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Proceedings of the American Mathematical Society

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On $ H\sp p(\bold R\sp n)$-multipliers of mixed-norm type

Authors: C. W. Onneweer and T. S. Quek
Journal: Proc. Amer. Math. Soc. 121 (1994), 543-552
MSC: Primary 42B15; Secondary 42B30, 46E99
MathSciNet review: 1204383
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Abstract: For a function m in $ {L^\infty }({{\mathbf{R}}^n})$, an appropriately chosen function $ \eta $ in $ {C^\infty }({{\mathbf{R}}^n})$ and $ \delta > 0$ we define $ {m_\delta }$ by $ {m_\delta }(\xi ) = m(\delta \xi )\eta (\xi )$. We show that if $ 0 < p \leq 1$ and if the sequence $ ((m_{2^n})\hat \emptyset )$ belongs to a certain mixed-norm space, depending on p, then m is a Fourier multiplier for the corresponding Hardy space $ {H^p}({{\mathbf{R}}^n})$. Moreover, we prove the sharpness of our multiplier theorem. Comparable results had been proved earlier for multipliers for Hardy spaces defined on a locally compact Vilenkin group.

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Keywords: Hardy spaces, Fourier multipliers, mixed norms
Article copyright: © Copyright 1994 American Mathematical Society