This question is post on [MSE](https://math.stackexchange.com/questions/1993560/the-minkowski-inequality-for-fractional-order) a week ago. I move it here to draw more attention.


Let $u\in C^\infty(\bar I)$ be given where $I=(0,1)$. Define
$$
t(\alpha):=\left(\int\_I\int\_I \frac{|u(x)-u(y)|^\alpha}{|x-y|^{1+s\alpha}}\right)^{\frac1\alpha}
$$
where $1<\alpha<2$, $0<s<1$ is fixed. Note the $t(\alpha)$ above defines the fractional order sobolev seminorm. See [the Wikipedia article on "Sobolev space", section "Sobolev spaces with non-integer $k$"](https://en.wikipedia.org/wiki/Sobolev_space#Sobolev_spaces_with_non-integer_k).


We know that for usual $L^p$ space, we have, for $p<q$, $\|u\|\_{L^p(I)}\leq \|u\|\_{L^q(I)}$ by using Hölder inequality. So I am wondering whether similar properties hold for sobolev fractional seminorm with non-integer $k$.


That is, I am wondering for $1<\alpha\_1<\alpha\_2<2$, do we have
$$
t(\alpha\_1)\leq Ct(\alpha\_2)
$$
hold, where $C$ is a constant does not depends on $u$. Just like what we usually have for $L^p$ norm. However, I tried to prove it by using Minkowski inequality or something like Hölder, but I can't do it… I think the domain $I=(0,1)$ would be important and I also tried to use Jensen inequality, but no lucky…


Any ideas? Thank you!



The Hölder inequality for fractional order Sobolev seminorm?
Your question can be rephrased by asking whether one has a Hölder estimate
$$
 |u|\_{W^{s, p}} \le C |u|\_{W^{s, q}},
$$
when $p < q$ or whether $W^{s, q} \subset W^{s, p}$.


**There is no such embedding or inequality.**


For proofs of this fact you can have a look at the paper Mironescu, Sickel, [*A Sobolev non embedding*, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. *26* (2015), no 3, 291—298](https://hal.archives-ouvertes.fr/hal-01162231).


One way to see this is to take 
$$
 u(x) = \sum\_{j \in \mathbb{N}} c\_j e^{i 2^j x},
$$
and to observe that 
$$
|u|\_{W^{s, p}}^p \simeq \sum\_{j \in \mathbb{N}} |2^{js} c\_j|^p,
$$
and then choose suitably $c\_j$.


