# Translation invariant operators

In this notes we show that bounded translation invariant operators from ${L^{p}(\mathbb{R}^{d})}$ to ${L^{q}(\mathbb{R}^{d})}$ are essentially convolution singular integral operators for certain $p,q$. More precisely, let ${A}$ be a bounded linear operator from ${L^{p}(\mathbb{R}^{d})}$ to ${L^{q}(\mathbb{R}^{d})}$, we say that ${A}$ is a translation invariant operator, if

$\displaystyle \tau_{y}A=A\tau_{y} \ \ \ \ \ (1)$

where ${(\tau_{y}\varphi)(x)=\varphi(x-y)}$ denotes the translation operator on , $\mathbb{R}^{d}$ and $\varphi$ is a  Schwartz function. The object of this notes is only find a tempered distribution ${T\in \mathcal{S}'(\mathbb{R}^{d})}$ such that

$\displaystyle (A\varphi)(x) =(\varphi\ast T)(x), \ \ \ \ \ (2)$

if ${A}$ is translation invariant operator and ${A}$ is bounded on certain Lebesgue ${L^{q}(\mathbb{R}^{d})}$ spaces. Firstly, let us recall some well know results.

• Let ${\alpha=(\alpha_1,\cdots,\alpha_d)\in (\mathbb{N}\cup\{0\}})^{d}$, ${x^{\alpha}=x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_d^{\alpha_d}}$. Then there exist a positive constant ${C_{d,\alpha}}$ such that

$\displaystyle \vert x^{\alpha}\vert \leq C_{d,\alpha}\vert x\vert^{\vert \alpha\vert}, \;\; \vert \alpha\vert =\alpha_1+\cdots+\alpha_{d}. \ \ \ \ \ (3)$

In fact, let ${\mathbb{S}^{d-1}}$ be an unity sphere and let ${f:\mathbb{S}^{d-1}\rightarrow \mathbb{R}}$ which is given by  ${f(x_1,\cdots,x_d)=\vert x_{1}^{\alpha_1}\cdots x_{d}^{\alpha_d}\vert}$. Then the function ${f}$ is a continuous function in ${\mathbb{R}^{d}}$ and for

$\displaystyle C_{d,\alpha}=\sup_{x\in \mathbb{S}^{d-1}}\vert f(x)\vert,\nonumber \ \ \ \ \ (4)$

we get ${\vert y^{\alpha}\vert \leq C_{d,\alpha}}$, for all ${y=\frac{x}{\vert x\vert}\in \mathbb{S}^{d-1}}$ and ${x\in\mathbb{R}^{d}}$. Since

$\displaystyle y^{\alpha}=\left(\frac{x_{1}}{\vert x\vert}\right)^{\alpha_1}\cdots \left(\frac{x_{d}}{\vert x\vert}\right)^{\alpha_d}=\frac{x_1^{\alpha_1}x_2^{\alpha_2}\cdots x_d^{\alpha_d}}{\vert x\vert^{\alpha_1+\cdots+\alpha_{d}}},\nonumber \ \ \ \ \ (5)$

we obtain (3).

• Let ${k\in \mathbb{Z}^{+}}$, we have

$\displaystyle \vert x\vert^{k} \leq C_{d,k}\sum_{\vert \beta\vert=k}\vert x^{\beta}\vert, \ \ \ \ \ (6)$

for all ${x\in \mathbb{R}^{d}}$.

• $\displaystyle \frac{C_{d, d+1}}{(2\pi) ^{d+1}}\sum_{\vert \alpha\vert \leq d+1}\vert (-2\pi i x)^{\alpha}\vert \geq (1+\vert x\vert)^{-d-1}. \ \ \ \ \ (8)$

Indeed, by inequality (6) we have

$\displaystyle \begin{array}{rcl} \frac{C_{d, d+1}}{(2\pi)^{d+1}}\sum_{\vert \alpha\vert \leq d+1}\vert (-2\pi x)^{\alpha}\vert \geq \vert x\vert ^{d+1}\geq \left(\frac{1}{2}\right)^{d+1}\geq (1+\vert x\vert)^{-d-1}. \end{array}$

Lemma 1 Let ${u}$ be a tempered distribution. Suppose that ${D^{\alpha} u\in L^{p}_{loc}(\mathbb{R}^{d})}$ for all ${\vert \alpha\vert \leq d+1}$. Then ${u}$ is a function and there exists a positive constant ${C}$ such that

$\displaystyle \vert u(x)\vert \leq C\sum_{\vert \alpha\vert \leq d+1}\left(\int_{\mathbb{R}^{d}}\vert D^{\alpha}u(y)\vert^{p}dy\right)^{\frac{1}{p}}, \;\;\vert x\vert\geq 2. \ \ \ \ \ (9)$

