Absolute convergence of Walsh-Fourier series and related results

We consider the Walsh orthonormal system on the interval [0, 1) in the Paley enumeration and the Walsh-Fourier coefficients $ hat f $ (n), n ∈ ?, of functions f ∈ L ^p for some 1 < p ≤ 2. Our aim is to find best possible sufficient conditions for the finiteness of the series Σ _n=1 ^∞ a _n | $ hat f $ (n)| ^r , where {a _n } is a given sequence of nonnegative real numbers satisfying a mild assumption and 0 < r < 2. These sufficient conditions are in terms of (either global or local) dyadic moduli of continuity of f. The sufficient conditions presented in the monograph [2] are special cases of our ones.