Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?

Huge chunks of the theory of nonlinear PDEs rely critically on analysis in $L^p$-spaces.

• Let's take the 3D Navier-Stokes equations for example. Leray proved in 1933 existence of a weak solution to the corresponding Cauchy problem with initial data from the space $L^2(\mathbb R^3)$. Unfortunately, it is still a major open problem whether the Leray weak solution is unique. But if one chooses the initial data from $L^3(\mathbb R^3)$, then Kato showed that there is a unique strong solution to the Navier-Stokes equations (which is known to exist locally in time). $L^3$ is the "weakest" $L^p$-space of initial data which is known to give rise to unique solutions of the 3D Navier-Stokes.

• In some cases the structure of the equations suggests the choice of $L^p$ as the most natural space to work in. For instance, many equations stemming from non-Newtonian fluid dynamics and image processing involve the $p$-Laplacian $\nabla\left(|\nabla u|^{p-2}\nabla u\right)$ with $1 < p < \infty.$ Here the $L^p$-space and $L^p$-based Sobolev spaces provide a natural framework to study well-posedness and regularity issues.

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• Yet another example from harmonic analysis (which goes back to Paley and Zigmund, I think). Let $$F(x,\omega)=\sum\limits_{n\in\mathbb Z^d} g_n(\omega)c_ne^{inx},\quad x\in \mathbb T^d,$$ where $g_n$ is a sequence of independent normalized Gaussians and $(c_n)$ is a non-random element of $l^2(\mathbb Z^d)$. Then the function $F$ belongs almost surely to any $L^p(\mathbb T^d)$, $2\leq p <\infty$ and it does not belong almost surely to $L^{\infty}(\mathbb T^d)$.

There have been very recent applications of this resut to the existence of solutions to the nonlinear Schrodinger equations with random initial data (due to Burq, Gérard, Tzvetkov et al).

I feel as though this question may have come up before. Anyhow, the $\ell_4$ norm, and more generally the $\ell_{2k}$ norm for any positive integer $k$, come up naturally in Fourier analysis, since the $\ell_{2k}$ norm of the Fourier transform of $f$ equals the sum of $f(x_1)...f(x_k)\overline{f(y_1)...f(y_k)}$ over all $x_1+...+x_k=y_1+...+y_k$. That sort of sum comes up a lot in additive combinatorics, especially when $f$ is closely related to the characteristic function of a set. And you can get other norms by duality -- for instance the $4/3$ norm is the dual of the 4-norm, and therefore comes up too.

Tim, I've got two words for you: interpolation theorems (e.g., Riesz-Thorin and Marcinkiewicz interpolation theorems). Such theorems let you pass from information about some operators on $L^1$ and $L^\infty$ to some operators on $L^2$ using all the intermediate exponents $p$.

The point here is not that one actually cares about $L^{37.24}$ for its own sake, but the interpolation theorems show you that such "exotic" $L^p$-spaces can be at the service of her majesty $L^2$. I think for a student, these interpolation theorems provide an attractive reason to care about $L^p$ for all $p \geq 1$.

This is not my area at all, so I welcome follow-up comments from analysts on this answer.