New Exceptional Sets and Convergence of the Square Partial Sums of Walsh–Fourier Series

For double Walsh–Fourier series and with f∈L2([0,1)×[0,1))we prove two almost orthogonality results relative to the linearized maximal square partial sums operator SN(x,y)f(x,y).Assumptions are N(x,y)non-decreasing as a function of x and of y and,roughly speaking,partial derivatives with approximately constant ratio■≌2n0 for all x and y,where n0 is any fixed non-negative integer.Estimates,independent of N(x,y)and n0,are then extended to Lr,1<r<2.We give an application to the family N(x,y)=λxy on[0,1)×[0,1),anyλ>10.