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In this paper both we establish the best constants for the Nash inequalitieson the standard unit sphere $\mathbb{S}^n$ of $\mathbb{R}^{n+1}$ and we giveanswers on the existence of extremal functions on the corresponding problems.Also we study the problem of the best constants in the case, where the data areinvariant under the action of the group $G=O(k)\times O(m)$, and we find thebest constants.

Sharp Nash Inequalities on the unit sphere. The influence of symmetries