Physical interpretation of Lebesgue norm $L^p$

Question

I want to know precisely what $L^p$ represent really and physically. I know that $L^p$ norm of a function is an attempt to measure its width and height but what distinguish Lebesgue norms from $C^m$ norm for example or any other functional spaces norm?

Answer 1

The $L^p$ norms are a way to measure the "energy" of a function. The most imporant ones to start with are the case where $p=1, p=2, p=\infty$.

In general, norms are ways of somehow measuring "length" of some element of a vector space.

for $p=1$, the definition is the area under $|f|$ must be finite. This can lead to some interesting cases.

Consider the Dirichlet function $D(x)$ which is $1$ on the rationals and $0$ at the irrationals. $||f||_{L^1([0,1])}=0$, so despite the fact this function is nowhere continuous, we still have a sense of how much "area" is under the graph. In this case $0$ because "most" of the points are $0$.

Intuitively, the $L^1$ norm measures now "spiky" a function is. The $L^{\infty}$ norm measures how "wide" a function is, and $L^2$ is in some sense an averaging of height and width, the $L^p$ spaces with $2<p<\infty$ are increasingly "biased" to how "wide" $f$ is.

How are the $L^p$ norms different from $C^m$ norms:

Lots of ways. A desirable property of function spaces is that the space be "complete" with respect to the norm. Complete normed spaces are so useful they have their own name: Banach Spaces, and are the main objects studied in functional analysis.

Functions which are $C^{\infty}$ (i.e. smooth) have derivatives of all orders, and in particular, are $m$ times differentiable. It turns out $C^{\infty}$ is not only a subset of $L^p$ but also dense in it (except for $p=\infty$). The $C^m$ spaces are finicky when it comes to completeness under various norms, so as a "main" function space, they have some problems. As such, it is better to consider them as dense subspaces of $L^p$.

Why is it useful to consider them as dense subspaces instead? Here are two key reasons:

$1$. Closed subspaces of complete metric spaces (normed spaces are always metric spaces by taking the metric to be $d(x,y)=||x-y||$) are also complete. Since $C^m$ is closed, it follows that it is complete under the $L^p$ norm.

Really what we are doing is all the nice properties of continuous functions as a collection of objects are better studied when they are viewed as a subspace of $L^p$.

$2$. In the modern theory of PDE, the function spaces being used are the Sobolev spaces $W^{k,p}$ where $k$ denotes the $k-th$ weak derivative and $p$ is the corresponding $L^p$ norm.

Ideally, when studying PDEs, we want strong solutions (solutions which are $C^{\infty}$), however this is wishful thinking, and in many cases, finding strong solutions is extremely difficult. As such, the concept of "weak derivative" was introduced. As expected, if a function has a strong derivative, it has a weak derivative. This allows to create a weak formulation for $PDE$ where the PDE has a solution if we consider weak derivatives rather than strong.

If we tried to use $C^m$ to study $PDEs$ we would not be able to find weak solutions for $PDEs$, which is a cornerstone of the modern theory. In addition, the Sobolev spaces $W^{k,p}$ take into account both the norm (often interpreted as energy, like I mentioned in the first sentence) and the weak derivative.

the $C^m$ spaces under various norms are very restritive compared to the Sobolev spaces in PDE, and even just regular $L^p$ spaces.

Conclusion

The $C^m$ spaces are significantly less useful to people who study PDEs, but even in analysis, many notions such as "almost everywhere" continuous or differentiable don't make sense in the $C^m$ spaces under any norm, even under norms which make $C^m$ complete. The Sobolev spaces allow you to measure the energy of a function and smoothness/differentiability properties in a more general setting by allowing notions of "weak" differentiability, which doesn't make sense without the notion of "almost everywhere."

Much more can be said, but I hope this gives some intuition!