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In this note we obtain the existence and the uniqueness of the solution of a variational inequality associated to the degenerate operator $$Lu=-\sum _{i,j=1}^n{(a_{ij}(x)u_{x_i}+d_{j}u)}_{x_j}+\sum _{i=1}^n{b}_i{u}_{x_i}+cu$$ assuming the coefficients of the lower terms and the known term belonging to a suitable degenerate Stummel-Kato class. The weight $w$, which gives the degeneration, belongs to the Muckenoupt class $A^2$.

We prove Harnack inequality for weak solutions to quasilinear subelliptic equation of the following kind $$\sum _{J=1}^m{X}_j^{*}{A}_j(x,u(x),Xu(x))+B(x,u(x),Xu(x))=0,$$ where $X_1,\mathrm{\dots },X_m$ are a system of non commutative locally Lipschitz vector fields. As a consequence, the weak solutions of (*) are continuous.

In this paper is proved a weighted inequality for Riesz potential similar to the classical one by D. Adams. Here the gain of integrability is not always algebraic, as in the classical case, but depends on the growth properties of a certain function measuring some local potential of the weight.