Remove the logarithmic dimension dependence for the heat maximal operator

Determine whether the logarithmic dependence on the dimension can be removed from the weak-type $(1,1)$ bound for the centered heat maximal operator $\mathcal{H}_{\ast}$ on $\mathbb{R}^n$; one possible route is to improve the Abel-power square-function estimate in Theorem 2.1 from dependence $N^{1/2}$ to $N^{1/2-\epsilon}$ for some $\epsilon>0$.

Background

The paper proves a weak-type $(1,1)$ estimate for the centered heat maximal operator $\mathcal{H}_{\ast}f(x)=\sup_{t>0}|H_tf(x)|$ with dimensional growth $O(\log(1+n))$. Combined with the pointwise comparison between the centered Hardy--Littlewood maximal function and the heat maximal operator, this yields an $O(\sqrt n\log(1+n))$ bound for the Hardy--Littlewood maximal function over Euclidean balls.

The logarithmic loss arises in the low Mellin-frequency estimate, which depends on a square-function bound for Abel powers whose current dimensional dependence is $\sqrt N$. The authors identify improving that dependence to $N^{1/2-\epsilon}$ as one possible way to eliminate the logarithmic factor, but explicitly state that their methods do not achieve this improvement.