  In this article, the authors present a new $L^p$- primal-dual weak Galerkin
method ($L^p$-PDWG) for convection-diffusion equations with $p>1$. The
existence and uniqueness of the numerical solution is discussed, and an
optimal-order error estimate is derived in the $L^q$-norm for the primal
variable, where $\frac 1p+\frac 1q=1$. Furthermore, error estimates are
established for the numerical approximation of the dual variable in the
standard $W^{m,p}$ norm, $0\le m\le 2$. Numerical results are presented to
demonstrate the efficiency and accuracy of the proposed $L^p$-PDWG method.
