caa a22 4500 605461317 CHVBK 20210128100242.0 cr unu---uuuuu 210128e20150701xx s 000 0 eng 10.1007/s10114-015-4488-x doi (NATIONALLICENCE)springer-10.1007/s10114-015-4488-x Poincaré and Sobolev inequalities for vector fields satisfying Hörmander's condition in variable exponent Sobolev spaces [Elektronische Daten] [Xia Li, Guo Lu, Han Tang] 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. Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg, 2015 Poincaré inequalities nationallicence the representation formula nationallicence fractional integrals on homogeneous spaces nationallicence vector fields satisfying Hörmander's condition nationallicence stratified groups nationallicence high order non-isotropic Sobolev spaces with variable exponents nationallicence Sobolev inequalities with variable exponents nationallicence Li Xia School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, 100875, Beijing, P. R. China aut Lu Guo Department of Mathematics, Wayne State University, 48202, Detroit, MI, USA aut Tang Han School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, 100875, Beijing, P. R. China aut Acta Mathematica Sinica, English Series Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 31/7(2015-07-01), 1067-1085 1439-8516 31:7<1067 2015 31 10114 https://doi.org/10.1007/s10114-015-4488-x text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s10114-015-4488-x text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Li Xia School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, 100875, Beijing, P. R. China aut NATIONALLICENCE 700 1- Lu Guo Department of Mathematics, Wayne State University, 48202, Detroit, MI, USA aut NATIONALLICENCE 700 1- Tang Han School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, 100875, Beijing, P. R. China aut NATIONALLICENCE 773 0- Acta Mathematica Sinica, English Series Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society 31/7(2015-07-01), 1067-1085 1439-8516 31:7<1067 2015 31 10114