TITLE: Qual problem in Analysis
QUESTION [5 upvotes]: I am having trouble with the following qual problem.  Some help would be awesome.  Thanks.
Let $f$ be a measurable function on $(0, ∞).$ Let $p > 1/2$ and define $g(x) = (x^p + x^{−p})f(x).$
Show that if $g ∈ L_2(0,∞)$ then $f ∈ L_1(0,∞).$

REPLY [4 votes]: Cauchy Schwarz:
$$\int_0^\infty |f(x)| dx = \int_0^\infty \frac{|g(x)|}{x^p + x^{-p}} \leq \left( \int_0^\infty |g(x)|^2 dx \right)^{1/2} \cdot \left(\int_0^\infty \frac{1}{(x^p+x^{-p})^2} dx \right)^{1/2} < \infty$$
when $p > 1/2$.