Remarks on a theorem by N. Yu. Antonov

Abstract

Abstract.We extend some results of N. Yu. Antonov on convergence of Fourier series to more general settings. One special feature of our work is that we do not assume smoothness for the kernels in our hypotheses. This has interesting applications to convergence with respect to general orthonormal systems, like the Walsh-Fourier system, for which we prove a.e. convergence in the class L log L log log log L. Other applications are given in the theory of differentiation of integrals.

Pointwise convergence of Cesàro and Riesz means on certain function spaces

“…Sjölin-Soria [13] extended results of [1] to more general settings. We can apply results of [13] to prove Theorem 2 for d ≥ 2.…”

“…Sjölin-Soria [13] extended results of [1] to more general settings. We can apply results of [13] to prove Theorem 2 for d ≥ 2. Indeed, we easily see that Theorem 2 for d ≥ 2 follows from Lemma 1 and methods of [13,Section 3] (see Remark at the end of Section 3 of [13]).…”

Carleson’s Theorem: Proof, Complements, Variations

“…One seeks to "extrapolate" these inequalities to the setting of the Theorem above and Antonov nicely exploits the explicit nature of the kernels involved in this maximal operator. Also see the work of P. Sjölin and F. Soria [90] who demonstrate that Antonov's approach extends to other maximal operator questions.…”

Partial sums of Fourier series

“…Sjölin and Soria (2003) extended this result to the case of D (series of Fourier-Walsh) and other orthogonal expansions [19]. On the other hand there exist integrable functions whose Fourier series diverge almost everywhere [7] and even everywhere [8]: these are Kolmogorov's examples of 1923 and 1926.…”