Absolute convergence of double series of Fourier–Haar coefficients for functions of bounded $p$-variation

We consider functions of two variables of bounded $p$-variation of the Hardy type on the unit square. For these functions we obtain a sufficient condition for the absolute convergence of series of positive powers of Fourier coefficients with power-type weights with respect to the double Haar system. This condition implies those for the absolute convergence of the Fourier–Haar series for functions of one variable, provided that they have a bounded Wiener $p$-variation or belong to the class $\operatorname{Lip}\alpha$. We show that the obtained results are unimprovable. We also formulate $N$-dimensional analogs of the main result and its corollaries.