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 PDF

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

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 $\text {BMO}$.

References

K. Astala and F. W. Gehring, Quasiconformal analogues of theorems of Koebe and Hardy-Littlewood, Michigan Math. J. 32 (1985), no. 1, 99–107. MR 777305, DOI 10.1307/mmj/1029003136
K. Astala and F. W. Gehring, Injectivity, the BMO norm and the universal Teichmüller space, J. Analyse Math. 46 (1986), 16–57. MR 861687, DOI 10.1007/BF02796572
B. Bojarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $\textbf {R}^{n}$, Ann. Acad. Sci. Fenn. Ser. A I Math. 8 (1983), no. 2, 257–324. MR 731786, DOI 10.5186/aasfm.1983.0806
Peter L. Duren, Theory of $H^{p}$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. MR 0268655
John B. Garnett, Bounded analytic functions, Pure and Applied Mathematics, vol. 96, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981. MR 628971
F. W. Gehring, Symmetrization of rings in space, Trans. Amer. Math. Soc. 101 (1961), 499–519. MR 132841, DOI 10.1090/S0002-9947-1961-0132841-2
F. W. Gehring, The $L^{p}$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265–277. MR 402038, DOI 10.1007/BF02392268
F. W. Gehring and O. Martio, Lipschitz classes and quasiconformal mappings, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 203–219. MR 802481, DOI 10.5186/aasfm.1985.1022
G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. II, Math. Z. 34 (1932), no. 1, 403–439. MR 1545260, DOI 10.1007/BF01180596
T. Iwaniec and C. A. Nolder, Hardy-Littlewood inequality for quasiregular mappings in certain domains in $\textbf {R}^n$, Ann. Acad. Sci. Fenn. Ser. A I Math. 10 (1985), 267–282. MR 802488, DOI 10.5186/aasfm.1985.1030
Vesa Lappalainen, $\textrm {Lip}_h$-extension domains, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 56 (1985), 52. MR 822591
O. Martio, S. Rickman, and J. Väisälä, Definitions for quasiregular mappings, Ann. Acad. Sci. Fenn. Ser. A I No. 448 (1969), 40. MR 0259114
Raimo Näkki and Bruce Palka, Lipschitz conditions and quasiconformal mappings, Indiana Univ. Math. J. 31 (1982), no. 3, 377–401. MR 652824, DOI 10.1512/iumj.1982.31.31033
C. A. Nolder, Conjugate functions and moduli of continuity, Illinois J. Math. 31 (1987), no. 4, 699–709. MR 909793
H. M. Reimann, Functions of bounded mean oscillation and quasiconformal mappings, Comment. Math. Helv. 49 (1974), 260–276. MR 361067, DOI 10.1007/BF02566734
Hans Martin Reimann and Thomas Rychener, Funktionen beschränkter mittlerer Oszillation, Lecture Notes in Mathematics, Vol. 487, Springer-Verlag, Berlin-New York, 1975 (German). MR 0511997
Jukka Sarvas, Symmetrization of condensers in $n$-space, Ann. Acad. Sci. Fenn. Ser. A. I. 522 (1972), 44. MR 348108
Susan G. Staples, $L^p$-averaging domains and the Poincaré inequality, Ann. Acad. Sci. Fenn. Ser. A I Math. 14 (1989), no. 1, 103–127. MR 997974, DOI 10.5186/aasfm.1989.1429
Matti Vuorinen, Conformal invariants and quasiregular mappings, J. Analyse Math. 45 (1985), 69–115. MR 833408, DOI 10.1007/BF02792546
Matti Vuorinen, Conformal geometry and quasiregular mappings, Lecture Notes in Mathematics, vol. 1319, Springer-Verlag, Berlin, 1988. MR 950174, DOI 10.1007/BFb0077904