  We study basic Dolbeault cohomology and find new Weitzenb\"ock formulas on a
transversely K\"ahler foliation. We investigate conditions on mean curvature
and Ricci curvature that impose restrictions on basic Dolbeault cohomology. For
example, we prove that on a transversely K\"ahler foliation with positive
transversal Ricci curvature, there are no nonzero basic-harmonic forms of type
$(r,0)$, among other results.
