## Asymptotically sharp multiplicative inequalities.(English)Zbl 0840.46018

Let $$\Omega$$ be a domain in $$\mathbb{R}^n$$. The limiting case of the Sobolev embedding theorem states that the Sobolev space $$W^{1, n}_0(\Omega)$$ is continuously embedded in the Lebesgue space $$L_q(\Omega)$$ for any $$q\geq n$$ and the inequality $$|u|_q\leq c(n, q, \Omega)|u|_{1, n}$$ holds for functions $$u\in W^{1, n}_0(\Omega)$$. If $$\Omega$$ is bounded, then the best constant satisfies $$c(n, q, \Omega)\leq c(n)|\Omega|^{1/q} q^{1-1/n}$$ and it follows from the theory of Orlicz spaces that the exponent $$1- 1/n$$ is the best possible. However, the ratio $$|u|_q/|\nabla u|_n$$ is not scale-invariant. The scale-invariant form of the embedding is the Gagliardo-Nirenberg inequality $$|u|_q\leq c(n,q, r)|\nabla u|^{1-r/q}_n|u|^{r/q}_r$$, where $$r\in [1,\infty)$$, $$q\in [r, \infty)$$ and $$u\in W^{1,n}_0 \cap L_r (\Omega)$$.
The authors proved that the best possible constant in the above Gagliardo-Nirenberg inequality satisfies the estimate $$c(n, q, r)\leq c'(n, r)q^{1- 1/n}$$ and that the exponent $$1-1/n$$ here is the best possible. This result improves the estimate $$c(n, q, r)\leq c(n, r)'q$$ given by O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’ceva [“Linear and quasilinear equations of parabolic type”, Trans. Math. Monographs 23 (1968; Zbl 0174.15403); Russian original (1967; Zbl 0164.12302)].

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 26D10 Inequalities involving derivatives and differential and integral operators

### Citations:

Zbl 0174.15403; Zbl 0164.12302
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