A note on strong summability of two-dimensional Walsh-Fourier series

The paper deals with the strong summability of Marcinkiewicz means with a variable power. Let $$H_n left( {f,x,y,A_n } right): = tfrac{1} {n}sumnolimits_{l = 1}^n {left( {e^{left. {A_n } right|left. {S_{ll} left( {f,x,y} right) - fleft( {x,y} right)} right|^{{1 mathord{left/ {vphantom {1 2}} right. kern-nulldelimiterspace} 2}} } - 1} right)} .$$ It is shown that if A _n ↑ ∞ arbitrary slowly, there exists f ∈ C(I ²) such that lim _n→∞ H _n (f, 0, 0, A _n ) = +∞. At the same time, for every f ∈ C (I ²) there exists A _n (f) ↑ ∞ such that lim _n→∞ H _n (f, x, y, A _n ) = 0 uniformly on I ².