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

TL;DR

This paper establishes weighted Poincaré and Sobolev inequalities with variable exponents on Carnot-Carathéodory spaces and groups, using advanced techniques to handle weights and variable exponents, with applications to degenerate p(·)-Laplacian problems.

Contribution

It introduces the first order and higher order weighted Poincaré inequalities with variable exponents on Carnot groups and spaces, extending previous results to more general weighted and variable exponent settings.

Findings

Proved boundedness of fractional integral operators on weighted variable exponent spaces.

Established existence and uniqueness of minimizers for degenerate p(·)-Laplacian energy problems.

Extended inequalities to cases with jump exponents and smaller Muckenhoupt weights.

Abstract

In this paper we will establish different weighted Poincar\'{e} inequalities with variable exponents on Carnot-Carath\'{e}odory spaces or Carnot groups. We will use different techniques to obtain these inequalities. For vector fields satisfying H\"{o}rmander'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 the first order weighted Poincar\'{e} inequalities with variable exponents. In the case of Carnot groups we also set up the higher order weighted Poincar\'{e} 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. Moreover, using other techniques, we extend some of these results when the exponent satisfies a jump condition and the weight is in a smaller Muckenhoupt class. Finally, we will use these weighted Poincar\'{e} 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.