Why are $L^p$-spaces so ubiquitous?

Question

It always baffled me why $L^p$-spaces are the spaces of choice in almost any area (sometimes with some added regularity (Sobolev/Besov/...)). I understand that the exponent allows for convenient algebraic manipulations, but is there more behind it than mathematical convenience?

What bugs me about $L^p$-spaces is that they don't build a scale (of inclusions) but still only allow for one parameter, so by making a choice of exponent you make a choice about two (to my current knowledge) unrelated properties of your function, a) its behavior at singularities (which get milder with high exponent) and b) its tail behavior (which gets less nice with high exponent). How can it still be a good idea to ask "does this operator map $L^p$ to $L^p$" rather than "what does this operator do with singularities and what does it do with tails"? Of course answers to the latter will be harder to formulate and prove, but is that all?

Answer 1

This is indeed a very good and natural question, as one usually learns that there is a whole spectrum of $L^p$-spaces but then in practice only $L^2$ (and, to a less extent, $L^1$ and $L^\infty$) seems to pop up. Why should we care about $L^{\frac{3}{2}}$? Of course this question has many possible answers, and I find that a convincing one comes from the context of nonlinear analysis.

The basic observation is the fact that $$\lVert u^k\rVert_{L^p}=\lVert u\rVert_{L^{kp}}^k.$$ So, when dealing with nonlinear problems, we can expect that we will have to play some trick with the index $p$. We won't be able to stay in the comfortable $L^2$-space all the time.

For an example of this let us consider the following PDE: $$\tag{1} -\Delta u (x)= u^2(x), \qquad x\in \mathbb{R}^3. $$ The associated linear inhomogeneous problem $$\tag{2} -\Delta u= h $$ can be solved very satisfactorily in the functional setting of $L^2$-space via the Fourier transform: assuming that everything lies in $L^2(\mathbb{R}^3)$, we can Fourier transform termwise in (2) and write $\hat{u}(\xi)=\lvert\xi\rvert^{-2}\hat{h}$, which can then be anti-transformed back to $$u(x)= \left(\lvert 4\pi y\rvert^{-1} \ast h\right) (x)\stackrel{\text{def}}{=}(-\Delta)^{-1} h.$$ (Note that $\lvert 4\pi y\rvert^{-1}$ is exactly the fundamental solution of the Laplace operator). Setting $h=u^2$, we can now reformulate the nonlinear equation (1) as follows: $$\tag{3} u=(-\Delta)^{-1}\left( u^2\right),$$ which is now an equation of fixed-point type. We want to approach it via the contraction mapping principle, by showing that the mapping $$\Phi(u)=(-\Delta)^{-1}\left( u^2\right)$$ is contractive on some complete metric space to be specified later. To do so we need some estimates on $\Phi$ and those can be provided by the Hardy-Littlewood-Sobolev inequality, which in our case ($\alpha=2,\ n=3$) reads $$\lVert (-\Delta)^{-1} f\rVert_{L^q(\mathbb{R}^3)} \le C \lVert f\rVert_{L^p(\mathbb{R}^3)}, \qquad 2+\frac{3}{q}=\frac{3}{p}. $$ (The condition on $p$ and $q$ can be recovered via the scaling argument, by observing that both sides of this inequality are homogeneous with respect to the scaling $f(x)\mapsto f(\lambda x)$, and therefore the degrees of homogeneity must match). With $f=u^2$ this inequality reads $$\tag{4} \lVert \Phi(u)\rVert_{L^q(\mathbb{R}^3)}\le C \lVert u\rVert_{L^{2p}(\mathbb{R}^3)}^2.$$ It is now clear that our hands are tied: the only way to get something meaningful is to have $q=2p$, which means that $q=\frac{3}{2}$. Thus the right functional setting for this problem is $L^\frac{3}{2}(\mathbb{R}^3)$-space.

