TITLE: Equivalent Norms on Sobolev Spaces
QUESTION [6 upvotes]: When $k$ is a positive integer and $1<p<\infty$, we know that there is some
$C>0$ such that for all $u\in W^{k,p}\left(\mathbb{R}^{N}\right)  :$
$$
\left\Vert \left(  -I+\Delta\right)  ^{\frac{k}{2}}u\right\Vert _{p}\leq
C\left(  \left\Vert \left(  -\Delta\right)  ^{\frac{k}{2}}u\right\Vert
_{p}+\left\Vert u\right\Vert _{p}\right)  .
$$
(Stein, Singular Integrals and Differentiability Properties of Functions; etc)
The question is that can we have the following estimate: for $\varepsilon>0$,
there exists $C\left(  \varepsilon\right)  >0$ such that for all $u\in
W^{k,p}\left(\mathbb{R}^{N}\right)  :$
$$
\left\Vert \left(  -I+\Delta\right)  ^{\frac{k}{2}}u\right\Vert _{p}%
\leq\left(  1+\varepsilon\right)  \left\Vert \left(  -\Delta\right)
^{\frac{k}{2}}u\right\Vert _{p}+C\left(  \varepsilon\right)  \left\Vert
u\right\Vert _{p}?
$$
Note that when $p=2$, the answer is yes via Fourier transform. But I have no idea in the general case.

REPLY [4 votes]: If $k \in (0, 2]$, we define the multiplier 
$$
  m (\xi) =  (1 + \vert \xi \vert^2)^\frac{k}{2} - \vert \xi \vert^k.
$$
We observe that if $\vert \xi \vert \ge 2$, then by differentiability
$$
  \big \vert (1 + \vert \xi \vert^2)^\frac{k}{2} - \vert \xi \vert^k \big\vert
= \vert \xi \vert^k \Big \vert \Big(1 + \frac{1}{\vert \xi \vert^2}\Big)^\frac{k}{2} - 1 \Big\vert
\le C \vert \xi \vert^{k - 2},
$$
so that $m$ is bounded on $\mathbb{R}^N$.
Similarly,
$$
\vert   D^\ell m (\xi)\vert \le \frac{C_\ell}{\vert \xi \vert^\ell}.
$$
By the classical Mikhlin multiplier theorem, this implies that 
$$
 \big\Vert(-\Delta + I)^\frac{k}{2}u - (-\Delta)^\frac{k}{2}u \big\Vert_p
\le C \Vert u \Vert_p,
$$
from which the estimate follows.
If $k \in (2, \infty)$, we define the multiplier
$$
  m (\xi) =  \frac{(1 + \vert \xi \vert^2)^\frac{k}{2} - \vert \xi \vert^k}{1 + \vert \xi \vert^{k - 2}}.
$$
It can be checked that the multiplier satisfies also the conditions of the Mikhlin multiplier theorem. Therefore,
$$
\big \Vert \big((- \Delta + I)^\frac{k}{2} - (-\Delta)^\frac{k}{2}\bigr)\big((-\Delta)^{\frac{k}{2} - 1} + I\big)^{-1} v\Vert_p
\le C\Vert v \Vert_p.
$$
If we set $v = (-\Delta)^{\frac{k}{2} - 1}u + u$,
then 
$$\big \Vert \big((- \Delta + I)^\frac{k}{2} - (-\Delta)^\frac{k}{2}\bigr)u\Vert_p
\le \Vert (-\Delta)^{\frac{k}{2} - 1}u + u \Vert_p
\le \Vert (-\Delta)^{\frac{k}{2} - 1}u\Vert_p + \Vert u \Vert_p
$$
By a classical interpolation
$$ 
\Vert (-\Delta)^{\frac{k}{2} - 1}u\Vert_p
\le C \Vert (-\Delta)^\frac{k}{2}u\Vert_p^\frac{k - 2}{k}
\Vert u \Vert_p^\frac{2}{k}.
$$
The requested inequality follows then from Young’s inequality.