Estimate of the \(L^{p}\)-Fourier transform norm on nilpotent Lie groups
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.
scientific article; zbMATH DE number 1915838