lder inequality;"/> lder inequality;"/> Degenerate Weakly (k<sub>1</sub>, k<sub>2</sub>)-Quasiregular Mappings | Korea Science
Degenerate Weakly (k1, k2)-Quasiregular Mappings

• Journal title : Kyungpook mathematical journal
• Volume 51, Issue 1,  2011, pp.59-68
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2011.51.1.059
Title & Authors
Degenerate Weakly (k1, k2)-Quasiregular Mappings
Gao, Hongya; Tian, Dazeng; Sun, Lanxiang; Chu, Yuming;

Abstract
In this paper, we first give the definition of degenerate weakly ($\small{k_1}$, $\small{k_2}$-quasiregular mappings by using the technique of exterior power and exterior differential forms, and then, by using Hodge decomposition and Reverse H$\small{\"{o}}$lder inequality, we obtain the higher integrability result: for any $\small{q_1}$ satisfying 0 < $\small{k_1({n \atop l})^{3/2}n^{l/2}\;{\times}\;2^{n+1}l\;{\times}\;100^{n^2}\;$2^l(2^{n+3l}+1)$\;(l-q_1)}$ < 1 there exists an integrable exponent $\small{p_1}$ = $\small{p_1}$(n, l, $\small{k_1}$, $\small{k_2}$) > l, such that every degenerate weakly ($\small{k_1}$, $\small{k_2}$)-quasiregular mapping f $\small{{\in}}$ $\small{W_{loc}^{1,q_1}}$ ($\small{{\Omega}}$, $\small{R^n}$) belongs to $\small{W_{loc}^{1,p_1}}$ ($\small{{\Omega}}$, $\small{R^m}$), that is, f is a degenerate ($\small{k_1}$, $\small{k_2}$)-quasiregular mapping in the usual sense.
Keywords
Degenerate weakly ($\small{k_1}$, $\small{k_2}$)-quasiregular mapping;exterior power;Hodge decomposition;Reverse H$\small{\"{o}}$lder inequality;
Language
English
Cited by
References
1.
B. Borjarski and T. Iwaniec, Analytical foundations of the theory of quasiconformal mappings in $R^n$. Ann. Acad. Sci. Fenn Ser. A.I. Math., 8(1983), 257-324.

2.
S. K. Donaldson and D. P. Sullivan, Quasiconformal 4-manifolds, Acta Math., 163(1989), 181-252.

3.
H. Gao, Regularity for weakly ($K_1,K_2$)-quasiregular mappings, Science in China, 46(2003), 499-505.

4.
M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems. Ann of Math Stud, 105. Princeton: Princeton Univ. Press, 1983.

5.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order. Berlin: Springer-Verlag, 1983.

6.
T. Iwaniec, G. Martin, Geometric function theory and non-linear analysis, Clarendon Press, Oxford, 2001.

7.
T. Iwaniec and G. Martin, Quasiregular mappings in even dimensions, Acta Math., 170(1993), 29-81.

8.
T. Iwaniec, p-Harmonic tensors and quasiregular mappings, Ann of Math., 136(1992), 589-624.

9.
Yu. G. Reshetnyak, Space mappings with bounded distortion, vol 73, Trans. Math. Monographs, Amer. Math. Soc., 1989.

10.
S. Rickman, Quasiregular mappings, Berlin, Heidelberg: Springer-Verlag, 1993.

11.
S. Zheng and A. Fang, $L^p$-integrability for ($K_1,K_2$)-quasiregular mappings, Acta Math. Sin., 41(5)(1998), 1019-1024.

12.
S. Zheng and A. Fang, On degenerate quasiregular mappings, Chn. Ann. of Math., 19A(6)(1998), 741-748.