Endpoint mapping properties of spherical maximal operators

For a function $f\in L^p({\mathbb{R}}^d)$, $d\ge 2$, let $A_{t}f(x)$ be the mean of $f$ over the sphere of radius $t$ centered at $x$. Given a set $E\subset (0,\infty)$ofdilationsweproveendpointboundsforthemaximaloperator$M_E$definedby$M_E f(x)=\sup_{t\in E} |A_t f(x)|$.