An inequality of J. Necas

Question

The above inequality belongs to J. Necas. Sur les normes equivalentes dans $W^k_p$ et sur la coercivite des formellement positives. Sem. math. sup. Universite Montreal, 102-128 (1966). But I can't get this article. Can any one help me on proving this? Or are there other references?

I find the proof of the case $k=1,p=2$ in the book Mathematical tools for the study of the imcompressible Navier-Stokes equations and related models. And there gives a hint on the case of $k=1,p\neq2,\Omega=\mathbb{R}^d$:characterise the Fourier Multipliers that continuously map $L^p(\mathbb{R}^d)$ into itself. But I'm nor familiar with Fourier Multiplier.

Answer 1

I've been looking for that article for weeks, without success, nevertheless concerning the inequality you write, you can find it for $p=2$ in

Necas, J. Direct Methods in the Theory of Elliptic Equations, Springer, 2012, p.186-190.

For $1<p<\infty$ I suppose you can use Marcinkiewicz interpolation, but I'm looking for that article, because I think there is a better method, without using Fouries multipliers directly, that's why I'm interested in this article.

If someone there has the paper, I'd be in debt.

Thanks a lot.

Answer 2

I found a complete proof for the case $p=2$.A reference is provided for other values of $p$. The reference in question is

Pierre Fabrie, Franck Boyer, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, 2012, Springer.

Section IV.1 is devoted to the Nečas inequality. This question has been asked and answered a long time ago. However, I hope it will help future visitors.