this turned out to be wrong, see the comment of fedja below for the counter example.

$\sum_{m>0,n>0,k>0,m+n>k}\frac{1}{(m+n-k)^{1/2}\dot (m+n)^{1/2}}a_m a_n b_k$$\le$ C$\\|A\|_{l^2}^2$*$\|B\|_{l^2}$.

where A is the sery $\{a_m\},m=1,2,3...$, B is the sery $b_k, k=1,2,3...$. C is a constant which is independent of A and B. $\| \|_{l^2}$ denotes the $l^2$ Hilbert norm of the sequence.