BOUNDS FOR THE EIGENVALUES OF THE FRACTIONAL LAPLACIAN

In this article, we extend Polya's legendary inequality for the Dirichlet Laplacian to the fractional Laplacian. Polya's argument is revealed to be a powerful tool for proving such extensions on tiling domains. As in the Dirichlet Laplacian case, Polya's inequality for the fractional Laplacian on any bounded domain is still an open problem. Moreover, we also investigate the equivalence of several related inequalites for bounded domains by using the convexity, the Lieb-Aizenman procedure (the Riesz iteration), and some transforms such as the Laplace transform, the Legendre transform, and the Weyl fractional transform.