THE STABILITY OF SOBOLEV NORMS FOR THE LINEAR WAVE EQUATION WITH UNBOUNDED PERTURBATIONS

The paper proves reducibility and uniform Sobolev-norm bounds for the 1D wave equation with quasi-periodic, order-1 pseudo-differential perturbations by first performing a delicate pseudo-differential regularization that makes the diagonal part smoothing while keeping the off-diagonal part bounded, and only then running a block-KAM scheme with refined Melnikov and measure estimates. The candidate solution instead applies a largely standard KAM reducibility as if commutators of matrix-valued pseudo-differential operators gained one derivative and treats the perturbation as order 0 without the paper’s critical preliminary transformation. This misses the central obstruction emphasized in the paper and yields an incorrect smoothing gain and oversimplified measure control.