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The two-dimensional classical Hardy spaces Hp(R×R) are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded fromHp(R×R) to Lp(R ) (1/2 < p ≤ ∞) and is of weak type (H 1(R×R), L1(R )) where the Hardy space H ♯ 1(R × R) is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ H 1(R × R) ⊃ L logL(R ) converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on Hp(R×R) whenever 1/2 < p <∞. Thus, in case f ∈ Hp(R×R), the Fejér means converge to f in Hp(R × R) norm (1/2 < p < ∞). The same results are proved for the conjugate Fejér means.

Fejér means of two-dimensional Fourier transforms on $H_p(ℝ × ℝ)$