Generalized Hölder estimates via generalized Morrey norms for some ultraparabolic operators

We consider a class of hypoelliptic operators of the following type \(\begin{aligned} {\mathcal {L}}=\sum \limits _{i,j=1}^{p_0} a_{ij} \partial _{x_i x_j}^2+\sum \limits _{i,j=1}^{N} b_{ij} x_i \partial _{x_j}-\partial _t, \end{aligned}\) $$\begin{array}{c}\hfill \mathcal{L}=\sum _{i,j=1}^{p_0}{a}_{\mathit{ij}}{\partial }_{x_i{x}_j}^2+\sum _{i,j=1}^N{b}_{\mathit{ij}}{x}_i{\partial }_{x_j}-{\partial }_t,\end{array}$$ where \((a_{ij})\) $(a_{\mathit{ij}})$ , \((b_{ij})\) $(b_{\mathit{ij}})$ are constant matrices and \((a_{ij})\) $(a_{\mathit{ij}})$ is symmetric positive definite on \({\mathbb {R}}^{p_0}\) ${\mathbb{R}}^{p_0}$ \((p_0\le N)\) $(p_0\le N)$ . We obtain generalized Hölder estimates for \({\mathcal {L}}\) $\mathcal{L}$ on \({\mathbb {R}}^{N+1}\) ${\mathbb{R}}^{N+1}$ by establishing several estimates of singular integrals in generalized Morrey spaces.