  We extend the resolvent estimate on the sphere to exponents off the line
$\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$. Since the condition
$\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$ on the exponents is necessary for a
uniform bound, one cannot expect estimates off this line to be uniform still.
The essential ingredient in our proof is an $(L^{r}, L^{s})$ norm estimate on
the operator $H_{k}$ that projects onto the space of spherical harmonics of
degree $k$. In showing this estimate, we apply an interpolation technique first
introduced by Bourgain [2]. The rest of our proof parallels that in Huang-Sogge
[8].
