Journal of Fourier Analysis and Applications

, Volume 2, Issue 5, pp 487–505 | Cite as

Hardy Spaces on the Plane and Double Fourier Transforms

  • Dang Vu Giang
  • Ferenc Moricz


We provide a direct computational proof of the known inclusion \({\cal H}({\bf R} \times {\bf R}) \subseteq {\cal H}({\bf R}^2),\) where \({\cal H}({\bf R} \times {\bf R})\) is the product Hardy space defined for example by R. Fefferman and \({\cal H}({\bf R}^2)\) is the classical Hardy space used, for example, by E.M. Stein. We introduce a third space \({\cal J}({\bf R} \times {\bf R})\) of Hardy type and analyze the interrelations among these spaces. We give simple sufficient conditions for a given function of two variables to be the double Fourier transform of a function in \(L({\bf R}^2)\) and \({\cal H}({\bf R} \times {\bf R}),\) respectively. In particular, we obtain a broad class of multipliers on \(L({\bf R}^2)\) and \({\cal H}({\bf R}^2),\) respectively. We also present analogous sufficient conditions in the case of double trigonometric series and, as a by-product, obtain new multipliers on \(L({\bf T}^2)\) and \({\cal H}({\bf T}^2),\) respectively.


Fourier Series Hardy Space Hardy Inequality Hardy Type Double Fourier Series 
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Copyright information

© Birkhäuser Boston 1995

Authors and Affiliations

  • Dang Vu Giang
    • 1
  • Ferenc Moricz
    • 2
  1. 1.Institute of Mathematics, University of Veszprem, Egyetem U. 10, 8201 VeszpremHungary
  2. 2.Bolyai Institute, University of Szeged, Aradi Vertanuk Tere 1, 6720 SzegedHungary

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