Let $\Omega$ be any domain(open and connected) in $\Bbb R^d$.Define the $\text{BMO}(\Omega)$ space as
$$\text{BMO}(\Omega)= \big\{u\in L^1_{loc}(\Omega)\,\,:\,\, |u|_{\text{BMO}(\Omega)} <\infty \big\},$$with the seminorm,$$|u|_{\text{BMO}(\Omega)}=\sup_{B_r(x_0) \subset \Omega} \frac{1}{|B_r(x_0)|} \int_{B_r(x_0)} \Big| u(x) - \frac{1}{|B_r(x_0)|} \int_{B_r(x_0)} u(z) \,d z \Big| d x.$$
Let us consider the seminorm
$$|u|^*_{\text{BMO}(\Omega)}=\sup_{x_0\in \Omega,\, r>0} \frac{1}{|\Omega\cap B_r(x_0)|} \int_{\Omega\cap B_r(x_0)} \Big| u(x) - \frac{1}{|\Omega\cap B_r(x_0)|} \int_{ \Omega\cap B_r(x_0)} u(z) \,d z \Big| d x.$$In particular,$$ |u|^*_{\text{BMO}(\Bbb R^d)}=|u|_{\text{BMO}(\Bbb R^d)}.$$
Question:1- Are the seminorms $|\cdot|_{\text{BMO}(\Omega)}$ and $|\cdot|^*_{\text{BMO}(\Omega)}$ when $\Omega$ is a $d$-set? That is there is $C>0$ such that$$ |u|^*_{\text{BMO}(\Omega)}\leq C|u|_{\text{BMO}(\Omega)}$$for $u\in \text{BMO}(\Omega)$Note that we clearly have $$ |u|_{\text{BMO}(\Omega)}\leq |u|^*_{\text{BMO}(\Omega)}$$
Recall that Assume $\Omega$ is a $d$-set if there is $c>0$ such that $\Omega\cap B_r(a)|\geq cr^d$ for all $a\in \Omega$ and $r>0$. For instance an example of a d-set, is a Lipschitz domain. $\Omega=\Bbb R^d_+=\{x\in \Bbb R^d\,\,:\,\, x_d>0\}$ is also a $d$-set.
2- Is the John Nuremberg inequality true on $(\text{BMO}(\Omega)|\cdot|_{\text{BMO}(\Omega)})$? That is there are $a>0, b>0$ such that
such that for any ball $B\subset \Omega$ and $u\in \text{BMO}(\Omega)$ there holds that\begin{align*}|\{x \in B : |u(x) - u_B| > t\}| &\leq a |B|\exp\Big(\frac{-b t}{|u|_{\text{BMO}(\Omega)}}\Big).\end{align*}for any ball $B\subset \Omega$ and $t>0$
Any reference or answer is welcome.