Regularity for degenerate/singular normalized $p$-Laplacian equations with Hamiltonian terms

This paper focuses on the regularity of viscosity solutions to normalized $p$-Laplacian equations with variable-exponent double phase type degeneracy/singularity and Hamiltonian terms. Based on a new improved oscillation-type estimate combined with a localized analysis, we establish sharp interior $C^{1,\alpha}$ regularity estimates in a unified way. In addition, in the degenerate case, we obtain improved gradient H\"{o}lder regularity results at points where the Hamiltonian coefficient and source term vanish, and establish a Schauder-type estimate at local extrema. Notably, our results are still novel even restricted to single power-type singularity or degeneracy law.