Fejér means of two-dimensional Fourier transforms on ${\displaystyle H_p(ℝ\times ℝ)}$

The two-dimensional classical Hardy spaces ${\displaystyle H_p(ℝ\times ℝ)}$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from ${\displaystyle H_p(ℝ\times ℝ)}$ to ${\displaystyle L_p\left({ℝ}^2\right)}$ (1/2 < p ≤ ∞) and is of weak type ${\displaystyle (H^{\#}\_1(ℝ\times ℝ),L_1\left({ℝ}^2\right))}$ where the Hardy space ${\displaystyle H^{\#}\_1(ℝ\times ℝ)}$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ ${\displaystyle H_1^{\#}(ℝ\times ℝ)}$ ⊃ ${\displaystyle L\log L\left({ℝ}^2\right)}$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on ${\displaystyle H_p(ℝ\times ℝ)}$ whenever 1/2 < p < ∞. Thus, in case f ∈ ${\displaystyle H_p(ℝ\times ℝ)}$, the Fejér means converge to f in ${\displaystyle H_p(ℝ\times ℝ)}$ norm (1/2 < p < ∞). The same results are proved for the conjugate Fejér means.