Consider the following maximal function : in dimension $n$, if $f\in L^1(B^n(0,1))$, $\alpha\geq 0$ :
$$
M_\alpha f : x \mapsto \sup_{r>0} \frac{1}{r^n} \left( \int_{B(x,r)} |f| \right)^{1+\alpha}
$$
If $\alpha=0$, then $M_0 : L^1 \to L^{(1,\infty)}$ is bounded, where $L^{(1,\infty)}$ is the weak $L^1$ space. If $\alpha>0$, we still have a bound of the form
$$
\| M_\alpha f\|_{L^{(1,\infty)}} \leq C_{n,\alpha} \|f\|_{L^1}^{1+\alpha}
$$
The question is : can we improve this inequality if $\alpha>0$ ? More precisely, can we find a strict subspace $X\subset L^{(1,\infty)}$ such that $M_\alpha : L^1 \to X$ is bounded ?