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.



$BMO$-property via a John-Nirenberg type estimate?
In the simpler dyadic setting the answer to your question is no.
Consider independent Bernoulli ($\pm 1$) random variables, aka Rademacher functions, $X\_1,X\_2,\dots$. Take any function that is quadratic in them:


$$f := \sum\_{i < j} c\_{ij} X\_i X\_j,$$
$$\sum\_{i<j} |c\_{ij}|^2 < \infty$$


(it can include diagonal terms too, but they are not important).


Take the dyadic partition, $\mathcal F\_n := \sigma(X\_{\le n})$ and decompose $f$ conditionally on $\mathcal{F}\_n$ into the "conditionally linear" part and the "conditionally quadratic" part:


$$f - \mathsf{E}\left[f \middle| \mathcal F\_n\right] = \sum\_{i \le n} X\_i \sum\_{j>n} c\_{ij} X\_j + \sum\_{n<i<j} c\_{ij} X\_i X\_j$$


By our construction, conditionally on $\mathcal{F}\_n$, $f$ is still a quadratic polynomial in $X$, and it is known that on those the $L^1$ norm is equivalent to the exponential Orlicz norm (this follows from hypercontractivity, see e.g. Theorem 6.7 in Janson "Gaussian Hilbert spaces"), so the John-Nirenberg-type bound does hold. On the other hand, the conditional variance of $f$ given $\mathcal{F}\_n$ is bounded from below by the conditional variance of the linear part, which is:


$$\sum\_{j > n} \left(\sum\_{i \le n} c\_{ij} X\_i \right)^2$$


This has no reason to be bounded, so $f$ is generally not in $\mathrm{BMO}$.


