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Hardy and Littlewood found a relation between the smoothness of the radial limit of an analytic function on the unit disk $D \subset {\mathbb C}$ and the growth of its derivative. It is reasonable to expect an analytic function to be smooth on the boundary if its derivative grows slowly, and conversely. Gehring and Martio showed this principle for uniform domains in ${\mathbb R}^{2}$. Astala and Gehring proved quasiconformal analogue of this principle for uniform domains in ${\mathbb R}^n$. We consider \textit{$\alpha$-quasihyperbolic metric}, $k_D^{\alpha}$ and we extend it to proper domains in ${\mathbb R}^n$.
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참고문헌 (12)
참고문헌 신청
K. Astala / 1985 / Quasiconformal analogues of theorems of Koebe and Hardy-Littlewood / Michigan Math. J. 32(1):99~107
P. L. Duren / 1970 / Theory of Hp spaces / Academic Press
F. W. Gehring / 1983 / Quasidisks and the Hardy-Littlewood property / Com-plex Variables Theory Appl 2(1):67~78
F. W. Gehring / 1979 / Uniform domains and the quasi-hyperbolic metric / J. Anal. Math 36:50~74
G. H. Hardy / 1932 / Some properties of fractional integrals. II / Math. Z. 34(1):403~439
P. L. Duren / 1970 / Theory of Hp spaces / Academic Press
F. W. Gehring / 1983 / Quasidisks and the Hardy-Littlewood property / Com-plex Variables Theory Appl 2(1):67~78
F. W. Gehring / 1979 / Uniform domains and the quasi-hyperbolic metric / J. Anal. Math 36:50~74
G. H. Hardy / 1932 / Some properties of fractional integrals. II / Math. Z. 34(1):403~439
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