I have started learning about $L^p$ spaces and it seems very abstract at the moment. I was hoping someone could give me some concrete examples of functions that are, or are not, in certain $L^p$ spaces so I can gain some intution that helps when dealing with the theory.
- For example, I understand that we if we have $0 < p < q < \infty$ then $L^q(U) \subset L^p(U)$. So what is an example of a function $f$ such that $f \in L^1(U)$ but $f \notin L^2(U)$?
- How about functions in $L^\infty(U)$, it seems from the above that as you increase the indices $p$ and $q$ it places tighter constraints on the functions, so that seems to imply that $L^\infty(U)$ will restrict more functions than any $L^p(U)$ with $0 < p < \infty$? Or is $L^\infty$ a special case? What is a concrete example of a function that is (or isn't) in $L^\infty(U)$ but not (or is) in $L^p(U)$?
It is not true in general that $p < q$ implies $L^q \subset L^p$. For example, if $U = (1,\infty)$ and $f(x) = 1/x$, then $f \in L^2$ but $f \not \in L^1$. However, it is true if $U$ has finite measure.
For an example of a function which is in $L^1$ but not $L^2$, consider $f(x) = 1/\sqrt{x}$, say on $U = (0,1)$. Note that this function is also not in $L^{\infty}$.
If $U$ has finite measure, then any $L^{\infty}$ function is also in every $L^p$ for $p < \infty$, because we have $|f(x)| \leq \|f\|_{\infty}$ for almost every $x$, and therefore $$\int_U |f|^p \leq \int_U \|f\|_{\infty}^p = \|f\|_{\infty}^p \mu(U) < \infty$$
On the other hand, if $U$ has infinite measure, then $f$ can be in $L^{\infty}$ but not in any $L^p$ for $p < \infty$. For a simple example, let $f$ be the constant function $1$.