Bounds for moments of cubic and quartic Dirichlet $L$-functions

We study the $2k$-th moment of central values of the family of primitive cubic and quartic Dirichlet $L$-functions. We establish sharp lower bounds for all real $k \geq 1/2$ unconditionally for the cubic case and under the Lindel\"of hypothesis for the quartic case. We also establish sharp lower bounds for all real $0 \leq k<1/2$ and sharp upper bounds for all real $k \geq 0$ for both the cubic and quartic cases under the generalized Riemann hypothesis (GRH). As an application of our results, we establish quantitative non-vanishing results for the corresponding $L$-values.