On the H\"older regularity of signed solutions to a doubly nonlinear equation. Part III

We establish the local H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is $$\partial_t\big(|u|^{q-1}u\big)-\Delta_p u=0,\quad 1<p<2,\quad 0<p-1<q.$$ The proof exploits the space expansion of positivity for the singular, parabolic $p$-Laplacian and employs the method of intrinsic scaling by carefully balancing the double singularity.