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In this paper, our primary objective is to provide a fresh perspective on therelationship between the $(\overline{N},p,q)$ method, which is a product ofrelevant one-dimensional summability methods, and $P$-convergence for doublesequences. To accomplish this objective, we establish certain Tauberianconditions that control the behavior of a double sequence in terms of both$O_L$-oscillation and $O$-oscillation in certain senses, building a bridgebetween $(\overline{N},p,q)$ summability and $P$-convergence, while imposingcertain restrictions on the weight sequences. As special circumstances of ourfindings, we demonstrate that Landau-type $O_L$ condition with respect to$(P_m)$ and $(Q_n),$ as well as Hardy-type $O$ condition with respect to$(P_m)$ and $(Q_n),$ serve as Tauberian conditions for $(\overline{N},p,q)$summability under particular additional conditions. Consequently, these resultsencompass all classical Tauberian theorems, including conditions such as slowdecrease or slow oscillation in certain senses.

The novel Tauberian conditions associated with the $(\overline{N},p,q)$ summability of double sequences