Advertisement

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
Article

Abstract

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.

Keywords

Fourier Series Hardy Space Hardy Inequality Hardy Type Double Fourier Series 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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

Personalised recommendations