A correction theorem and the dyadic space H1, ∞

Abstract

It is shown that for every l∞-function f and for every ɛ, ɛ>0, there exists a function g such that mes {t=g} <ɛ, while the partial sums of the Fourier and Fourier-Walsh series of the function g are uniformly bounded by the number C log (e−1)∥f∥∞. In the proof we make use of the characterization of the dyadic space H1, ∞ in terms of atomic decompositions (it is, apparently, new).