Weighted $p(\cdot)$-Poincaré and Sobolev inequalities for vector fields satisfying Hörmander’s condition and applications
Volume 274 / 2024
Abstract
We generalize different weighted Poincaré inequalities with variable exponents on Carnot-Carathéodory spaces or Carnot groups, using different techniques. For vector fields satisfying Hörmander’s condition in variable non-isotropic Sobolev spaces, we consider a weight in the variable Muckenhoupt class $A_{p(\cdot ),p^{\ast }(\cdot )}$, where the exponent $p(\cdot )$ satisfies appropriate hypotheses, and in this case we obtain first order weighted Poincaré inequalities with variable exponents. For Carnot groups we also establish higher order weighted Poincaré inequalities with variable exponents. For these results the crucial part is proving the boundedness of the fractional integral operator on Lebesgue spaces with weighted and variable exponents on spaces of homogeneous type. These results extend those obtained by X. Li, G. Lu and H. L. Tang [Acta Math. Sinica English Ser. 31 (2015), 1067–1085], by considering weighted inequalities. They can also be viewed as the extension of weighted Poincaré and Sobolev inequalities widely studied by many authors to the case of variable exponents.
Finally, we use these weighted Poincaré inequalities to establish the existence and uniqueness of a minimizer to the Dirichlet energy integral for a problem involving a degenerate $p(\cdot )$-Laplacian with zero boundary values in Carnot groups.
