Almost everywhere divergence of double Fourier series along shrinking conical regions

Abstract

We study almost everywhere divergence of rectangular partial sums of double trigonometric Fourier series along variable regions concentrated near the diagonal. Fefferman's theorem shows that, in two variables, unrestricted rectangular summation is radically different from the one-dimensional Carleson-Hunt theory: there are continuous functions whose rectangular Fourier sums diverge everywhere. Bakhvalov proved that this phenomenon persists even when the indices are restricted to a fixed cone-shaped neighbourhood of the diagonal. On the other hand, the diagonal summation results of Tevzadze and Fefferman, and the later theorem of Antonov for shrinking cones, show that convergence is restored when the aperture is of order $1/n$. We prove that Antonov's aperture condition is sharp in the $L^{2}$-scale. For every positive nonincreasing sequence $\{λ_{n}\}$ with $% \sup_{n}nλ_{n}=\infty $, we construct a function $f\in L^{2}(\mathbb{T% }^{2})$ whose symmetric rectangular Fourier partial sums fail to converge almost everywhere along the corresponding variable cone. The construction combines a Fefferman-type analytic block with frequency separation and independent random translations. We also observe that the sufficiency part of Antonov's theorem does not require monotonicity of $\{λ_{n}\}$.