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

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 $p\ne 2$.

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.