When reading the characterization of Besov space with $L\_p$-modulus of continuity in the 7th chapter “Fractional Order Space” of Sobolev space written by Adams(Page 243), I encounter some small specific problem. We define 
$$\Delta\_hw(x):=w(x)-w(x-h)$$ 
We want to prove 
\begin{equation}||\Delta\_hw||\_p\leq |h||w|\_{1,p}\qquad(\*)\end{equation}
The original words in the book are “We majorize $|w(x-h)-w(x)|$ by the integral of $|{\rm{grad}} w|$ along the line segment joining $x-h$ to $x$, and use Holder's inequality to majorize that by $|h|^{1/p’}$ times the one-dimensional $L\_p$ norm of the restriction of $|{\rm{grad}} w|$ to that segment. Finally, we take $p$-th powers, integrate with respect to $x$, and take a $p$-th root to get that $||\Delta\_hw||\_p\leq |h||w|\_{1,p}$”.
I know through Holder’s inequality,
$$|w(x-h)-w(x)|=\int\_{x-h}^x|grad w|\cdot1\leq(\int\_{x-h}^x|grad w|^pds)^{1/p}\cdot|h|^{1/p’}$$
Then
$$||\Delta\_h w||\_p=(\int\_\mathbb{R}|w(x-h)-w(x)|^p)^{1/p}\leq|h|^{1/p’}(\int\_\mathbb{R}\int\_{x-h}^x|grad w|^pdsdx)^{1/p}$$
It seems that through integral mean value theorem, 
$$(\int\_\mathbb{R}\int\_{x-h}^x|grad w|^pdsdx)^{1/p}=(\int\_\mathbb{R}\int\_{x-h/2}^{x+h/2}|grad w|^pdsdx)^{1/p}=|h|^{1/p}(\int\_\mathbb{R}|grad w(\delta\_x)|^pdx)^{1/p}$$
Where $\delta\_x\in(x-h/2,x+h/2)$. But I can not convince myself it is bounded above by $|h|^{1/p}||gradw||\_p$. Another idea is to use maximal function? Since 
\begin{align\*}
(\int\_\mathbb{R}\int\_{x-h/2}^{x+h/2}|grad w|^pdsdx)^{1/p}&=|h|^{1/p}(\int\_\mathbb{R}\frac{1}{|h|}\int\_{x-h/2}^{x+h/2}|grad w|^pdsdx)^{1/p}\\
&\leq |h|^{1/p}(\int\_\mathbb{R}\mathrm{M}(|gradw|^p)(x)dx)^{1/p}
\end{align\*}
And I want to use $||\mathrm{M}(f)||\_p\leq C||f||\_p$ when $p\geq2$. But I do not know how to use it to solve this problem. So any idea to prove equation $(\*)$? Thanks in advance!



Characterization of Besov space with Lp-modulus of continuity
1) One way would be to apply Tonelli's theorem 


\begin{align\*}
&\int\_{\mathbb{R}} \int\_{x-h}^{x}|w'(s)|^{p}ds dx = \int\_{\mathbb{R}} \int\_{\mathbb{R}} |w'(s)|^{p} \chi\_{[x-h,x]}(s) dsdx =\\
& \int\_{\mathbb{R}} \int\_{\mathbb{R}}|w'(s)|^{p}\chi\_{[x-h,x]}(s) dxds= |h| \int\_{\mathbb{R}}|w'(x)|^{p}dx
&
\end{align\*}
You can notice that when $p=1$ you are not loosing too much (in this case there is no Holder's inequality). 


2) Another more sophisticated way (which only works for $p>1$) would be to apply maximal function
\begin{align\*}
|w(x-h)-w(x)|\leq \int\_{x-h}^{h} |w'(s)|ds \leq |h| \tilde{M}(w') (x)
\end{align\*}
where $\tilde{M}$ is the *uncentered maximal function operator*. Next, raise to the power $p$ ($p>1$) and apply Hardy--Littlewood, you will gain the constant $C\_{p}$ which behaves as $\frac{p}{p-1}$. 


