Do $L^p$ integrals over fixed radius balls satisfy this reverse Hölder-like inequality? For measurable $f:\mathbb R^n \to \mathbb R$, define$$M_qf(x) =\left(\int_{B_{1/2}(0)} |f(x+y)|^{q} dy  \right)^{1/q} = \|f\|_{L^q(B_{1/2}(x))} $$
and define the (inhomogeneous) dyadic annuli $ D_0 := B(0,1)$, $D_j = \{x\in\mathbb R^n: 2^{j-1} \le |x| \le 2^j\} $.
Hölder’s inequality trivially gives for $1\le q\le q’ \le \infty $ the pointwise bound $M_qf(x) \le M_{q’}f(x)$.
I’m interested in proving the following inequality, for all $q\in[1,\infty]$ and some $C>0$, (which I have on good authority is true): setting $\tilde D_j = D_{j-1}
\cup D_j \cup D_{j+1}$,
$$\fbox{$x\in D_j \implies M_q f(x)\le C\int_{
\tilde D_j} M_q f$}$$
and I think $C$ can be taken independent of $j$? Note that this is equivalently asking for $\|M_q f\|_{L^\infty(D_j)} \le C\|M_q f\|_{L^1(\tilde D_j)} $ which looks like a reverse of the relation for $q,q’$ above, but you need to take a second norm, and it’s at the cost of changing the set slightly. (In fact, this proves after an interpolation argument that $\|M_q f\|_{L^{p’}(D_j)} \le C\|M_q f\|_{L^p(\tilde D_j)} $ for all $1\le p\le p’ \le \infty$.)
(There was an attempt here, but I've removed it because it was long and I don't feel it is useful. The interested can see the edit history)
 A: I got some help, so I'll post a self-answer. My $C=C(n)$ is indeed independent of $j$ but it is rather large for large $n$; I would like to know if the constant can be taken independent of $n$, but I kind of doubt it. I believe the result is false for $N=2^n$. Anyway:
Let $\alpha\in(0,1)$ be arbitrary. First note that there exists $x_0\in D_j$ such that
$$ \left(\int_{ B_{1/2}(0)} |f(x_0+y)|^{q} dy  \right)^{1/q} = M_q f(x_0) \ge \alpha \| M_q f\|_{L^\infty(D_j)}.$$
Covering $ B_{1/2}(0)$ with a large number $N=N(n)$ (to be chosen shortly) of balls $B_k$ of equal radius $O(1/N^{1/d})$ (their total volume is therefore $O(1)$ which far exceeds  $|B_{1/2}(0)|=C2^{-d}$), we obtain by an easy contradiction argument that at least one of the sets $S=B_{1/2}(0)\cap B_k$ satisfies$$ \left(\int_{ S} |f(x_0+y)|^{q} dy \right)^{1/q} \ge \frac\alpha N \|  M_q f\|_{L^\infty(D_j)}$$
Now, choosing  $N$ large enough,   we have that $ S\subset  B_{1/2}(x)$ for each fixed $x\in S$. Thus,
$$  \frac  \alpha N \|  M_q f\|_{L^\infty(D_j)} \le \left(\int_{ B_{1/2}(x)}|f(x_0+y)|^q dy\right)^{1/q} =  M_qf(x_0+x)$$
Since all such points $x_0+x$ as $x$ varies over $S$ belong to $\tilde { D}_j :=  D_{j-1}\cup  D_{j} \cup  D_{j+1}, $ we obtain
$$ \frac{\alpha |S| }{N} \|M_q f\|_{L^\infty(D_j)} \le \int_{S} M_q f(x_0+x)dx \le \int_{\tilde D_j} M_q f(z)dz$$
Since  we can arrange all such sets $S$ to satisfy $|S|\gtrsim 1/N$  the result holds with some $C<\infty$. The above constructed $C$ satisfies
$C \lesssim N^2 $.
