Extension to more singular generalized SQG equations

Establish whether anomalous regularization and anomalous integrability extend to generalized SQG equations with constitutive-law homogeneity parameter \(\mathfrak{h}\in(0,1)\), beyond the range \(\mathfrak{h}\in(-1,0)\) treated by the paper.

Background

The paper explains that its methods extend naturally from the Biot–Savart and Calderón–Zygmund cases to generalized SQG equations with $\mathfrak{h}\in(-1,0)$, after adjusting the scaling numerology. In that range, the authors expect analogous anomalous regularization and anomalous integrability results under the corresponding subcritical assumptions.

The more singular range $\mathfrak{h}\in(0,1)$ is not addressed. Extending the same turbulent phenomenologies to this range is explicitly left unresolved.