Decay and regularity properties for the dispersion generalized Benjamin-Ono equation

We study spatial decay properties of solutions to the dispersion generalized Benjamin-Ono equation. We first establish persistence properties in weighted Sobolev spaces under a sharp condition between decay and regularity, improving previous results. Additionally, extending recent works on the Benjamin-Ono equation by Linares and Ponce [27], we also show that decay at two distinct times implies additional regularity of solutions. The proof relies on weighted energy estimates and pseudo-differential methods, which build on an iterative mechanism that trades decay for regularity.