Hölder continuity and Harnack inequality for De Giorgi classes related to Hörmander vector fields (Q1913394)

After introducing De Giorgi classes \(\text{DG}^2\) related to Hörmander's square operators, the author proves the Hölder continuity of the functions belonging to these classes and Harnack inequality. Then the obtained results are applied to the Q-minima problem for the functional of the type \[ J(u,\Omega)= \int_\Omega f(x,u,Y(u))dx,\;Y(u)=\{Y_j(u)\}^\nu_{j=1}. \] Here \(X=\{X_i\}^m_{i=1}\) is a collection of \(C^\infty\) vector fields on \(\mathbb{R}^n\), satisfying a Hörmander condition of type \(k\), \(Y\)-commutators of \(X_1,X_2,\dots,X_m\) up to order \(k\). Under the growth and Carathéodory conditions on \(f\), the quasi minima belong to De Giorgi classes. Theorem 4.1 concerns the example of the equation \(\sum^m_{i=1} X_iX^*_i-B(x,u,Y(u))= 0\). The operator \(B\) satisfies, among others, the condition \(|B(x,s,w)|\leq L\sum^m_{i=1} |w_i|^2+l(x)\), \(L\) is a suitable positive constant, and \(l\in L^1(\Omega)\). It is proved that a bounded weak solution \(u\) \((|u|\leq M)\) of the equation is a Q-minimum for the functional \(J(u,\Omega)=\int_\Omega(\sum^m_{i=1}|X_i(u)|^2+l(x))dx\), provided \(L<1/2 M\).