Hardy-Littlewood property and $\alpha$-quasihyperbolic metric
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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 ${\mathbb R}^n$.
Abstract
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 ${\mathbb R}^n$.
참고문헌 (12)
- K. Astala, "Quasiconformal analogues of theorems of Koebe and Hardy-Littlewood", Michigan Math. J., 32(1), pp.99~107 (1985)[DOI]
- P. L. Duren, "Theory of Hp spaces" (1970), Academic Press
- F. W. Gehring, "Quasidisks and the Hardy-Littlewood property", Com-plex Variables Theory Appl, 2(1), pp.67~78 (1983)[DOI]
- F. W. Gehring, "Uniform domains and the quasi-hyperbolic metric", J. Anal. Math, 36, pp.50~74 (1979)[DOI]
- G. H. Hardy, "Some properties of fractional integrals. II", Math. Z., 34(1), pp.403~439 (1932)[DOI]
- J. Heinonen, "Lectures on Analysis on Metric Spaces, Universitext" (2001), Springer-Verlag
- 김기원, "INNER UNIFORM DOMAINS, THE QUASIHYPERBOLIC METRIC AND WEAK BLOCH FUNCTIONS", 대한수학회보, 49(1), pp.11~24 (2012), 대한수학회
- P. Koskela, "An inverse Sobolev lemma", Rev. Mat. Iberoamericana, 10(1), pp.123~141 (1994)[DOI]
- N. Langmeyer, "The quasihyperbolic metric, growth, and John domains" (1996), ProQuest LLC
- N. Langmeyer, "The quasihyperbolic metric, growth, and John domains", Ann. Acad. Sci. Fenn. Math, 23(1), pp.205~224 (1998)
- E. Soultanis, "Distortion of quasiconformal maps in terms of the quasihyperbolic metric", J. Math. Anal. Appl., 402(2), pp.623~634 (2013)[DOI]
- J. Vaisala, "Lectures on n-dimensional quasiconformal mappings", Lecture Notes in Math-ematics, 229 (1971), Springer-Verlag[DOI]