Show that $K(x,y)=(2^{nk}\mathcal{F}^{-1}(2^kx))_{k\in\mathbb{Z}}$ is a singular kernel

Question

I want to show that for a bump function $\psi$ with support in the annulus $\{\frac{1}{2}\leq\vert x\vert\leq2\}$ the kernel $K(x,y)=(2^{nk}\mathcal{F}^{-1}\psi(2^k(x-y)))_{k\in\mathbb{Z}}$ is a vector valued singular kernel, i.e. it must be shown that $\Vert K(x,y)\Vert_{\ell^2(\mathbb{Z})}\leq C\frac{1}{\vert x-y\vert^n}$ with a constant C depending only on n. Unfortunately I'm not able to show this, so I'm hoping for your help. greets Lukas

Answer 1

Presumably, you mean a $C_{c}^{\infty}$ function by "bump function." I am going to change your notation for convenience: I will write $\widehat{\psi}$ where you have written $\psi$. For each $k$, set $\psi_{k}(x):=2^{kn}\psi(2^{k}x)$. For each $x$, define an operator-valued ($\mathbb{C}\rightarrow\ell^{2}(\mathbb{Z})$) by $\vec{K}(x)=(\psi_{k}(x))_{k}$. Since $\psi$ is Schwartz, we have that

$$|\psi(x)|\lesssim_{N}(1+|x|)^{-N}, \quad |\nabla\psi(x)|\lesssim_{N}(1+|x|)^{-N}$$ for $N>0$ arbitrarily large (say $N=n+1$). The size condition follows from

\begin{align*} \|\vec{K}(x)\|_{\mathbb{C}\rightarrow\ell^{2}}&=\left(\sum_{2^{k}|x|<1}|\psi(x)|^{2}+\sum_{2^{k}|x|\geq 1}|\psi_{k}(x)|^{2}\right)^{1/2}\\ &\leq\sum_{2^{k}|x|\leq 1}|\psi_{k}(x)|+\sum_{2^{k}|x|>1}|\psi_{k}(x)|\\ &\lesssim\sum_{2^{k}|x|<1}2^{kn}+\sum_{2^{k}|x|\geq 1}2^{kn}(|2^{k}x|)^{-N}\\ \end{align*} where we use the fact that $\ell^{1}$-norm dominates $\ell^{2}$-norm. Let $k_{0}$ and $k_{0}'$ respectively denote the maximal and minimal integers such that $2^{k_{0}}<|x|^{-1}$ and $2^{k_{1}}\geq|x|^{-1}$. Then the last expression above is majorized by $$\lesssim_{n}2^{k_{0}n}+2^{k_{1}(n-N)}|x|^{-N}\lesssim_{n}|x|^{-n}+2^{k_{1}(n-N)}|x|^{-N}\lesssim |x|^{-n}$$

by our choice of $N$. You said you didn't to verify smoothness, so I won't do it. By a modification of the argument given, you can verify that $\vec{K}$ satisfies the Hormander smoothness condition.