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Lieb-Thirring inequalities on the $$N$$-sphere and in the plane, and some applications. (English) Zbl 0789.58079
We prove the Lieb-Thirring inequalities for a family of scalar functions defined on a sphere $$S^ n$$, which are orthonormal in $$L_ 2(S^ n)$$ and have zero mean value, for $$n\geq 1$$. We give explicit values of all the constants involved. In the case of the two-dimensional sphere, we prove the Lieb-Thirring inequalities for an orthonormal family of non- divergent (or irrotational) vector fields with the explicit value of the constant as well. For non-divergent (or irrotational) vector fields defined on the plane $$\mathbb{R}^ 2$$ we prove the Lieb-Thirring inequalities with the value of the constant less than was known before. Finally, the rate of growth of the constant is estimated, when a parameter $$p$$ tends to its limit, and embeddings in the exponential Orlicz spaces are proved. Applications to the dimension of attractors are given.
Reviewer: A.A.Ilyin (Moskva)

##### MSC:
 58J90 Applications of PDEs on manifolds 26D10 Inequalities involving derivatives and differential and integral operators 35Q30 Navier-Stokes equations
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