Hilbert's double series theorem $\sum_{n,m}\frac{a_n b_m}{m+n}$ and its generalizations.

Let $l_p$ be the space of all complex sequences $x=\{x_n\}$ equipped with the finite norm $\|x\|_p=\left(\sum_n |x_n|^p\right)^{1/p}$. Hilbert’s double series theorem states that
$$\sum_{n,m}\frac{a_n b_m}{m+n}<\frac{\pi}{\sin(\pi/p)} \|a\|_p \|b\|_q$$
for non negative $a=\{a_n\}\in l_p$ and $b=\{b_n\}\in l_q$. In literature there are known some generalizations of this theorem, e.g. when
$$\sum_{n,m}\frac{a_n b_m}{(m+n)^\lambda}$$
but, for example, I did not find a lower bound for
$$\sum_{n,m}\frac{a_n b_m}{m+n}$$
or an upper bound for
$$\sum_{n,m}\frac{a_n b_m}{(m+n)^\lambda}$$
when $\lambda=0$.

Any suggestions please?

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