On the possibility of strengthening the Lieb-Thirring inequality

In 1976, Lieb and Thirring obtained an upper bound for the square of the normon L ²(?²) of the sum of the squares of functions from finite orthonormal systems via the sum of the squares of the norms of their gradients. Later, a series of Lieb-Thirring inequalities for orthonormal systems was established by many authors. In the present paper, using the standard theory of functions, we prove Lieb-Thirring inequalities, which have applications in the theory of partial differential equations.