I'm thinking about a recent result by M. Lacey which says that a dyadic spherical maximal function satisfies a sparse bilinear bound. To be precise, define the unit scale dyadic spherical maximal function $$Mf(x) := \sup_{1<r<2} |A_rf(x)|$$ for functions $f : \mathbb{R}^d \to \mathbb{C}$ where $A_rf$ is the spherical average of radius $r$. Lacey claims that this satisfies a bilinear sparse bound; see (1.1) in that paper for the definition of a sparse bound and see Theorem 1.4 for the statement of his theorem.

Lacey immediately reduces to working with a function $f$ spatially localized to a cube $Q$ with side-lengths 1. This is a typical reduction in the area. Lacey then uses some stopping time argument that appears unmotivated to me via some $L^p$-improving continuity bounds. I have not really understood his proof and instead I came up with the following alternative proof. It is so simple that I am worried I made a simple/silly mistake somewhere.

**Question:** Is the following proof for Lacey's theorem correct, or have I misunderstood something?

**Begin proof:**
The localization implies that we have
$$< M(f1_Q),g > = < 1_{5Q}M(f1_Q),g > = < 1_{5Q}M(f1_Q),1_{5Q}g >$$
where $<\cdot,\cdot >$ denotes inner product.
Holder's ineqality implies
$$< 1_{5Q}M(f1_Q),1_{5Q}g > \leq \|1_{5Q}M(f1_Q)\|_p \|1_{5Q}g\|_{p'}.$$
Since $M$ satisfies $L^p$-improving properties, this means that $M: L^p \to L^q$ for some range of $Q$. Therefore,
$$< M(f1_Q),g > \leq C_{p,q} \|f1_Q\|_{L^q(5Q)} \|g\|_{L^{p'}(5Q)}.$$
Writing out these norms we have
$$\|f1_Q\|_{L^q(5Q)}^q = \int_Q |f|^q$$
So in Lacey's norms where $<f>_{Q,p} = (|Q|^{-1} \int_Q |f|^p)^{1/p}$ this becomes
$$\|f1_Q\|_{L^q(5Q)} = |Q|^{1/q} <f>_{Q,q}.$$
Therefore,
\begin{align*}
< M(f1_Q),g >
& \leq C_{p,q} |Q|^{1/q} <f>_{Q,q} |Q|^{1/p'} <g>_{5Q,p'}
\\ & = C_{p,q} |Q|^{1/q + 1/p'} <f>_{Q,q} <g>_{5Q,p'}.
\end{align*}
But $Q$ has volume comparable to 1 so that this is simply
$$< M(f1_Q),g > \leq C_{p,q} |Q| <f>_{Q,q} <g>_{5Q,p'}.$$

Now the essential point is that this is almost a bilinear sparse bound. If the dilated cube $5Q$ were actually $Q$, then I would have the simplest such possible sparse bound since it only involves a single cube! Fortunately one can easily turn this bound into a sparse bound and replace $5Q$ with $Q$ by another standard move in the area.

**End Proof**

*Side question*: Lacey actually claims the bilinear sparse bound for the larger maximal function defined where the supremum above is taken over all $r>0$. However I don't think that he proves this result. Instead I think he means to claim that he can obtain a sparse bound such that the same constant applies to each scale dyadic maximal function. Or is there a way to extend this to the full spherical maximal function?