An interpolation argument about homogeneous Sobolev spaces

Question

I am studying the book Fourier Analysis and Nonlinear Partial Differential Equations by Hajer Bahouri, Jean-Yves Chemin and Raphaël Danchin. I am trying to understand a statement in the Remark 1.44 (section 1.3.2).

Let us recall that we define $$ \dot{H}^s(\mathbb{R}^d)=\{u \in S'(\mathbb{R}^d): \hat{u} \in L^1_{loc} \mbox{ such that } \|u\|_{\dot{H}^s(\mathbb{R}^d)} < +\infty\} $$ where $$ \|u\|_{\dot{H}^s(\mathbb{R}^d)}=\int_{\mathbb{R}^d}|\xi|^{2s}|\hat{u}(\xi)| d\xi. $$ Thanks to the following embedding theorem:

Theorem: $\|u\|_{L^p(\mathbb{R}^d)} \leq C_d \frac{p}{\sqrt{p-2}}\|u\|_{\dot{H}^s(\mathbb{R}^d)}$ for $p=\frac{2d}{d-2s}$ and where $C_d$ only depends on $d$.

the authors remark (and this is where I found problems) that observing $\|u \|_{L^2}=C \|u\|_{\dot{H}^0(\mathbb{R}^d)}$ we can conclude that if $2<p<4$: $$\|u\|_{L^p(\mathbb{R}^d)} \leq C' \sqrt{p} \cdot \|u\|_{\dot{H}^s(\mathbb{R}^d)}$$ for a constant $C'$ which does not depend on $p$. Thus the constant does not explode when $p$ goes to $2$ from above.

Answer 1

I do not know exactly why they obtain a constant with a $\sqrt p$ but it is indeed true that the constant does not blow-up at $p=2$ and indeed one can prove that by the method of complex interpolation:

• By the theorem you mention, the identity map $\mathrm{Id} : u\mapsto u$ is continuous from $\dot{H}^s$ to $L^p$ and with norm $M_p \leq C_d \frac{p}{\sqrt{p-2}}$. On the other hand, $\mathrm{Id} : u\mapsto u$ is of course continuous from $L^2$ to $L^2$ with norm $M_2 = 1$, hence by complex interpolation, for any $s\in(0,\sigma)$ and for $p = \frac{2d}{d-2s}$ and $q = \frac{2d}{d-2\sigma}$ $$ M_p \leq M_2^{s/\sigma} M_q^{1-s/\sigma} \leq C_d \left(\frac{q}{\sqrt{q-2}}\right)^{1-s/\sigma}. $$ Then indeed, by taking for example $q=4$ (and so $\sigma = d/4$) on obtains that for any $p\in[2,4]$, $M_p \leq \left(\frac{4\,C_d}{\sqrt{2}}\right)^{1-4s/d} = \left(2\sqrt{2}\,C_d\right)^{1-4s/d}$, and so $$ \|u\|_{L^p} \leq \left(2\sqrt{2}\,C_d\right)^{1-4s/d} \|u\|_{\dot{H}^s}. $$

Of course, up to replacing $C_d$ by $C_d+1$, I can assume that $2\sqrt{2}\,C_d \geq 1$, and write $(2\sqrt{2}\,C_d)^{1-4s/d}\leq \sqrt{2}\,C'_d$. Moreover, $\sqrt{2} \leq \sqrt{p}$ so indeed one can rewrite the eqaution as $\|u\|_{L^p} \leq C_d\,\sqrt p\,\|u\|_{\dot{H}^s}$, but there is nothing special about this $\sqrt p$ as far as I can see?