Let 0 < q < ∞. In this study, we introduce the spaces and of q-bounded variation double sequences and q-summable double series as the domain of four-dimensional backward difference matrix Δ and summation matrix S in the space ℒ_q of absolutely q-summable double sequences, respectively. Also, we determine their α- and β-duals and give the characterizations of some classes of four-dimensional matrix transformations in the case 0 < q ≤ 1.

Keywords: summability theory, double sequence, difference sequence space, double series, alpha-dual, beta-dual, matrix domain of 4-dimensional matrices, matrix transformations