Proof: Let ${C_{0}^{\infty}(\mathbb{R}^{d})}$ be the space of all bump functions ${\varphi_{r}:\mathbb{R}^{d}\rightarrow\mathbb{R}}$ with support on  euclidean sphere ${\mathbb{S}_r}$ of radius ${r}$. Since ${u\in L^{p}(\mathbb{R}^{d})}$, it follows from Hölder inequality that ${w:=\varphi_{r}u\in L^{1}(\mathbb{R}^{d})}$. Moreover, we have that  ${\hat{w}}$ lives in ${L^{1}(\mathbb{R}^{d})}$ too. This follows by means of the inequality (8) and some properties of Fourier transform on euclidean spaces. Indeed,

$\displaystyle \begin{array}{rcl} \vert\widehat{w}(x)\vert &\leq & C_{d}(1+\vert x\vert)^{-(d+1)}\sum_{\vert\alpha\vert\leq d+1}\vert (-2\pi i x)^{\alpha}\widehat{w}(x)\vert\\ \\ &\leq &C_{d}(1+\vert x\vert)^{-(d+1)}\sum_{\vert\alpha\vert\leq d+1}\Vert\widehat{D^{\alpha}w}\Vert_{\infty}\\ \\ &\leq &C_{d}(1+\vert x\vert)^{-(d+1)}\sum_{\vert\alpha\vert\leq d+1}\Vert D^{\alpha}w\Vert_{L^{1}(\mathbb{R}^{d})}\\ \\ &\leq &C_{d}(1+\vert x\vert)^{-(d+1)}\sum_{\vert\alpha\vert\leq d+1}\Vert \sum_{\gamma\leq \alpha}\binom{\alpha}{\gamma}(D^{\gamma}u)(D^{\alpha-\gamma}\varphi_{r})\Vert_{L^{1}}\\ \\ &\leq &C_{d,\alpha, p'}(1+\vert x\vert)^{-(d+1)}\sum_{\vert\alpha\vert\leq d+1}\Vert D^{\alpha}u\Vert_{L^{p}}. \end{array}$

Integrating the last inequality, we obtain

$\displaystyle \Vert \widehat{w}\Vert_{L^{1}}\leq C\sum_{\vert\alpha\vert\leq d+1}\Vert D^{\alpha}u\Vert_{L^{p}(\mathbb{R}^{d})}.\nonumber \ \ \ \ \ (10)$

It follows that Fourier inverse of ${\widehat{w}}$ agrees, almost everywhere, with an uniformly continuous function (see Folland p. 239). Since ${\varphi_{r}(x)=1}$ on ${\mathbb{S}_r}$, so the tempered distribution ${u}$ can be re-defined as an uniformly continuous function, namely ${\psi}$, which is defined on ${B_{r}(y)}$ for all ${r>0}$. Since the choice of ${r}$ is not relevant, we get

$\displaystyle \begin{array}{rcl} \Vert u\Vert_{\infty}=\Vert \psi \Vert_{L^{\infty}}\leq \Vert \widehat{\psi}\Vert_{L^{1}}= \Vert \widehat{w}\Vert_{L^{1}}\leq C\sum_{\vert\alpha\vert\leq d+1}\Vert D^{\alpha}u\Vert_{L^{p}(\mathbb{R}^{d})}. \end{array}$

$\Box$

Lemma 2 Let ${u\in\mathcal{S}'(\mathbb{R}^{d})}$ and let  ${D^{\alpha}u\in L^{p}(\mathbb{R}^{d})}$. If ${A:L^{p}(\mathbb{R}^{d})\rightarrow L^q(\mathbb{R}^{d})}$ is a bounded linear operator which is translation invariant, then ${A}$ commutes with derivatives, that is,

$\displaystyle A(D^{\alpha}u)=D^{\alpha}(Au), \ \ \ \ \ (11)$

for all multi-index ${\alpha}$.

Proof: For readers convenience, let ${\alpha=e_j:=(0,\cdots,0, 1,0,\cdots,0)}$. The general case, follows by interactions. Firstly, let ${h\in\mathbb{R}}$ and set

$\displaystyle u_{h}(x):=\tau_{-he_j}(x)=u(x_1,\cdots, x_{j}+h,\cdots,x_{d}) .\nonumber \ \ \ \ \ (12)$

Let $v=Au$ and note that $Au_{h}=A\tau_{-he_j}=\tau_{-he_j}A=v_{h}$. It follows that,

$\displaystyle A(\frac{u_h-u}{h})=\frac{v_h-v}{h}\nonumber \ \ \ \ \ (14)$

which converges to ${D_{x_j}v}$ in ${L^q(\mathbb{R}^{d})}$ as ${h\rightarrow 0}$. The continuity of ${A}$ in join with ${D_{x_j} u\in L^{p}(\mathbb{R}^{d})}$ give us

