The author proves the estimates for the tangential Cauchy-Riemann operator \(\bar\partial_b\) on 3 dimensional compact weakly pseudoconvex abstract CR manifold \(M\) under the assumptions that the range of \(\bar\partial_b\) is closed and that there is a strongly plurisubharmonic function \(\lambda\) at weakly pseudoconvex points. Such a \(\lambda\) always exists when \(M\) is embedded in some \(\mathbb C^n.\) If \(M\) is the smooth boundary of a bounded pseudoconvex manifold in \(\mathbb C^n,\) then the range of \(\bar\partial_b\) is closed. If \(f\) is in the range of \(\bar\partial_b,\) then one can find a solution \(u\) of \(\bar\partial_b u =f\) satisfying \(\| u\|_s\leq C_s\| f\|_s.\) To do these, a pseudodifferential operator \(G^t\) of order \(0\) whose symbol depends exponentially on \(\lambda\) is introduced. A generalized Sobolev \(s\)-norm \(\|\;\|_{G_{, s}^{t}}\) is defined and an estimate in this norm via microlocal analysis are obtained. This new norm can be considered as a Sobolev norm ‘weighted’ by a pseudodifferential operator \(G^t.\)