Sufficiency conditions for 𝐿^{𝑝}-multipliers with power weights

Muckenhoupt, Benjamin; Wheeden, Richard L.; Young, Wo-Sang

doi:10.1090/S0002-9947-1987-0876461-9

Sufficiency conditions for $L^ p$-multipliers with power weights
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by Benjamin Muckenhoupt, Richard L. Wheeden and Wo-Sang Young PDF

Trans. Amer. Math. Soc. 300 (1987), 433-461 Request permission

Abstract:

Weighted norm inequalities in ${R^1}$ are proved for multiplier operators with the multiplier function of Hörmander type. The operators are initially defined on the space ${\mathcal {S}_{0,0}}$ of Schwartz functions whose Fourier transforms have compact support not including 0. This restriction on the domain of definition makes it possible to use weight functions of the form ${\left | x \right |^\alpha }$ for $\alpha$ larger than usually considered. For these weight functions, if $(\alpha + 1)/p$ is not an integer, a strict inequality on $\alpha$ is shown to be sufficient for a norm inequality to hold. A sequel to this paper shows that the weak version of this inequality is necessary.

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Necessity conditions for

multipliers with power weights

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