Let f be a function on the real line R such that both f and xf are in L^2(R). Prove that f ∈ L^1(R) and the L^1 norm of f(x) is less than or equal to 8 times (the L^2 norm of f(x)) times the L^2 norm of xf(x).
I'm sorry I don't know how to use Latex to post the problem. The origional problem is here: http://www.math.purdue.edu/~bell/MA598R/advanced1.pdf. It's Problem 18. It's one of the problems created by a teacher to help students prepare qualifying exams of real analysis.
I think the purpose of the question is to ask students to apply Plancherel Theorem, but I don't know where to start. I know all the properties of fourier transforms, but I couln't relate them here. I think the trick of the problem is to break f into two parts, but I couldn't figure out how to do it. Also, the form of the inequality is somewhat similar to the uncertainty principle, but I couldn't get any hints from the proof of the uncertainty principle beacause the hypothesis is different here.
I believe the question is not hard unless one knows where to start. I've tried my best and I'm still blind. Could anyone tell me how to do it? Any hints would be appreciated.
It is just an application of the Cauchy-Swartz inequality. For any $a>0$ $$\begin{align} \int_\mathbb{R}|f|&=\int_{|x|\le a}|f|+\int_{|x|>a}|x\,f|\,\frac{1}{|x|}\\ &\le\Bigl(\int_{|x|\le a}|f|^2\Bigr)^{1/2}\Bigl(\int_{|x|\le a}1\Bigr)^{1/2}+ \Bigl(\int_{|x|>a}|x\,f|^2\Bigr)^{1/2}\Bigl(\int_{|x|>a}\frac{1}{|x|^2}\Bigr)^{1/2}\\ &\le\sqrt{2\,a}\,\|f\|_2+\frac{\sqrt2}{\sqrt{a}}\|x\,f\|_2. \end{align}$$ Choose $a=b^2\|x\,f\|_2/(2\,\|f\|_2)$ to get $$ \|f\|_1\le\Bigl(b+\frac2b\Bigr)\sqrt{\strut\|f\|_2\,\|x\,f\|_2}\ . $$ Choose $b=\sqrt2$ and square to get $$ \|f\|_1^2\le8\,\|f\|_2\,\|x\,f\|_2\ . $$