Interpolation inequality

In the field of mathematical analysis, an interpolation inequality is an inequality of the form

\| u_{0} \|_{0} \leq C \| u_{1} \|_{1}^{\alpha_{1}} \| u_{2} \|_{2}^{\alpha_{2}} \dots \| u_{n} \|_{n}^{\alpha_{n}}, \quad n \geq 2,

valid for all u₀, ..., u_n in some (subsets of) vector spaces X₀, ..., X_n equipped with norms ‖·‖₀, ‖·‖₁, ..., ‖·‖_n, and where C is a constant independent of u₀, ..., u_n and α₁, ..., α_n are some real powers. Usually, the elements u₀, ..., u_n are all the same element u and only the norms differ (as in Ladyzhenskaya's inequality below), but some interpolation inequalities use different u₀, ..., u_n (as in Young's inequality for convolutions below).

The main applications of interpolation inequalities lie in the theory of Sobolev spaces, where spaces of functions that have a non-integer number of derivatives are interpolated from the spaces of functions with integer number of derivatives. The abstract structure of interpolation inequalities is formalized in the notion of an interpolation space.

A simple example of an interpolation inequality — one in which all the u_k are the same u, but the norms ‖·‖_k are different — is Ladyzhenskaya's inequality for functions u: ℝ² → ℝ, which states that whenever u is a compactly supported function such that both u and its gradient ∇u are square integrable, it follows that the fourth power of u is integrable and

\int_{\mathbb{R}^{2}} | u(x) |^{4} \, \mathrm{d} x \leq 2 \int_{\mathbb{R}^{2}} | u(x) |^{2} \, \mathrm{d} x \int_{\mathbb{R}^{2}} | \nabla u(x) |^{2} \, \mathrm{d} x,

i.e.

\| u \|_{L^{4}} \leq \sqrt[4]{2} \, \| u \|_{L^{2}}^{1/2} \, \| \nabla u \|_{L^{2}}^{1/2}.

(Since Ladyzhenskaya's inequality considers compactly supported functions u, Friedrichs' inequality implies that the L² norm of ∇u is equivalent to the H¹ Sobolev norm of u, and so Ladyzhenskaya's inequality really does only treat a single function u, not distinct functions u₀ = u₁ = u and u₂ = ∇u.)

Another simple example of an interpolation inequality — one in which the u_k and the norms ‖·‖_k are different — is Young's inequality for the convolution of two functions f, g: ℝ^d → ℝ:

\|f \star g\|_{L^{s}} \leq \|f\|_{L^{r}} \|g\|_{L^{p}},

where the exponents p, r and s ≥ 1 are related by

\frac{1}{r}+\frac{1}{p}=1+\frac{1}{s}.

Examples of interpolation inequalities

Agmon's inequality
Gagliardo–Nirenberg interpolation inequality
Ladyzhenskaya's inequality
Landau–Kolmogorov inequality
Marcinkiewicz interpolation theorem
Nash's inequality
Riesz–Thorin theorem
Young's inequality for convolutions