Divergence of the (C, 1) means of d-dimensional Walsh-Fourier series

In 1992, Móricz, Schipp and Wade [MSW] proved for functions in L log+ L(I2) (I2 is the unit square) the a.e. convergence of the double (C, 1) means of the Walsh-Fourier series σnf → f as min(n1, n2) → ∞, n = (n1, n2) ∈ N2. In the same paper, they also proved the restricted convergence of the (C, 1) means of functions in L(I2): σ(2n1,2n2) f → f a.e. as min(n1, n2) → ∞ provided |n1 - n2| < C. The aim of this paper is to demonstrate the sharpness of these results of Móricz, Schipp and Wade with respect to both the space L log+ L(I2) and the restrictedness |n1 - n2| < C.