Using Sobolev embedding for a high frequency tail

Question

Suppose $f\in L^p(\mathbb{R}^n)$ for some $1\leq p<2$. Let $\chi$ be a smooth function which is 1 outside the ball of radius 2 and 0 inside the unit ball (we can take it to be radial if desired). I want to show the estimate $$\left\Vert \frac{\chi\cdot\hat{f}}{|\cdot|^2} \right\Vert_{L^2}\leq C_{p,n}\Vert f\Vert_{L^p}$$ for some constant $C_{p,n}$ that doesn't depend on $f$. It is apparently obtained using a Sobolev embedding estimate, although I don't see how that works. I'm interested in seeing how the details work in this example so that I can generalize this method to another problem.

Answer 1

The estimate you want to prove is not correct in general. I will only present a counterexample for the case $p=1$ and large $n$, but I think that the claim should also fail for other $1 < p < 2$, once $n$ is large enough.

Counterexample: Let $f = 1_{[-1/2, 1/2]^n}$, and define $f_k (x) := k^n \cdot f(k x)$. Note that $f, f_k \in L^1$ with $\|f_k\|_{L^1} =1=\|f\|_{L^1}$ for all $k \in \Bbb{N}$. Furthermore, $\widehat{f}(0) = 1$, and $\widehat{f_k}(\xi) = \widehat{f}(\xi/k) \to \widehat{f}(0) = 1$ as $k \to \infty$ for each $\xi \in \Bbb{R}^n$.

Thus, if your estimate was indeed true, we would get by Fatou's lemma that $$ \Big\|\frac{\chi}{|\cdot|^2}\Big\|_{L^2}^2 = \int \liminf_{k \to \infty} \Big(\frac{\chi(\xi)}{|\xi|^2}\Big)^2 |\widehat{f_k}(\xi)|\, d \xi \leq \liminf_k \Big\| \frac{\chi \widehat{f_k}}{|\cdot|^2} \Big\|_{L^2}^2 \leq \liminf_k C_{1,n}^2 \|f_k\|_{L^1}^2 = C_{1,n}^2 < \infty. $$ But since $\chi$ is $1$ outside of the ball of radius $2$, we see by introducing polar coordinates that $$ \Big\|\frac{\chi}{|\cdot|^2}\Big\|_{L^2}^2 \geq c_n \cdot \int_2^\infty r^{n-1} \frac{1}{r^2} \, dr = \infty, $$ as soon as $n \geq 2$. Here, $c_n$ is the surface area of the unit sphere in $\Bbb{R}^n$.