4 problems

Let $d\ge 3$, let $p_d=d/(d-1)$, and let $E$ be the set defining the spherical maximal operator $\cM_E$. For each integer $k$ and relevant $0<\delta<1$, write $N(E^k,2^k\delta)$ fo…

Endpoint conjecture. The term $(1+j)^{b_3}$ in Corollary can be dropped; moreover, a restricted weak type estimate holds at $Q_{4,\gamma}$.

Optimal-exponent conjecture. The exponent $p_0(d,k)$ is conjectured to equal

Sharp-range conjecture. The linear degree-$k$ spherical maximal function is conjectured to be bounded on the sharp range $p > \frac{d}{d-k}$. If true, this would make the paper's c…