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Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition in variable exponent Sobolev spaces

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Abstract

In this paper, we will establish Poincaré 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 Poincaré inequalities for vector fields satisfying Hörmander’s condition in variable non-isotropic Sobolev spaces. We also set up the higher order Poincaré 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 Hörmander’s condition, but they also hold for Grushin vector fields as well with obvious modifications.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, P. R. China

    Xia Li & Han Li Tang

  2. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Guo Zhen Lu

Authors
  1. Xia Li
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  2. Guo Zhen Lu
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  3. Han Li Tang
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Correspondence to Han Li Tang.

Additional information

The first and third authors are partly supported by NSFC (Grant No. 11371056 ) and the second author is partly supported by a US NSF grant

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Li, X., Lu, G.Z. & Tang, H.L. Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition in variable exponent Sobolev spaces. Acta. Math. Sin.-English Ser. 31, 1067–1085 (2015). https://doi.org/10.1007/s10114-015-4488-x

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