On a theorem of Hardy and Littlewood on the polydisc

Kim, Hong Oh

doi:10.1090/S0002-9939-1986-0840619-X

On a theorem of Hardy and Littlewood on the polydisc
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by Hong Oh Kim

Proc. Amer. Math. Soc. 97 (1986), 403-409

DOI: https://doi.org/10.1090/S0002-9939-1986-0840619-X

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Abstract:

We prove the polydisc version of the theorem of Hardy and Littlewood on the fractional integral: If $0 < \alpha < 1/p$ and if $f \in {H^p}$, then ${I^\alpha }f \in {H^q}$ with $q = p/(1 - \alpha p)$ where ${I^\alpha }f$ is the fractional integral of $f$ of order $\alpha$.

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