The Fourier transforms of Lipschitz functions on the Heisenberg group (Q1582715)

The Heisenberg group \(H^n\) is the \((2n+1)\)-dimensional nilpotent Lie group with its underlying manifold \(R\times C^n= R^{2n+ 1}\), \(R\) and \(C\) being real and complex Euclidean spaces, respectively. The \(n\)th difference \(\Delta^n_hf(x)\) is given by standard definition. Definition 1. Let \(f\in L^p(H^n)\). Then \(f\) is said to belong to the Lipschitz class \(\text{Lip}(\alpha, p) >0\) if \[ \|\Delta^{2n+ 1}_h f\|_p= 0(|h|),\quad h\to 0,\tag{1} \] where \(\|\cdot\|_p\) is the usual \(L^p\)-norm on \(H^n\). Theorem 1. Let \(f\in L^p(H^n)\), \(1< p\leq 2\), such that (1) is satisfied. Then \(\|\widehat f\|^r_{\text{HS}}\) belongs to \(L^1(0,\infty)\) for \[ {(2n+ 1)p\over \alpha p+ (3n+ 1)p- (3n+ 1)}< r\leq q= {p\over p-1}, \] where \(\|\widehat f\|^r_{\text{HS}}\) is the Hilbert-Schmidt norm of \(\widehat f\). Theorem 2. Let \(f\) belong to \(L^2(H^n)\). Then the conditions \[ \|\Delta^{2n+1}_h f\|_2= 0(|h|^\alpha),\quad \alpha> 0,\quad h\to 0, \] \[ \int^\infty_X|\lambda|^n\|\widehat f\|^2_{\text{HS}} d\lambda= 0(X^{-2\alpha}),\quad \int^\infty_X\|\widehat f\|^2_{\text{HS}} d\lambda= 0(X^{-n- 2\alpha}), \] as \(X\to\infty\) are equivalent.