  In this paper we study the worst-case error of numerical integration on the
unit sphere $\mathbb{S}^{d}\subset\mathbb{R}^{d+1}$, $d\geq2$, for certain
spaces of continuous functions on $\mathbb{S}^{d}$. For the classical Sobolev
spaces $\mathbb{H}^s(\mathbb{S}^d)$ ($s>\frac d2$) upper and lower bounds for
the worst case integration error have been obtained By Brauchart, Hesse, and
Sloan earlier in papers. We investigate the behaviour for $s\to\frac d2$ by
introducing spaces $\mathbb{H}^{\frac d2,\gamma}(\mathbb{S}^d)$ with an extra
logarithmic weight. For these spaces we obtain similar upper and lower bounds
for the worst case integration error.
