Summability of double Fourier series on quantum tori

In this paper, we study two general summability methods generated by a function θ for noncommutative Fourier series on quantum tori T q 2 . For the rectangular θ -summation, we establish the noncommutative weak type maximal inequality ‖ ( σ m , n θ ( f ) ) ( m , n ) ∈ Σ β ‖ Λ 1 , ∞ ( T q 2 , ℓ ∞ ) ≤ c β , θ ‖ f ‖ L 1 ( T q 2 ) , which generalizes the result due to Marcinkiewicz and Zygmund (Fundam Math 32:122–132, 1939). For the Marcinkiewicz θ -summation, we prove that ‖ ( F n θ ( f ) ) n ≥ 1 ‖ Λ 1 , ∞ ( T q 2 , ℓ ∞ ) ≤ c θ ‖ f ‖ L 1 ( T q 2 ) . Both noncommutative weak type maximal inequalities imply the bilateral almost uniform convergence. The θ -summation contains almost all well known summability methods, such as the Fejér, Weierstrass, Riesz, Picard, Bessel, Riemann, Rogosinski and de La Vallée–Poussin summations.