$\displaystyle \begin{array}{rcl} \Vert \frac{v_h-v}{h} -A(D_{x_j} u)\Vert_{L^q(\mathbb{R}^{d})}&=&\Vert A(\frac{u_h-u}{h} -D_{x_j} u)\Vert_{L^q(\mathbb{R}^{d})}\\\\&\leq & \Vert A\Vert\, \Vert \frac{u_h-u}{h} -D_{x_j} u\Vert_{L^p(\mathbb{R}^{d})}. \end{array}$

Therefore, by uniqueness of limit in ${L^q(\mathbb{R}^{d})}$ one has

$\displaystyle D_{x_j}Au =A(D_{x_j} u),\nonumber \ \ \ \ \ (15)$

as we desired. $\Box$

Let us to introduce the reflection operator in ${\mathcal{S}(\mathbb{R}^{d})}$ which is given by,

$\displaystyle (R\varphi)(x)=\varphi(-x), \ \ \ \ \ (16)$

for all ${\varphi\in\mathcal{S}(\mathbb{R}^{d})}$. Notice that, for ${\varphi\in L^{1}_{loc}(\mathbb{R}^{d})}$ the integral

$\displaystyle \begin{array}{rcl} (\varphi\ast u)(x)&=&\int u(y) \varphi(-(y-x)) dy\\\\ &=&\int u(y) (R\varphi)(y-x) dy\\\\ &=&\int u(y) (\tau_{x}R\varphi)(y) dy \end{array}$

for all ${u\in C_{0}^{\infty}(\mathbb{R}^{d})}$. This motivates to  define convolution operator between a Schwartz function ${\varphi\in\mathcal{S}(\mathbb{R}^{d})}$ and a tempered distribution ${u\in\mathcal{S}'(\mathbb{R}^{d})}$ as follows

$\displaystyle (\varphi\ast u )(x)= u(\tau_{x}R\varphi). \ \ \ \ \ (17)$

Theorem 3 Let ${A: L^p(\mathbb{R}^{d})\rightarrow L^q(\mathbb{R}^{d})}$ be a bounded linear operator. If ${A}$ is translation invariant, there exists ${T\in \mathcal{S}'(\mathbb{R}^{d})}$ such that (2) holds.

Proof: Let ${\varphi\in\mathcal{S}(\mathbb{R}^{d})}$ be a Schwartz function such that ${D^{\alpha}\varphi\in L^{p}(\mathbb{R}^{d})}$. In view of Lemma 1 and Lemma 2, ${A\varphi}$ is a continuous function which satisfy the estimate

$\displaystyle \begin{array}{rcl} \vert (A\varphi)(0)\vert &\leq& C \sum_{\vert \alpha\vert\leq d+1} \Vert D^{\alpha} (A\varphi)\Vert_{L^{q}(\mathbb{R}^{d})}\\ \\&\leq & C\,\Vert A\Vert \sum_{\vert \alpha\vert\leq d+1} \Vert D^{\alpha}\varphi\Vert_{L^{p}(\mathbb{R}^{d})}\\ \\&\leq & C\,\Vert A\Vert \left(\int (1+\vert x\vert^{2})^{-Np}\right)^{1/p}\sum_{\vert \alpha\vert\leq d+1} \Vert \varphi\Vert_{(N,\alpha)}\\ \\&\leq & C\,\Vert A\Vert \sum_{\vert \alpha\vert\leq d+1} \Vert \varphi\Vert_{(N,\alpha)} \end{array}$

for ${N>d/p}$. Here ${\Vert \cdot\Vert_{(N,\alpha)}}$ stands for Schwartz norm

$\displaystyle \Vert \varphi\Vert_{(N,\alpha)}=\sup_{x\in \mathbb{R}^{d}}\,(1+\vert x\vert^{2})^{N}\vert D^{\alpha}\varphi(x)\vert. \ \ \ \ \ (18)$

In other words, there exists a tempered distribution ${T}$ such that $\displaystyle (A\varphi)(0)=T(R\varphi)$. As  ${A}$ is translation invariance, in view of (17) we have

$\displaystyle \begin{array}{rcl} (A\varphi)(x)&=&\tau_{-x}(A\varphi)(0)=(A\tau_{-x}\varphi)(0)\\ \\&=&T(R\tau_{-x}\varphi)=T(\tau_{x}R\varphi)=(\varphi\ast T)(x) \end{array}$

because of ${(R\tau_{x}\varphi)(z)=(\tau_{-x}R\varphi)(z)}$. $\\\\\\\Box$