On the maximal Fejér operator for double Fourier series of functions in Hardy spaces

• Autorzy: Ferenc Móricz
• Języki publikacji: EN

Abstrakty

EN

We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1,0)}(𝕋^2)$, $H^{(0,1)}(𝕋^2)$, or $H^{(1,1)}(𝕋^2)$. We prove that the maximal Fejér operator is bounded from $H^{(1,0)}(𝕋^2)$ or $H^{(0,1)}(𝕋^2)$ into weak-$L^1(𝕋^2)$, and also bounded from $H^{(1,1)}(𝕋^2)$ into $L^1(𝕋^2)$. These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1} log^{+} L(𝕋^2)$, $L^1(log^{+}L)^2(𝕋^2)$, and $L^μ(𝕋^2)$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures.

Kategorie tematyczne

• 47B38: Operators on function spaces (general)
• 42A50: Conjugate functions, conjugate series, singular integrals
• 46J15: Banach algebras of differentiable or analytic functions, H p -spaces
• 42B08: Summability

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1995
• Tom: 116
• Numer: 1
• Strony: 89-100

Daty

wydano

1995

otrzymano

1995-03-29

Twórcy

autor

Ferenc Móricz

• Bolyai Institute, University of Szeged, Aradi Vértanúk Tere 1, 6720 Szeged, Hungary

Bibliografia

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• [5] B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 25 (1935), 217-234.
• [6] J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, ibid. 32 (1939), 112-132.
• [7] F. Móricz, The maximal Fejér operator is bounded from $H^1(𝕋)$ into $L^1(𝕋)$, Analysis, submitted.
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• [9] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, 1959.