Triangular-\(\theta \) summability of double Fourier series on quantum tori

Abstract

We study the triangular \(\theta \)-mean of the partial sums of \(f \in L_{p}({\mathbb {T}}_{q}^{2})\) and prove the following noncommutative weak and strong type maximal inequalities:

$$\begin{aligned} \Vert (\sigma _n^{\Delta ,\theta }(f))_{n\ge 1}\Vert _{\Lambda _{1,\infty }({\mathbb {T}}_q^2,\ell _{\infty })}\le c_\theta \Vert f\Vert _{L_1({\mathbb {T}}_{q}^2)},\quad p=1 \end{aligned}$$

and

$$\begin{aligned} \left\| \left( \sigma _{n}^{\Delta ,\theta }(f)\right) _{n \ge 1}\right\| _{L_p({\mathbb {T}}_q^2, \ell _{\infty })} \le c_{p, \theta }\Vert f\Vert _{L_p({\mathbb {T}}_q^2)},\quad 1<p<\infty , \end{aligned}$$

where \({\mathbb {T}}_{q}^{2}\) is a 2-dimensional quantum torus. As a consequence, we obtain the bilateral almost uniform convergence of \(\sigma _n^{\Delta ,\theta }(f)\) provided \(f \in L_{p}({\mathbb {T}}_{q}^{2}).\)