On the convergence of parabolically scaled two-dimensional Fourier series in the linear phase setting
Volume 237 / 2017
Studia Mathematica 237 (2017), 101-117
MSC: 42B20, 42B08.
DOI: 10.4064/sm8182-10-2016
Published online: 24 February 2017
Abstract
For $$Sf(x,y)=\int ^\pi _{-\pi }\int ^\pi _{-\pi } {{e^{iM^2(x,y) y’}}\over {y’} }\, {{e^{iM(x,y) x’}}\over {x’}}f(x-x’,y-y’)\, dx’\, dy’ ,$$ the linearized maximal operator of the rectangular partial sums of the kind $(M,M^2)$ for double Fourier series, we prove a weak-type $(L^r, L^{r-\varepsilon })$ estimate for $1 \lt r\leq 2$ and any $\varepsilon \gt 0$ in case $M^2(x,y)=Ax+By$ with $x,y \in [0,2\pi ],$ uniformly with respect to $A, B\geq 0.$
