Riesz inequality for systems of root vector functions of the Dirac operator (Q443982)

Let \(D\) be a Dirac operator on a finite interval \(G\) and let \((\varphi _k)_{k=1}^\infty\) be its normalized eigenfunctions and associated functions with corresponding eigenvalue sequence \((\lambda _k)_{k=1}^\infty \). No boundary conditions are given; presumably, any boundary conditions which lead to a discrete spectrum will do. It is assumed that the imaginary parts of the eigenvalues as well as the lengths of the chains of eigenfunctions and associated functions are uniformly bounded. The authors show that \((\varphi _k)_{k=1}^\infty\) satisfies the Riesz (or Hausdorff-Young) inequality, i.e., there is \(M>0\) such that \[ \sum_{k=1}^\infty |(\varphi _k,f)|^q\leq M\|f\|_{p,2}^q,\;f\in L_p^2(G), \] holds for \(1<p\leq 2\), \(q=p/(p-1)\), if and only if the number of indices \(k\) with \(|\text{Re\,}\lambda _k-\nu|<1\) is uniformly bounded with respect to \(\nu \in \mathbb R\).