Let W2n [ f ] denote the 2 th partial sums of the Walsh-Fourier series of an integrable functionf. Let pn(x) represent the ratio W2n[ f, x]/2 , for x E [0, 1], and let T(f) represent the function (pn2)1/2. We prove that T(f) belongs to LP[0, 1] for all 0 <p < oo. We observe, using inequalities of Paley and Sunouchi, that the operatorf -T(f ) arises naturally in connection with dyadic differentiation. Namely, if f is strongly dyadically differentiable (with derivative Df ) and has average zero on the interval [0, 1], then the LP norms of f and T(Df) are equivalent when 1 < p < oo. We improve inequalities implicit in Sunouchi's work for the case p = 1 and indicate how they can be used to estimate the L1 norm of T(Df ) and the dyadic H1 norm of f by means of mixed norms of certain random Walsh series. An application of these estimates establishes that if f is strongly dyadically differentiable in dyadic H1, then Jj12N= II WN[f, x] GN[f, x]/Ndx C o.
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