Beurling's theorem for nilpotent Lie groups (Q531282)

Beurling's original theorem says that if \(f\in L^2(\mathbb R)\) satisfies \[ \int_{\mathbb R}\int_{\mathbb R}|f(x)|\,|\hat{f}(x)|e^{2\pi|xy|}\,dx\,dy<\infty, \] then \(f=0\) almost everywhere (\(\hat{f}\) stands for the Fourier transform of \(f\)). The central result of the paper under review is an analogue of this theorem for connected, simply connected nilpotent Lie groups. More precisely, the author proves that if \(G\) is such a group and \(f\in L^2(G)\) satisfies \[ \int_{\mathcal W} \int_G|f(x)|\, \|\pi_l(f)\|_{HS} e^{2\pi\|x\|\, \|l\|}|Pf(l)|\, dx\, dy<\infty, \tag \(*\) \] then \(f=0\) almost everywhere. Some words for the notation will be helpful: Choosing a strong Mal'tsev basis \({\mathcal B}=\{X_1,\dots,X_n\}\) of \({\mathfrak g}= \text{Lie}(G)\) and setting \({\mathfrak g}_i=\mathbb R-\text{span}\{X_1,\dots,X_i\}\) one defines the jump indices of an \(l\in{\mathfrak g}^*\) as the indices \(j\in\{1,\dots,n\}\) such that \({\mathfrak g}(l)+{\mathfrak g}_j\neq{\mathfrak g}(l)+{\mathfrak g}_{j-1}\), where \({\mathfrak g}(l)=\{X\in{\mathfrak g}\mid l([X,Y])=0\), \(\forall Y\in{\mathfrak g}\}\). For a fixed \(l\), denote as \(e(l)\) the set of jump indices for \(l\) and as \(e'(l)\) its complementary in \(\{1,\dots,n\}\). Let then \(V_{e'(l)}=\mathbb R-\text{span}\{X_{i^*}\mid i\in e'(l)\}\), \(V_{e(l)}=\mathbb R-\text{span}\{X_{i^*}\mid i\in e(l)\}\). If \({\mathcal U}:=\{l'\in{\mathfrak g}^*\mid e(l')=e(l)\}\), then \(\mathcal{W}:=V_{e'(l)}\cap{\mathcal U}\) intersects each coadjoint \(G\)-orbit \(G\cdot l=:{\mathcal O}_l\) (corresponding to a unitary irreducible representation \(\pi_l\)) in a single point making it a suitable cross-section for the generic coadjoint orbits in \({\mathfrak g}^*\). Furthermore \(\|\cdot\|_{HS}\) denotes the norm of the corresponding Hilbert-Schmidt operator and \(\pi_l\) is the equivalence class of the unitary irreducible representation \(\text{Ind}^G_H\chi_l\), that is, the induced representation from \(l\in{\mathfrak g}^*\) and an arbitrary polarization \({\mathfrak h}\) with respect to \(l\). Finally \(H=\exp{\mathfrak h}\) and \(Pf(l)\) stands for the Pfaffian of the skew-symmetric matrix \(M(l)=(l([X_i,X_j]))_{i,j\in e(l)}\). In order prove his theorem the author uses as a final argument the Plancherel formula \(\|f\|_2^2=\int_G|f(g)|^2\,dg=\int_{\mathcal W}\|\pi_l(f)\|_{HS}^2\,d\mu(l)\) where \(f\in L^1(G)\cap L^2(G)\), \(d\mu=|Pf(l)|\,dl\) is a Plancherel measure for the unitary dual space \(\widehat{G}\) and \(\pi_l(f)=\int_Gf(g)\pi_l(f)\,dg\). For this he computes explicitly the Hilbert-Schmidt norm \(\|\pi_l(f)\|^2_{HS}\) and the matrix coefficients of the group Fourier transform \(\pi_l(f)\) in two separate cases, namely \({\mathfrak g}(l)\subset[{\mathfrak g},{\mathfrak g}]\) and \({\mathfrak g}(l)\not\subset[{\mathfrak g},{\mathfrak g}]\). This computation, along with some supporting lemmas, takes up Section 3. In Section 4, the proof of the main result is given and in Section 5 the author explains how his theorem sharpens the Cowling-Price type theorem proved by A. Baklouti and N. Ben Salah. It also reduces to a version of the Gelfand-Shilov theorem by showing that the hypotheses of the above mentioned theorems imply the above condition \((*)\).