Two Corrections to the Heisenberg Example and Sobolev Formulation in The Work of Hong Wang

Abstract

This correction note identifies two local formulation errors in Christopher D. Sogge’s survey The Work of Hong Wang, arXiv:2609.40161v1 (30 September 2026). The displayed four-real-parameter Heisenberg family has only three real parameters of complex directions, so it cannot provide the asserted number of direction-separated tubes; the classical obstruction belongs to a complex Wolff-axiom formulation. Separately, a low-frequency dilation disproves the unrestricted local smoothing inequalities under the survey’s explicit homogeneous Sobolev norm. Inhomogeneous norms, or a fixed positive lower frequency cutoff, remove this obstruction. The note also clarifies exponent improvement and endpoint closure, acknowledging the survey’s open-and-closed argument. These observations concern the survey’s formulations and do not establish a failure of the underlying Kakeya or local smoothing theorems. This record contains the final revised PDF and a matching TeX source archive. Both author affiliations and an AI-assistance and author-responsibility statement are included.