Embeddings of Sobolev spaces

Question

Let $s_1,s_2\in \mathbb R$ such that $-\frac12<s_1\le s_2$. There exists $C>0$ such that for all smooth functions $w$ , for all $r>0$, $$\operatorname{supp} w \subset(-r,r)\Longrightarrow \Vert{w}\Vert_{H^{s_1}}\le r^{s_2-s_1}C\Vert{w}\Vert_{H^{s_2}}. $$

I want to prove that this is no longer true for $s_1<s_2=-\frac12$. Although it is quite easy to disprove the above statement when $s_1<s_2<-\frac12$, essentially by taking the Dirac mass, in the case $s_1<s_2=-\frac12$, some logarithmic term must be introduced while keeping the compact support of $w$.

Answer 1

Take $s_2=-1/2$, $s_1=-1/2-\delta$, and apply the to-be-disproved estimate to a standard delta sequence $\rho_\epsilon=\epsilon^{-1}\rho(x/\epsilon)$. Since we can take $r=\epsilon$, on the Fourier side the estimate is equivalent to $$ \int\frac{|\widehat{\rho}(\eta)|^2}{\epsilon^{1+2\delta}+|\eta|^{1+2\delta}}d\eta \lesssim \int\frac{|\widehat{\rho}(\eta)|^2}{\epsilon+|\eta|}d\eta $$ which is clearly false as long as $\widehat\rho(0)\neq0$ (the rhs grows like $\log\epsilon$ while the lhs grows like $\epsilon^{-2\delta}$)