I am reading this lemma from Pazy's semigroup of linear operators
I am having a hard time understanding
1- Why does the highlighted part hold?
2- How is the Hölder's inequality applied and how the coefficient $q = 2p/(2+p)$ is chosen? The way I see it is that we want $1 \leq r,t \leq \infty$ such that
$$ \left(\int_{\mathbb{R}^n} \left|( 1+ |\xi|^2)^{-p} ( 1+ |\xi|^2)^{q} |u(\xi)|^q\right| d \xi \right)^{1/q} \leq \left( \int_{\mathbb{R}^n} ( \left|1+ |\xi|^2)^{-p}\right| d \xi \right)^{1/r} \cdot \left(\int_{\mathbb{R}^n} \left| 1+ |\xi|^2)^{q} |u(\xi)|^q\right|^t d \xi \right)^{1/t} $$
but I don't know where to go from here.
3- How is the last inequality deduced? Assuming we apply Parseval's identity $ \int |f(x)|^2 dx = \int |\widehat{f}(\xi)|^2 d \xi$ for $f \in L^2$, the second to last line reads
$$ \left(\int_{\mathbb{R}^n} ( 1+ |\xi|^2)^{-p} d \xi \right)^{1/p} \left(\int_{\mathbb{R}^n} ( 1+ |\xi|^2)^{2}) |u(x)|^2 d x\right)^{1/2} $$
but I still don't see how the last inequality holds
Question 1: Depending on what definition you use, this is precisely the definition of $H^2(\mathbb{R}^n)$. See this question for more details: Sobolev spaces fourier norm equivalence
Question 2: Matching Hölder exponents is something you get a feeling for and need to try out for yourself, there is no highroad to just guessing the exponents. In this case, you want to find a $q$ such that $$ \frac{1}{p}+\frac{1}{2}=\frac{1}{q} $$ and rearranging for $q$ yiels $q=\frac{2p}{p+2}$. The rest is simplifying exponents after applying the following version of Hölders inequality $$ ||fg||_q \leq ||f||_p ||g||_2 $$ for $\frac{1}{p}+\frac{1}{2}=\frac{1}{q}$.
Question $3$: $$ \bigg( \int_{\mathbb{R}^n}(1+|\xi|^2)^{-p}d\xi \bigg)^{\frac{1}{p}} $$ is just a constant, independent of $u$. Then, like in the first part, we have $$ ||u||_{H^2(\mathbb{R}^n)}^2=\int_{\mathbb{R}^n}(1+|\xi|^2)^2|\hat{u}(\xi)|^2 d\xi. $$ But now you have $$ ||u||_{H^2(\mathbb{R}^n)} \leq C(||u||_2+||\Delta u||_2) $$ for some $C$ independent of $u$ by the other definition of Sobolev spaces involving weak derivatives. You have to use the (infamous) lemma $$ ||D^2u||_2=||\Delta u||_2 $$ here. However, I don't really see how you would apply Parsevals here directly, however, it is implicitly used in part 1 again.