Abstract
In this paper we construct solutions in L-p-space to the (partial derivative) over bar -equation with prescribed supports in strictly q-linearly convex domains with piecewise smooth boundaries in complex projective space. This result leads to a (partial derivative) over bar -closed extension of L-p-forms and consequently the a-equation can be solved on annuli type domains. Analogue results are proved on strictly q-pseudoconcave (strictly q* - pseudoconvex) domains in complete intersection subvarieties in the complex projective space.