## Some inequalities of Sobolev type on two-dimensional spheres.(English)Zbl 0652.26020

General inequalities 5, 5th Int. Conf., Oberwolfach/FRG 1986, ISNM 80, 401-408 (1987).
[For the entire collection see Zbl 0621.00004.]
Let $$S^ 2$$ be the two-dimensional sphere in $$R^ 3$$ and let $$H^ 2(dx)$$ be the usual measure on $$S^ 2$$. The main results read as follows: The (exact) inequalities $\int_{S^ 2}[u(x)-H^ 2(S^ 2)^{-1}\int_{S^ 2}u(y)H^ 2(dy)]^ 2H^ 2(dx)$
$\leq (4\pi)^{-1}(\int_{S^ 2}| \text{grad} u(x)| H^ 2(dx))^ 2$ and $\max u-\min u\quad \leq \quad c(\int_{S^ 2}| \text{grad} u(x)|^ pH^ 2(dx))^{1/p}$ hold for any smooth real function u on $$S^ 2$$, $$p>2$$ and $c = (2\pi)^{-1/p}[\pi^{1/2}\Gamma (1-p/(2p- 2))/\Gamma (1-1/(2p-2))]^{1-1/p}.$
Reviewer: J.Rákosník

### MSC:

 26D10 Inequalities involving derivatives and differential and integral operators

Zbl 0621.00004