Generalized Hölder estimates via generalized Morrey norms for some ultraparabolic operators

We consider a class of hypoelliptic operators of the following type

$$\begin{aligned} {\mathcal {L}}=\sum \limits _{i,j=1}^{p_0} a_{ij} \partial _{x_i x_j}^2+\sum \limits _{i,j=1}^{N} b_{ij} x_i \partial _{x_j}-\partial _t, \end{aligned}$$

where \((a_{ij})\), \((b_{ij})\) are constant matrices and \((a_{ij})\) is symmetric positive definite on \({\mathbb {R}}^{p_0}\) \((p_0\le N)\). We obtain generalized Hölder estimates for \({\mathcal {L}}\) on \({\mathbb {R}}^{N+1}\) by establishing several estimates of singular integrals in generalized Morrey spaces.