Gagliardo-Nirenberg and Sobolev-Trudinger type bounds for general Fourier multiplies and their relation to semi-classical Lieb-Thirring and Cwikel-Lieb-Rosenblum bounds for Schrödinger operators with general kinetic energies

We prove Gagliardo-Nirenberg and Sobolev-Trudinger type bounds for general Fourier multipliers (an $L^2$ based version). The extension of these bounds from functions to density matrices implies semiclassical bounds, such as the Lieb-Thirring and the Cwikel-Lieb-Rozenblum bounds, for Schrödinger operators with very general kinetic energies.
The main focus of our approach is not to get sharp constants but is the simplicity of the proofs. The bounds also extend to magnetic Schrödinger operators with a constant magnetic field (which are outside the framework of Fourier multipliers).
This is joint work with Michael Hofacker and Sylvain Zalczer (also from KIT).