If $F$ is an $r-$variate homogeneous polynomial of degree $/m$, then
$G:=|grad F|^2$ satisfies $||G||=/m^{2}||F||$ in the uniform norm on the
unit ball in $/R^r$. In other words, the norm of the map $F/mapsto G$ equals
to $/m2$. This remarkably plain result of Kellogg (1927)
encourages us to investigate the norm of the map $F/mapsto G:=/Dmr F$, where
$/Dmr$, $r,/m/in/N/setminus/{1/}$, is the restriction of the
Laplace operator to the corresponding polynomial space. It turns out that
the situation is now much more complicated, in particular,
$||G||=||/Dmr||/cdot||F||$ holds for very particular $F$, only. We describe
$||/Dmr||$ by the help of the family of polynomials $T_{/m}(/si x)$, which
we call, for good reasons, Zolotarev polynomials of the second kind. We
determine lower and upper bounds for $||/Dmr||$, as well as the asymptotics
for $/m/rightarrow/infty$. The lower bound is often, but by no means always
attained, which gives the problem a particular flavour. Some Bessel
functions and the $/phi/cot/phi$ - expansion are involved.
AMS(MOS) Subject Classification: 26D05, 26D10, 41A17, 41A63, 33C25, 47A30.
Keywords: Homogeneous polynomials, Laplace operator,
Zolotarev polynomials, Bessel functions, operator norm, best constants.