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In this paper we consider a question on existence of double Walsh seriesuniversal in weighted $L_\mu^1[0,1]^2$ spaces. We construct a weighted function$\mu(x,y)$ and a series by double Walsh system of the form$$\sum_{n,k=1}^\infty c_{n,k}W_n(x)W_k(y)\ \ \mbox{with} \ \\sum_{n,k=1}^\infty \left | c_{n,k} \right|^q <\infty\ \mbox{for all}\ q>2,$$which is universal in $L_\mu^1[0,1]^2$ concerning subseries with respect toconvergence, in the sense of both spherical and rectangular partial sums.

Existence of double Walsh series universal in weighted $L_\mu^1[0,1]^2$ spaces