Practical applications where one needs $L^p$ with $p\not\in\{1,2,\infty\}$

Question

It appears to me that in practical applications one only ever needs the $L^1$, $L^2$ and $L^\infty$ norms, which are rather special cases among the $L^p$ norms. However, I am virtually sure that this impression must be wrong and based only on my ignorance. Therefore I would like to ask you for examples of practical applications where one genuinely needs to approximate or minimize with respect to an $L^p$ with $p\not\in\{1,2,\infty\}$.

Note that I'm not asking about results whose proofs rely on other values of $p$ — I'm aware that the ability to consider other $L^p$ is often crucial theoretically. What my question is about is specifically the case where a problem from applied mathematics leads one to approximate a function in an $L^p$ norm with $p\not\in\{1,2,\infty\}$, or where the natural functional to minimize in order to solve an applied problem is such an $L^p$ norm.

• ho boon suan: math.stackexchange.com/questions/200564/…
• Jochen Glueck: Could you please specify what you mean by "practical application"?
• mathworker21: $L^4$ is pretty common, e.g. in additive combinatorics
• Iosif Pinelis: It might of interest that the similar question, asked about 15 years ago, garnered 164 upvotes.
• mathworker21: @gmvh One has $$\bigl\|\widehat{1_A}\bigr\|_4^4 = \#\{(a,b,c,d) \in A^4 : a + b = c + d\},$$ where $\widehat{1_A} : \mathbb{T} \to \mathbb{C}$ is defined by $\widehat{1_A}(\theta) = \sum_{a \in A} e^{2\pi i a \theta}$. The RHS is called the 'additive energy' of $A$ and is related to the doubling constant $\frac{|A+A|}{|A|}$. Additive energy came up in e.g. the recent breakthrough work of Guth-Maynard.

Answer 1

Links mentioned in comment mention applications to non-linear PDE. I mention an important application to linear PDE. It is the existence and analytic dependence of parameters of solution of Beltrami equation $$f_\overline{z}=\mu f_z,$$ also known as the "Measurable Riemann Mapping Theorem".

The theorem says that when $\|\mu\|_\infty<1$ there exists a homeomorphic solution in the complex plane, it is unique when properly normalized, for example fixes 0ドル$ and 1ドル$, and this unique solution depends holomorphically on $\mu$.

For the proof, see

Lars V. Ahlfors, Lectures on quasiconformal mappings , Princeton, N.J.–Toronto–New York–London: D. Van Nostrand Company, Inc. 146 p. (1966), MR200442, Zbl 0138.06002. Second, revised edition: AMS, 2006, Zbl 1103.30001.

The proof, due to Bogdan Bojarski (1957) uses Calderón–Zygmund estimates with $p\in(2,\infty)$.

This theorem, is a cornerstone of the theory of quasiconformal mappings in dimension 2 and holomorphic dynamics in complex dimension 1.

Remark. In 2000, a proof using only $L^2$ estimates was found:

Douady, Adrien; Buff, X. Le théorème d'intégrabilité des structures presque complexes, The Mandelbrot set, theme and variations, 307–324, London Math. Soc. Lecture Note Ser., 274, Cambridge Univ. Press, Cambridge, 2000.

Answer 2

One example is the famous fourth moment theorem by Nualart and Peccati on the convergence of the Wiener chaos of any order to normality -- see e.g. this post.

Another example is that, in order to get an asymptotic expansion in powers of 1ドル/\sqrt n$ of the c.d.f. of the standardized sum of $n$ i.i.d. random variables $X_i$ with a remainder $o(n^{(k-2)/2})$ for $k=3,4,\dots$, we need the condition $E|X_1|^k<\infty$ -- see e.g. Theorems 1 and 6 in Chapter VI of Petrov's book.

Answer 3

Converting the comments by @mathworker21 to an answer:

$L^4$ is pretty common, e.g. in additive combinatorics.

One has $$ \lVert \widehat{1_A}\rVert_4^4=\#\{(a,b,c,d)\in A^4~:~a+b=c+d\}, $$ where $\widehat{1_A}:\mathbb{T}\to\mathbb{C}$ is defined by $\widehat{1_A}(\theta)=\sum_{a∈A}e^{2πiaθ}$. The RHS is called the 'additive energy' of $A$ and is related to the doubling constant $\frac{|A+A|}{|A|}$. Additive energy came up in e.g. the recent breakthrough work of Guth-Maynard.

Answer 4

Stable laws in probability theory are in $L^p$, 0ドル< p \leq 2$. The case $p=2$ is the gaussian distribution. They are usually defined through their characteristic functions and do not have well-defined variance if $p\neq 2$.

Among these laws, the Haltsmark distribution is used in Plasma physics and astrophysics. This is one of the few stable law with an explicit density. Applications to other fields of physics such as spectroscopy are mentioned on the wikipedia page.

Stable laws are used in Economics under the name Pareto-Levy distributions to describe stock and commodity prices. Of course, stable laws appears in numerous problems internal to probability, statistics, dynamical systems (intermittency) and are associated to the central limit theorem for iid variables without finite second moment. So they are as widespread as these variables.