Norm inequalities for the Littlewood-Paley function 𝑔*_{𝜆}

Muckenhoupt, Benjamin; Wheeden, Richard L.

doi:10.1090/S0002-9947-1974-0387973-X

Norm inequalities for the Littlewood-Paley function $g^{\ast } _{\lambda }$
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by Benjamin Muckenhoupt and Richard L. Wheeden

Trans. Amer. Math. Soc. 191 (1974), 95-111

DOI: https://doi.org/10.1090/S0002-9947-1974-0387973-X

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Abstract:

Weighted norm inequalities for ${L^p}$ and ${H^p}$ are derived for the Littlewood-Paley function $g_\lambda ^ \ast$. New results concerning the boundedness of this function are obtained, by a different method of proof, even in the unweighted case. The proof exhibits a connection between $g_\lambda ^\ast$ and a maximal function for harmonic functions which was introduced by C. Fefferman and E. M. Stein. A new and simpler way to determine the behavior of this maximal function is given.

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