Pointwise convergence of lacunary partial sums of almost periodic Fourier series

Abstract:

It is a classical result that lacunary 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 operators corresponding to lacunary partial summation of almost periodic Fourier series are shown to be 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 “$S^p$, $p \in (1, \infty )$” is also seen to follow from a natural conjecture on the boundedness of the Hilbert transform.