Estimate of the \(L^{p}\)-Fourier transform norm on nilpotent Lie groups (Q1874703)

Let \(G\) be a connected and simply connected nilpotent Lie group and let \(m\) be the dimension of the generic orbits. The authors prove that for \(1 <p \leq 2\) the norm of the \(L^p\) Fourier transform is equal or less than \(A_p^{\frac{2 \dim G-m}{2}}\) where \(A_p =(\frac{p^{\frac 1p}}{q^{\frac 1q}})^{\frac 12}\) is the \(n\)-root of the norm of the \(L^p\) Fourier transform of the \(n\)-dimensional additive abelian Lie group. To this end they make use of the orbit method and the estimation they obtain seems to be precise. In fact the number \(\frac{2\dim G-m}{2}\) is the dimension of the polarizing subalgebras of generic linear forms and then it is strictly greater than the dimension of the center of \(G\) if this is not abelian.