The Hölder inequality for fractional order Sobolev seminorm?

Question

This question is post on MSE 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 $k$".

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!

Answer 1

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.

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 $c_j$.