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Let $S$ be a two-dimensional sphere, $\Delta$ be the Laplace-Beltrami operator on $S$ and $L^p(S)$, $p\geq 1$, be the usual $L^p$ space of real-valued functions on $S$. We also set $\|f\|_{H^\alpha(S)}:=\|(1-\Delta)^{\alpha/2}f\|_{L^2(S)}$, $\alpha\in\mathbb{R}$.

Let $p\in[2,\infty)$ and $\alpha=1-2/p$. Is it true that there exists a constant $C$ such that for all $f\in H^\alpha(S)$ it holds

$$ \|f\|_{L^p(S)} \leq C \|f\|_{H^\alpha(S)}? $$

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  • $\begingroup$ It might be overkill, but surely spherical harmonic expansions (with their sup-norm and L^2 norm comparisons, as in Stein-Weiss, e.g.) would resolve such issues? $\endgroup$
    – paul garrett
    Commented Dec 8, 2022 at 17:26
  • $\begingroup$ Is $S$ the round sphere, or the Riemannian metric is arbitrary? $\endgroup$
    – YCor
    Commented Dec 8, 2022 at 23:15
  • $\begingroup$ S is the "round" sphere embedded in $\mathbb{R}^3$ with the metric induced from $\mathbb{R}^3$. $\endgroup$
    – user72829
    Commented Dec 9, 2022 at 6:36
  • $\begingroup$ This can be proved using the heat semigroup generated by the Laplace Beltrami. I guess I can fill the details but I cannot do right now.... Probably in 2-3 days, if you still need it. $\endgroup$
    – Giorgio Metafune
    Commented Dec 9, 2022 at 11:49

1 Answer 1

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Today I could check, finally. The proof I had in mind works in any dimension with $\alpha >(N-1)(1/2-1/p)$ (in your case $N=3$) which is not optimal. The optimal result with equality is proved in Theorem II.2.7, Varopoulos, Saloff-Coste, Coulhon, Analysis and Geometry on groups.

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