This is my first question here, so I apologize for any mistakes being made, in formatting, tags, or otherwise.
I'm taking a course in Fourier Analysis using Gerald Folland's "Fourier Analysis and Its Applications". Chapter 7.1 introduces convolutions, and the first exercise in this chapter is as stated in the question title, where the convention in the book is that $L^n$ is taken to mean $L^n(-\infty, \infty)$.
I should perhaps point out that I am an undergraduate with no prior knowledge of functional analysis or measure theory; this is also not a requirement for the course or the book.
I got frustrated and looked at the answer; the function is $L^1$ but not $L^2$, so I'm trying to show that it is indeed $L^1$. Trying to show that $\int_{-\infty}^\infty |\sin(x)| / |x|^{3/2} dx < \infty$ by evaluating the integral directly doesn't seem like the right approach, especially since this is a chapter on convolutions.
My idea was to write the function as a product of two functions, say as $f(x) g(x)$, and look at the convolution $f * g(x)$ at $x = 0$. From the information in the chapter, I know that \begin{equation*} \int_{-\infty}^\infty |f(y) g(y)| dy < \infty, \end{equation*} if $f(x)$ is $L^1$ and $g(x)$ is bounded, or vice versa. It's also true if both functions are $L^1$, or if $f(x)$ is piecewise continuous and $g(x)$ is bounded and vanishes outside a finite interval.
I have tried different approaches and different ways to define $f(x)$ and $g(x)$ so that these conditions are satisfied, but I'm not making any progress. Have I got completely the wrong idea? Have I misunderstood what it means for a function to be $L^1$?
Any hints are appreciated!
Write the integrals of interest as
$$\int_{-\infty}^\infty\left|\frac{\sin(x)}{x^{3/2}}\right|^{n}\,dx=\int_{|x|\le 1}\left|\frac{\sin(x)}{x^{3/2}}\right|^{n}\,dx+\int_{|x|\ge 1}\left|\frac{\sin(x)}{x^{3/2}}\right|^{n}\,dx$$
for $n=1,2$.
Note that for $x\sim 0$, $\left|\frac{\sin(x)}{|x|^{3/2}}\right|\sim \frac{1}{\sqrt{|x|}}\in L^1[-1,1]$.
However, for $x\sim 0$, $\left|\frac{\sin(x)}{|x|^{3/2}}\right|^2 \sim \frac{1}{|x|}$ which in not integrable on $[-1,1]$.