operator with multiplier $\left(4\sum_{j=1}^{d}\sin^{2}(\pi\xi_{j})\right)^{it}$

3.3 Strichartz type estimates for discrete wave equation

We define the $d$-dimensional discrete wave equation by

[TABLE]

where $f$ and $g$ are a given suitable functions on $\mathbb{Z}^{d}$. Considered as a discrete counterpart of the continuous wave equation, many authors have been interested in studying this equation see, for example, [6, 5, 7] and the references therein. Putting

[TABLE]

and applying the discrete Fourier transform, considering t as a parameter, we deduce that the solution of (3.1) can be written (formally) in the form

[TABLE]

We will prove the following version of the Strichartz estimates.

[TABLE]

for all $1<p\leq 2\leq q<\infty$.

Let us obseve first that $\xi\rightarrow\sin(t\phi(\xi))/\phi(\xi)$ is a $C^{\infty}$- function on $\mathbb{T}^{d}$ and then in view of Theorem 2.7 we have

[TABLE]

whenever $1<p\leq q<\infty$. To prove (3.2) it suffices to show that

[TABLE]

We write

[TABLE]

As

[TABLE]

it follows from Theorem 2.2 that

[TABLE]

and by using $\ell^{p}$-boundedness of $R_{j}$,

[TABLE]

which conclude the proof of (3.2).