$BMO$-property via a John-Nirenberg type estimate?

Question

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.

Answer 1

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 $\mathrm{BMO}$.