Do we have the pointwise bound $\left|\tilde{f}\right| \lesssim_d Mf$?

Question

Edit. I know I am missing a mean value on all integrals, but unfortunately I do not know if it is possible to make an integral sign with a horizontal slash through it with MathJax.

For any locally integrable function $f: \mathbb{R}^n \to \mathbb{C}$, define the function $\tilde{f}: \mathbb{R}^n \to \mathbb{R}^+$ as$$\tilde{f}(x) := \sup_{B \ni x} \int_B \left|f - \int_B f\right|,$$where the supremum ranges over all balls containing $x$. In particular we see that$$\left\|\tilde{f}\right\|_{L^\infty(\mathbb{R}^n)} = \left\|f\right\|_{\text{BMO}(\mathbb{R}^n)}.$$I am curious as to whether or not we have the pointwise bound$$\left|\tilde{f}\right| \lesssim_d Mf.$$Could anybody help? Thanks in advance.

Answer 1

A while back, Eric Thoma told me the following over a well-known social medium. I may expand this into a complete solution in the near future.

The constant is $2$. This is just the triangle inequality. It is referenced without proof on page 184 of Muscalu-Schlag's Classical and Multilinear Harmonic Analysis, Volume I. This is the sharp maximal function, right? We have$$\int \left( f - \int f\right) \leq \int f + \int \int f = 2 \int f \leq 2 Mf.$$What is cool is that a converse holds for $f$ in $L^p$, but in $L^p$ norms rather than pointwise, which allows for interpolation with BMO.