Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition in variable exponent Sobolev spaces. (English) Zbl 1339.46033
Authors’ 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 hold for Grushin vector fields as well with obvious modifications.
Reviewer: Jiří Rákosník (Praha)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 42B35 | Function spaces arising in harmonic analysis |
| 26D10 | Inequalities involving derivatives and differential and integral operators |
Keywords:
Poincaré inequalities with variable exponents; representation formula; fractional integrals on homogeneous spaces; vector fields satisfying Hörmander’s condition; stratified Lie groups; non-isotropic Sobolev spaces with variable exponents; Sobolev inequalities with variable exponentsReferences:
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