Are bilinear sparse bounds for local operators trivial?

Question

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?

Answer 1

You are simply restating the $L^p$-improving property of the single scale spherical maximal operator. Lacey's paper proves a sparse bound for the larger maximal function $$ M_{\mathrm{full}} f= \sup_{r>0} A_r f $$ and its lacunary counterpart, which are multiscale objects. If you think of the estimate you wrote down as the bilinear estimate describing the behavior of a single scale of the operator $M_{\mathrm{full}}$, then Lacey's result tells you that the bisublinear form induced by $M_{\mathrm{full}}$ can be controlled by a superposition of single scale positive bilinear estimates over essentially disjoint cubes.

The proof is based on the continuity with respect to small translations of the $L^p$ improving property of $A_r$, which allows (via stopping time arguments) to exploit cancellation and sum over scales. This continuity was unobserved before, and for this reason sparse bounds for maximal spherical averages and general maximal radon transforms were not deemed to be possible before -- standard methods of Lerner type based on mean oscillation formulas are very far from being applicable.