Schauder estimates for parabolic nondivergence operators of Hörmander type (Q870101)

In the very interesting paper under review the authors obtain Schauder estimates for solutions of parabolic nondivergence operators of Hörmander type. Precisely, let \(\{X_j\}_{j=1}^q\) be a system of real smooth vector fields in a bounded domain \(\Omega\subset \mathbb R^n\) which satisfies Hörmander's rank condition that the Lie algebra generated by the fields \(\{X_j\}_{j=1}^q\) with respect to the multiplication \([X,Y]=XY-YX\) has rank \(n\) at each point of \(\Omega.\) Consider a symmetric, uniformly positive definite matrix \(\{a_{ij}(t,x)\}_{i,j=1}^q\) of real functions over \(U\subset \mathbb R\times \Omega\) and define the operator \[ H=\partial_t - \sum_{i,j=1}^q a_{ij}(t,x)X_iX_j - \sum_{i}^q b_{i}(t,x)X_i-c(t,x). \] The authors derive local a~priori estimates of Schauder type in the parabolic Hölder spaces \(C^{k,\alpha}(U)\) defined by the fields \(X_j\) and the Carnot--Carathédory distance induced by them. Namely, suppose \(a_{ij},\;b_i,\;c\in C^{k,\alpha}(U)\) for some integer \(k\geq0\) and \(\alpha\in(0,1).\) Then, for each domain \(U'\Subset U\) there exists a constant \(C\) under control, such that \[ \| u\| _{C^{k+2,\alpha}(U')}\leq C\left(\| Hu\| _{C^{k,\alpha}(U)}+\| u\| _{L^{\infty}(U)}\right) \] for each \(u\in C^{k+2,\alpha}_{\text{loc}}(U)\) such that \(Hu\in {C^{k,\alpha}(U)}.\) The technique adopted relies mainly on \(C^\alpha\)-continuity of singular and fractional integrals over spaces of homogeneous type and could be considered also of independent interest.