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| Operator-Valued Fourier Multipliers on Multi-dimensional Hardy Spaces |
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Citation： |
Shangquan BU.Operator-Valued Fourier Multipliers on Multi-dimensional Hardy Spaces[J].Chinese Annals of Mathematics B,2011,32(2):293~302 |
| Page view： 1733
Net amount： 1286 |
Authors： |
Shangquan BU; |
Foundation： |
the National Natural Science Foundation of China (No. 10731020) and the Specialized
Research Fund for the Doctoral Program of Higher Education (No. 200800030059). |
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| Abstract： |
The author establishes operator-valued Fourier multiplier theorems on
multi-dimensional Hardy spaces $H^p(\BT^d; X)$, where $1\leq p <
\infty$, $d\in\BN$, and $X$ is an AUMD Banach space having the
property $(\alpha)$. The sufficient condition on the multiplier is a
Marcinkiewicz type condition of order $2$ using Rademacher
boundedness of sets of bounded linear operators. It is also shown that
the assumption that $X$ has the property $(\alpha)$ is necessary
when $d\geq 2$ even for scalar-valued multipliers. When the
underlying Banach space does not have the property $(\alpha)$, a
sufficient condition on the multiplier of Marcinkiewicz type of order
$2$ using a notion of $d$-Rademacher boundedness is also given. |
Keywords： |
Hp-Spaces, Fourier multiplier, Rademacher boundedness,
d-Rademacher boundedness |
Classification： |
42A45, 42B15, 43A17, 46B20 |
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