Poincare and Sobolev Inequalities for Vector Fields Satisfying Hormander＇s Condition in Variable Exponent Sobolev Spaces

In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequ...

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Bibliographic Details
Published in:数学学报：英文版 2015 (7), p.1067-1085
Main Author: Xia LI Guo Zhen LU Han Li TANG
Format: Article
Language:English
Subjects:
Lebesgue空间
Sobolev空间
向量场
庞加莱
索伯列夫不等式
非各向同性
非线性偏微分方程
齐型空间
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Summary:In this paper, we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces. The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type. We obtain the first order Poincare inequalities for vector fields satisfying Hormander＇s condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups. Moreover, we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups. These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p（x）-subLaplacian. Our results are only stated and proved for vector fields satisfying Hormander＇s condition, but they also hold for Grushin vector fields as well with obvious modifications.
ISSN:1439-8516
1439-7617