Gagliardo–Nirenberg interpolation inequality

Statement of the inequality

1. every function u: Rⁿ → R that lies in L^q with m^th derivative in L^r also has j^th derivative in L^p;
2. and, furthermore, there exists a constant C depending only on m, n, j, q, r and α such that

1. If j = 0, mr < n and q = ∞, then it is necessary to make the additional assumption that either u tends to zero at infinity or that u lies in L^s for some finite s > 0.
2. If 1 < r < ∞ and m − j − n/r is a non-negative integer, then it is necessary to assume also that α ≠ 1.

Consequences

• When α = 1, the L^q norm of u vanishes from the inequality, and the Gagliardo–Nirenberg interpolation inequality then implies the Sobolev embedding theorem.
• Another special case of the Gagliardo–Nirenberg interpolation inequality is Ladyzhenskaya's inequality, in which m = 1, j = 0, n = 2 or 3, q and r are both 2, and p = 4.
• In the setting of the Sobolev spaces, with, a special case is given by.