Projective descriptions of the (LF)-spaces of type $LB(lambda_p(A),F)$

Let $1\leq p<+\infty$ or $p=0$ and let $A=(a_n)_n$ be an increasing sequence of strictly positive weights on $I$. Let $F$ be a Fréchet space. It is proved that if $\lambda _p(A)$ satisfies the density condition of Heinrich and a certain condition $(C_t)$ holds, then the (LF)-space $LB_i(\lambda _p(A),F)$ is a topological subspace of $L_b(\lambda _p(A),F)$. It is also proved that these conditions are necessary provided $F=\lambda _q(A)$ or $F$ contains a complemented copy of $l_q$ with $1<p\leq q <+\infty$.