Indeed, if we let $B_R\subset L^{\frac{3}{2}}(\mathbb{R}^3)$ denote the closed ball of radius $R$, we see from (4) that $\Phi(B_R)\subset B_R$ if $R< 1/C$. Then, again by (4), we see that $$ \begin{split} \lVert \Phi(u)-\Phi(v)\rVert_{L^{\frac{3}{2}}(\mathbb{R}^3)}&\le C \lVert u^2-v^2\rVert_{L^{\frac{3}{4}}(\mathbb{R}^3)} \\ &=C\lVert (u+v)(u-v)\rVert_{L^{\frac{3}{4}}(\mathbb{R}^3)} \\ &\le C \lVert u+v\rVert_{L^{\frac{3}{2}}(\mathbb{R}^3)}\lVert u-v\rVert_{L^{\frac{3}{2}}(\mathbb{R}^3)} \\ &\le 2RC\lVert u-v\rVert_{L^{\frac{3}{2}}(\mathbb{R}^3)}. \end{split} $$ This means that the map $\Phi\colon B_R\to B_R$ is contractive if $R<\frac{1}{2 C}$.

As a final remark, let us observe that we have actually proven two facts:

1. the existence of a unique solution to (1) in small neighborhoods of the origin in $L^\frac{3}{2}(\mathbb{R}^3)$-space;
2. the fact that the sequence $$\begin{cases} u_{n+1}=(-\Delta)^{-1}\left( u_n^2 \right) \\ \lVert u_0\rVert_{L^{\frac{3}{2}}(\mathbb{R}^3)} \le R \end{cases}$$ converges in the $L^{\frac{3}{2}}(\mathbb{R}^3)$ topology to the solution to (1), no matter which initial condition $u_0$ we choose (provided that $R$ is sufficiently small).

Fact 2. justifies also the necessity to deal with convergence issues in $L^p$-spaces with $p\ne 2$.

Answer 2

Because $L^{p}$ spaces expose the subtle nature of arguments. You have reflexive, non-reflexive, separable, non-separable, algebra, Hilbert, Banach, etc.. And, interpolation works between such spaces because of the exponent. They're good spaces for testing conjectures. They're the original spaces that firmly established the need to separate a space from its dual, and they remain an important part of the foundation of Functional Analysis.

Answer 3

Since this is an open-ended question, I will answer this for finite-dimensional spaces, i.e., spaces $\mathbb{R}^d$ for some positive integer $d$, and for discrete optimization settings.

Let's note several spaces properties of $L_p$ spaces:

• $L_2$ norm is the only norm that is rotation invariant. That is, for any unitary operator $U$, $\|Ux\|_2 = \|x\|_2$ for all $x \in \mathbb{R}^d$.
• $L_2$ norm is the only norm that is its own dual. This leads to the Cauchy-Schwarz inequality.
• $L_1$ norm represents a natural measure of vector $x$: it is the sum of its (absolute) coordinate values, i.e., $\|x\|_1 = \sum_{i \in [d]}|x_i|$. This makes it very useful in combinatorial settings. Indeed, many combinatorial optimization problems can be written as $L_1$ minimization problems (e.g. minimum weight matchings, $k$-median clustering, shortest path problem, traveling repairman problem, minimum spanning tree etc).
• $L_\infty$ norm represents another natural measure of vector $x$: it is the maximum (absolute) coordinate value, i.e., $\|x\|_\infty = \max_{i \in [d]} |x_i|$. Many combinatorial problems are $L_\infty$ norm minimization problems (e.g., $k$-center, makespan minimization, traveling salesman problem, set cover etc).
• The class of $L_p$ norms for $p \in [1, \infty]$ is its own dual (up to appropriate normalization). This leads to many beautiful theorems about $L_p$ norms (e.g. Holder's inequality).
• $L_p$ norms provide a way to 'smoothly interpolate' between $L_1$ and $L_\infty$ norms using a single parameter $p$.

A discussion about $L_p$ norms leads to a discussion of other rich classes of norms. In combinatorial settings, many other norm classes such as 'top-$k$' norms, ordered norms etc are an active area of research.