Gagliardo–Nirenberg interpolation inequality

The inequality concerns functions u: Rⁿ → R. Fix 1 ≤q, r ≤ ∞ and a natural number m. Suppose also that a real number α and a natural number j are such that

${\displaystyle {\frac {1}{p}}={\frac {j}{n}}+\left({\frac {1}{r}}-{\frac {m}{n}}\right)\alpha +{\frac {1-\alpha }{q}}}$

and

${\displaystyle {\frac {j}{m}}\leq \alpha \leq 1.}$

Then

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

${\displaystyle \|\mathrm {D} ^{j}u\|_{L^{p}}\leq C\|\mathrm {D} ^{m}u\|_{L^{r}}^{\alpha }\|u\|_{L^{q}}^{1-\alpha }.}$

The result has two exceptional cases:

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.

For functions u: Ω → R defined on a bounded Lipschitz domain Ω ⊆ Rⁿ, the interpolation inequality has the same hypotheses as above and reads

${\displaystyle \|\mathrm {D} ^{j}u\|_{L^{p}}\leq C_{1}\|\mathrm {D} ^{m}u\|_{L^{r}}^{\alpha }\|u\|_{L^{q}}^{1-\alpha }+C_{2}\|u\|_{L^{s}}}$

where s > 0 is arbitrary; naturally, the constants C₁ and C₂ depend upon the domain Ω as well as m, n etc.

• 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. (Note, in particular, that r is permitted to be 1.)
• 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 ${\displaystyle H^{s}}$, with ${\displaystyle 0\leq s_{1}<s<s_{2}}$, a special case is given by ${\displaystyle \|u\|_{H^{s}}\leq \|u\|_{H^{s_{1}}}^{\frac {s_{2}-s}{s_{2}-s_{1}}}\|u\|_{H^{s_{2}}}^{\frac {s-s_{1}}{s_{2}-s_{1}}}}$. This can also be derived via Plancherel theorem and Hölder's inequality.

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• Nirenberg, L. (1959). "On elliptic partial differential equations". Ann. Scuola Norm. Sup. Pisa (3). 13: 115–162.
• Haïm Brezis, Petru Mironescu. Gagliardo-Nirenberg inequalities and non-inequalities: the full story. Annales de l’Institut Henri Poincaré - Non Linear Analysis 35 (2018), 1355-1376.
• Leoni, Giovanni (2017). A First Course in Sobolev Spaces: Second Edition. Graduate Studies in Mathematics. 181. American Mathematical Society. pp. 734. ISBN 978-1-4704-2921-8
• Nguyen-Anh Dao, Jesus Ildefonso Diaz, Quoc-Hung Nguyen (2018), Generalized Gagliardo-Nirenberg inequalities using Lorentz spaces and BMO, Nonlinear Analysis, Volume 173, Pages 146-153.