A quasiregular analogue of a theorem of Hardy and Littlewood

Nolder, Craig A.

doi:10.1090/S0002-9947-1992-1036007-3

A quasiregular analogue of a theorem of Hardy and Littlewood
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by Craig A. Nolder

Trans. Amer. Math. Soc. 331 (1992), 215-226

DOI: https://doi.org/10.1090/S0002-9947-1992-1036007-3

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Abstract:

Suppose that $f$ is analytic in the unit disk. A theorem of Hardy and Littlewood relates the Hölder continuity of $f$ over the unit disk to the growth of the derivative. We prove here a quasiregular analogue of this result in certain domains in $n$-dimensional space. We replace values of the derivative with a local integral average. In the process we generalize a result on the continuity of quasiconformal mappings due to Nakki and Palka. We also present another proof of the relationship between the growth of the derivative and quasiregular mappings in $\text {BMO}$.

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