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

In this paper we consider a question on existence of double Walsh series universal 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 $$\sum _{n,k=1}^{\mathrm{\infty }}{c}_{n,k}{W}_n(x)W_k(y)with\sum _{n,k=1}^{\mathrm{\infty }}{\mid c_{n,k}\mid }^q<\mathrm{\infty }forallq>2,$$ which is universal in $L_{\mu }^1[0,1{]}^2$ concerning subseries with respect to convergence, in the sense of both spherical and rectangular partial sums.