Subelliptic $p$-Laplacian spectral problem for Hörmander vector fields

Based on variational methods, we study the spectral problem for the subelliptic $p$-Laplacian arising from smooth Hörmander vector fields. We derive the smallest eigenvalue, prove its simplicity and isolatedness, establish the positivity of the first eigenfunction and show Hölder regularity of eigenfunctions. Moreover, we determine the best constant for the $L^p$-Poincaré-Friedrichs inequality for Hörmander vector fields as a byproduct.