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!

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!

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 here, roll down for the 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 holder 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 sth like holder, 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!

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!

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 $\alpha\geq 1$, $0<s<1$ is fixed. Note the $t(\alpha)$ above defines the fractional order sobolev seminorm. See here, roll down for the 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 holder inequality. So I am wondering whether similar properties hold for sobolev fractional seminorm with non-integer $k$.

That is, I am wondering for $1\leq\alpha_1<\alpha_2$, 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 sth like holder, 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!

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 here, roll down for the 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 holder 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 sth like holder, 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!