Singular measures with small \(H(p,q)\)-projections (Q1810326)

Let \(S=S_n\subset \mathbf {C}^n\), \(n\geq 2\), be the complex unit sphere and \(H(p,q)\) the space of complex spherical harmonics. For a given probability measure \(\mu\in M(S)\) on \(S\) let \(\mu_{p,q}\), \(p,q\in \mathbf {Z}^2_+\), denote the projection of \(\mu\) on \(H(p,q)\). A function \(h: \mathbf {Z}^2_+\rightarrow \mathbf {R}_+\) is called \(S\)-admissible if there exists a continuous singular probability measure \(\mu\in m(S)\) such that \(\|\mu_{p,q}\|_2=O(h(p,q))\). It is proved that if \(h(p,0)=h(0,p)=p^{-1/2}\) and \(h(p,q)=0\) if \(pq\not= 0\) then \(h\) is \(S\)-admissible.