Quantum pseudo-telepathy

In mathematical analysis, Agmon's inequalities, named after Shmuel Agmon, consist of two closely related interpolation inequalities between the Lebesgue space $L^{\mathrm{\infty }}$ and the Sobolev spaces $H^s$. It is useful in the study of partial differential equations.

Let $u\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ where $\mathrm{\Omega }\subset {\mathbb{ℝ}}^3$. Then Agmon's inequalities in 3D state that there exists a constant $C$ such that

$\Vert u{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C\Vert u{\Vert }_{H^1(\mathrm{\Omega })}^{1/2}\Vert u{\Vert }_{H^2(\mathrm{\Omega })}^{1/2},$

and

$\Vert u{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C\Vert u{\Vert }_{L^2(\mathrm{\Omega })}^{1/4}\Vert u{\Vert }_{H^2(\mathrm{\Omega })}^{3/4}.$

In 2D, the first inequality still holds, but not the second: let $u\in H^2(\mathrm{\Omega })\cap H_0^1(\mathrm{\Omega })$ where $\mathrm{\Omega }\subset {\mathbb{ℝ}}^2$. Then Agmon's inequality in 2D states that there exists a constant $C$ such that

$\Vert u{\Vert }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le C\Vert u{\Vert }_{L^2(\mathrm{\Omega })}^{1/2}\Vert u{\Vert }_{H^2(\mathrm{\Omega })}^{1/2}.$

See also

• Ladyzhenskaya inequality

• Brezis–Gallouet inequality

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

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