I'm trying to understand why this inequality is true
$$ \Vert f\Vert_{L^2}^{p}\cdot h\left(\Vert f\Vert_{\infty}\right) \leq c \Vert f\Vert_{k+1}^{p} \cdot h\left(\Vert f\Vert_{k+1}\right), $$
where $ \Vert \cdot\Vert_{k+1} $ denotes the usual norm of the Sobolev Space $ H^{k+1}(R^n) $, $ h(y) $ denotes a nondecreasing nonnegative and continuous function, c is a costant, $k \geq 0 $ and $ p> 1$.
Maybe i can use an embedding theorem? which one? Can anyone please help me?
Best Answer
Using a bit of Fourier analysis, one can show that, for $s>\frac{d}{2}$, $$H^s(\mathbb{R}^d) \hookrightarrow L^\infty(\mathbb{R}^d).$$ In particular, consider $$\|f\|_{L^\infty} \leq \|\hat{f}\|_{L^1} \leq \|\langle \xi\rangle ^{-s}\|_{L^2} \|\langle \xi\rangle ^{s}\hat{f}\|_{L^2} \lesssim \|f\|_{H^s}, $$ where the first inequality follows, pretty much directly, from the Fourier inversion formula.
Now, $\|f\|_{L^2} \leq\|f\|_{H^{k+1}}$ for every $k\geq 0$, so that part is fine, too. But, you still have some work to do. In particular, you have $$h(\|f\|_{L^\infty}) \leq h(c\|f\|_{H^s}),$$ but you need to get to $$h(\|f\|_{L^\infty}) \lesssim h(\|f\|_{H^s}).$$ You may need some additional assumptions on $h$ (particularly on how $h$ scales) to do this.
As a bit of a tangent, it's actually the case that for $s > \frac{d}{2} + k$, we have $H^s(\mathbb{R}^d) \hookrightarrow C^k(\mathbb{R}^d) \cap L^\infty(\mathbb{R}^d)$.