Decay of Walsh series and dyadic differentiation

Abstract:

Let ${W_2} n [f]$ denote the ${2^n}{\text {th}}$ partial sums of the Walsh-Fourier series of an integrable function $f$. Let ${\rho _n}(x)$ represent the ratio ${W_2}n[f,x]/{2^n}$, for $x \in [0,1]$, and let $T(f)$ represent the function ${(\Sigma \rho _n^2)^{1/2}}$. We prove that $T(f)$ belongs to ${L^p}[0,1]$ for all $0 < p < \infty$. We observe, using inequalities of Paley and Sunouchi, that the operator $f \to T(f)$ arises naturally in connection with dyadic differentiation. Namely, if $f$ is strongly dyadically differentiable (with derivative $\dot Df$) and has average zero on the interval [0, 1], then the ${L^p}$ norms of $f$ and $T(\dot Df)$ are equivalent when $1 < p < \infty$. We improve inequalities implicit in Sunouchi’s work for the case $p = 1$ and indicate how they can be used to estimate the ${L^1}$ norm of $T(\dot Df)$ and the dyadic ${H^1}$ 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 ${H^1}$, then $\int _0^1 {\Sigma _{N = 1}^\infty |{W_N}[f,x] - {\sigma _N}[f,x]/N dx < \infty }$.