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Transactions of the American Mathematical Society

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Necessity conditions for $ L\sp p$ multipliers with power weights

Author: Benjamin Muckenhoupt
Journal: Trans. Amer. Math. Soc. 300 (1987), 503-520
MSC: Primary 42A45; Secondary 42B15
MathSciNet review: 876463
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Abstract: It is shown that if multiplier operators are bounded on with weight $ {\left\vert x \right\vert^\alpha }$ for all functions in the space $ {\mathcal{S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0 and all multiplier functions in a standard Hörmander type multiplier class, then $ \alpha $ must satisfy certain inequalities. This is a sequel to a previous paper in which conditions on $ \alpha $ that were almost the same were shown to be sufficient for the norm inequality to hold.

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