• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.59-68
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• 2011

In this paper, we first give the definition of degenerate weakly ($k_1$, $k_2$-quasiregular mappings by using the technique of exterior power and exterior differential forms, and then, by using Hodge decomposition and Reverse H$\"{o}$lder inequality, we obtain the higher integrability result: for any $q_1$ satisfying 0 < $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 $p_1$ = $p_1$(n, l, $k_1$, $k_2$) > l, such that every degenerate weakly ($k_1$, $k_2$)-quasiregular mapping f ${\in}$ $W_{loc}^{1,q_1}$ (${\Omega}$, $R^n$) belongs to $W_{loc}^{1,p_1}$ (${\Omega}$, $R^m$), that is, f is a degenerate ($k_1$, $k_2$)-quasiregular mapping in the usual sense.

• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.29-36
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• 2011

Some global estimates for the Jacobians of quasiregular mappings f = ($f^1$, $f^2$, ${\cdots}$, $f^n$) of the Sobolev class $W^{1,n}$(${\Omega}$, $R^n$) in $L^{\varphi}({\mu})$-domains and John domains are established.