Hölder estimates for subelliptic operators (Q1874691)

This paper deals with the equation \(L_\lambda u= u_{xx}+|x|^{2\lambda} u_{yy}= f(x,y)\), \(\lambda> 0\). In the case of \(\lambda\)-integer, \(L_\lambda\) is sum of squares of vector fields operator, and Hörmander has proved subelliptic estimates for it in the scale of Sobolev spaces. The author of the paper under consideration studies optimal Hölder estimates for the same operator. To do this he introduces an appropriate intrinsic Hölder space \(C^\alpha_*\) with the following distance \(d\) between the points \(X= (x_1, y_1)\), \(Y= (x_2,y_2)\): \[ d(X,Y)= (x_1- x_2)+ {|y_1- y_2|\over|x_1|^\lambda+|x_2|^\lambda+|y_1- y_2|^{{\lambda\over 1+\lambda}}}, \] \(\lambda< \alpha< (2\lambda)\wedge 1\) and proves subelliptic type estimates in this scale of spaces.