I am now studying functional analysis, especially $L^p$ spaces and I'm wondering what kind of property of functions the $L^p$ norms measure.
I know that when $\Omega$ has a finite measure, there exists the relation, $L^q(\Omega)\subset L^p(\Omega)$ for $1 \leq p < q \leq \infty $. But in general, it seems that there is no such a simple relation, and I ask this question.
(If there would be such a relation, it would be safe to say "That is the resolving power of each $L^p$ norms.", but there is not....)
Anything will help, thanks!
Assume we are considering $\Omega = \mathbb{R}$. It may help to look at some examples: for instance, $f(x) := \inf\left\{1,\frac{1}{|x|}\right\}$ is not in $L^1$, but in $L^p$ for $p > 1$; in this sense, $L^p$ measures "how fast does a function decay". Consider $f(x) := \chi_{[-1,1]} \cdot \frac{1}{\sqrt{|x|}}$, then $f$ is in $L^p$ for $1 \leq p < 2$; in this case $L^p$ measures "how fast does a function blow up". In general, $p=1$ measures if a function is absolutely integrable. As $p$ increases, the integral $\int |f(x)|^p\, dx$ will, in a very vague sense, concentrate on the portion where $|f(x)|$ is large. In particular, $p = \infty$ corresponds to the $\sup$-norm, which measures the supremum of $|f(x)|$ on the entire domain.
One general (yet interesting) relation is that $L^p \cap L^q \subset L^r \subset L^p + L^q$ for any $p \leq r \leq q$. The first inclusion follows from interpolation between $L^p$ norms; the second inclusion is (roughly) because we can take the "smaller" and "larger" part of an $L^r$ function and argue that they are in the desired spaces.
Also $C_c^\infty$ is dense in any $L^p$ for all $1 \leq p < \infty$ (so often times proving properties of operators on $L^p$ reduces to proving them for $C_c^\infty$ functions). Hence the $L^p$ spaces all contain the same dense subset (for $p <\infty$).