Let $\Omega \subset \mathbb R^d, d\ge 2$, be bounded and denote a ball in $\mathbb R^d$ by $B$. Denote also $$ f_B:= \frac1{|B|}\int_B f \, dx. $$ Suppose $f \in L_{\rm loc}^p(\Omega)$ for all $1<p<\infty$ and $$\tag 1 \left|\{x \in B : |f(x)-f_{2B}| > \lambda\}\right| \le c_1 \exp\left(-c_2\frac\lambda{\left[\frac1{|2B|}\int_{2B} |f(x)-f_{2B}|^\delta \, dx\right]^\frac1\delta}\right)|B| $$ for all $\delta>0$ and for all balls $2B \subset \Omega$, with some constants $c_1, c_2=c_2(\delta) >0$. My question is whether this - or some suitable modification of this - would be enough to guarantee that $$ f \in BMO(\Omega), $$ where $$ BMO(\Omega) := \left\{f \in L^1(\Omega): \sup_{B\subset \Omega} \frac1{|B|}\int_B |f(x)-f_B| \, dx < \infty\right\}. $$

Any references which might be useful are greatly appreciated.