# How do the balls maximizing the maximal function depend on their centers?

Let $$\mu$$ be a finite Borel measure on $$\mathbb R^N$$ and let $$f\in L^1(\mu)$$ be a non-negative function. Let $$M_\mu f$$ denote the maximal function of $$f$$ relative to $$\mu$$, i.e. $$(M_\mu f)(x)=0$$ if $$\mu(B(x,r))=0$$ for some $$r>0$$ and $$(M_\mu f)(x) = \sup_{0 otherwise. (Here $$B(x,r)$$ denotes the open ball of radius $$r$$ centered at $$x$$.)

Suppose that $$a>0$$ and $$K\subset \mathbb R^N$$ is a compact such that $$M_\mu f > a$$ on $$K$$. Then for each $$x\in K$$ there exists $$r_x>0$$ such that $$\frac{1}{\mu(B(x,r_x))} \int_{B(x,r_x)}f \, d\mu > a. \tag{1}$$

Question. Is it possible to choose for each $$x\in K$$ the radius $$r_x>0$$ in such a way that (1) holds and the mapping $$x\mapsto r_x$$ is continuous, or upper semicontinuous, or at least Borel?

This question is inspired by another recent question about Besicovich type covering theorem.

• I have seen done in articles with measurable $x \mapsto r_x$. The $r_x$ is a metric analogue of what some people call "stopping time" in probability. I doubt that you can do this continuously. – Adrián González-Pérez May 19 '19 at 14:54
• @AdriánGonzález-Pérez thank you, interesting! I also doubt that $x \mapsto r_x$ can be chosen continuous in general, but it would be interesting to see an explicit counterexample. And if it can be chosen to be not too bad (upper semicontinuous or Borel), it would be interesting to see the proof. – Skeeve May 19 '19 at 15:01
• I will start by taking a radius $x \mapsto r_x$ that implements the supremum up to $\epsilon > 0$ and then take the limit over a countable sequence $\epsilon_n \to 0$. – Adrián González-Pérez May 19 '19 at 15:04

Consider any $$x\in K$$ and $$r>0$$ such that $$f_r(x):=\frac{1}{\mu(B(x,r))}\int\limits_{B(x,r)}f d\mu>a.$$ For any fixed $$d\in(0,\tfrac{r}{2})$$ and for any $$y\in B(x,d)$$ we have $$B(x,r-2d)\subset B(y,r-d)\subset B(x,r)$$. Therefore for any $$x\in K$$ and any $$y\in K\cap B(x,d)$$
$$f_{r-d}(y)=\frac{1}{\mu(B(y,r-d))}\int\limits_{B(y,r-d)} fd\mu\geq \frac{1}{\mu(B(x,r))}\int\limits_{B(x,r-2d)} fd\mu >a$$ if $$d$$ is small enough (say if $$0) just by the monotone convergence of the last integral. This means that for every $$x\in K$$ there exist $$r_x$$ and $$0 such that
$$f_{r_x-d_x}(y)>a\quad\text{ for every }y\in B(x,d_x)\cap K.$$ By compactness we can cover $$K$$ with a finite numer of balls $$B_i=B(x_i,d_{i})$$. Then any selection of the radii $$r_{i}-d_{i}$$ will work. In particular for every $$y\in K$$ we can take $$r(y):=\max_i\{r_{i}-d_{i}:y\in B_i\}=\max_i (r_i-d_i) \chi_{B_i}(y)$$ which gives a lower semicontinuous function (supremum of l.s.c. is l.s.c.). Choosing instead the minimum (and considering closed balls $$\overline{B(x,d_x)}$$ in the argument above) gives an upper semicontinuous function. Actually now that I think about it it's not necessary to extract a finite subfamily...in any case this choice gives even an essentially locally constant function.
• good point, but could you add some details on how you choose $r(y)$ when, say, $y\in B_1 \cap B_2$? As far as I understand, we would like to have $r(y) \in (r_1 - d_1, r_1) \cap (r_2 - d_2, r_2)$, however these intervals might be disjoint and I don't see why $r(y)=\max_{i}(r_i-d_i)$ should yield $f_{r(y)}(y)>a$. – Skeeve May 22 '19 at 19:27
• It is just sufficient for $r(y)$ to lie in the union of those intervals - either one would work – Del May 23 '19 at 8:27