Logarithmic vanishing theorems for effective $q$-ample divisors

Let $X$ be a compact K"ahler manifold and $D$ be a simple normal crossing divisor. If $D$ is the support of some effective $k$-ample divisor, we show $$H^q(X,{\mathrm{\Omega }}_X^p(\log D))=0,\text{for}p+q>n+k\mathrm{.}$$