Pointwise Convergence of Dyadic Partial Sums of Almost Periodic Fourier Series

It is a classical result that dyadic partial sums of the Fourier series of functions $f\in L^p(\mathbb{T})$ converge almost everywhere for $p \in (1, \infty)$. In 1968, E. A. Bredihina established an analogous result for functions belonging to the Stepanov space of almost periodic functions $S^2$ whose Fourier exponents satisfy a natural separation condition. Here, the maximal operator corresponding to dyadic partial summation of almost periodic Fourier series is bounded on the Stepanov spaces $S^{2^k}$, $k\in \mathbb{N}$ for functions satisfying the same condition; Bredihina’s result follows as a consequence. In the process of establishing these bounds, some general results are obtained which will facilitate further work on operator bounds and convergence issues in Stepanov spaces. These include a boundedness theorem for the Hilbert transform and a theorem of Littlewood–Paley type. An improvement of “$S^{2^k}$, $k\in \mathbb{N}$” to “$S^p$, $p\in (1,\mathrm{\infty })$” is also seen to follow from a natural conjecture on the boundedness of the Hilbert transform.