Lieb-Thirring Inequalities on the <i>N</i>-Sphere and in the Plane, and Some Applications

Abstract

In this paper we prove the Lieb-Thirring inequalities for a family of scalar functions defined on a sphere Sn, which are orthonormal in L2(Sn) and have zero mean value, for n ⩾ 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 R2 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.