Inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operators

Let $X_j,Y_j(j=1,\cdot \cdot \cdot ,n)$ be vector fields satisfying H"{o}rmander’s condition and ${\Delta_L} = \sum\limits_{j = 1}^n {(X_j^2 + Y_j^2)}$. In this paper, we establish some inequalities of Dirichlet eigenvalues for degenerate elliptic partial differential operator ${\mathrm{\Delta }}_L$ and ${\mathrm{\Delta }}_L^2$. These inequalities extend Yang’s inequalities for Dirichlet eigenvalues of Laplacian to the settings here and the forms of inequalities are more general than Yang’s inequalities. To obtain them, we give a generalization of the inequality by Chebyshev.