The classes $\mathcal{B}_{m,l}$ and Hölder estimates for quasilinear doubly degenerate parabolic equations

Inner and boundary Hölder estimates for nonnegative weak solutions of quasilinear doubly degenerate parabolic equations are established. The proof of these results is based on studing some classes $\mathcal{B}_{m,l}$ which can be considered as an extensions of the classes $\mathcal{B}_2$ introduced by Ladyzhenskaya–Uraltseva and the classes $\mathcal{B}_m$ introduced by DiBenedetto. Imbedding of the classes $\mathcal{B}_{m,l}$ in appropriate Hölder spaces is proved.