Weighted $p(\cdot )$-Poincar\'{e} and Sobolev inequalities for vector fields satisfying H\"{o}rmander's condition and applications

VALLEJOS LUCAS ALEJANDRO; VIDAL RAÚL EMILIO

STUDIA MATHEMATICA

POLISH ACAD SCIENCES INST MATHEMATICS

Lugar: VARSOVIA; Año: 2023

0039-3223

In this paper we will generalize different weighted Poincar´{e} inequalitieswith variable exponents on Carnot-Carath´{e}odory spaces or Carnot groups.We will use different techniques to obtain these inequalities. For vectorfields satisfying H"{o}rmander´s condition in variable non-isotropicSobolev spaces, we consider a weight in the variable Muckenhoupt class $%A_{p(cdot ),p^{ast }(cdot )}$, where the exponent $p(cdot )$ satisfiesappropriate hypotheses, and in this case we obtain the first order weightedPoincar´{e} inequalities with variable exponents. In the case of Carnotgroups we also set up the higher order weighted Poincar´{e} inequalitieswith variable exponents. For these results the crucial part is proving theboundedness of the fractional integral operator on Lebesgue spaces withweighted and variable exponents on spaces of homogeneous type. These results extend those obtained in [X. Li, G. Lu and H. L. Tang. extit{Poincar´e and Sobolev inequalities for vector fields satisfying H"ormander´s condition in variable exponent Sobolev spaces.} Acta Mathematica Sinica, English Series, 31(7), (2015), 1067--1085], by considering weighted inequalities. They can also be viewed as the extension of weighted Poincar´e and Sobolev inequalities widely studied by many authors to the case of variable exponent case. Finally, we will use these weighted Poincar´{e} inequalities to establishthe existence and uniqueness of a minimizer to the Dirichlet energy integralfor a problem involving a degenerate $p(cdot )$-Laplacian with zeroboundary values in Carnot groups.