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Given an elliptic diffusion operator $L$ defined on a compact and connectedmanifold (possibly with a convex boundary in a suitable sense) with an$L$-invariant measure $m$, we introduce the non-linear $p-$operator $L_p$,generalizing the notion of the $p-$Laplacian. Using techniques of the intrinsic$\Gamma_2$-calculus, we prove the sharp estimate $\lambda\geq (p-1)\pi_p^p/D^p$for the principal eigenvalue of $L_p$ with Neumann boundary conditions underthe assumption that $L$ satisfies the curvature-dimension condition BE$(0,N)$for some $N\in[1,\infty)$. Here, $D$ denotes the intrinsic diameter of $L$.Equality holds if and only if $L$ satisfies BE$(0,1)$. We also derive the lowerbound $\pi^2/D^2+a/2$ for the real part of the principal eigenvalue of anon-symmetric operator $L=\Delta_g+X\cdot\nabla$ satisfying$\operatorname{BE}(a,\infty)$.

Sharp Estimates for the Principal Eigenvalue of the p-Operator