论文标题
加权$ p（\ cdot）$ - Poincaré和Sobolev不平等] {加权$％p（\ cdot）$ - Poincaré和Sobolev的矢量领域不平等，满足Hörmander的状况和应用
Weighted $p(\cdot)$-Poincaré and Sobolev inequality]{Weighted $% p(\cdot )$-Poincaré and Sobolev inequalities for vector fields satisfiying Hörmander's condition and applications
论文作者
论文摘要
在本文中，我们将建立不同的加权庞加莱不平等现象，并在Carnot-Carathéodory空间或Carnot组上具有可变的指数。我们将使用不同的技术来获得这些不平等。 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 the first order weighted Poincaré指数可变的不平等。对于Carnot组，我们还设置了具有可变指数的高阶加权庞加莱不平等现象。对于这些结果，至关重要的部分证明了在同质类型空间上具有加权和可变指数的Lebesgue空间上的分数积分运算符的界限。此外，使用其他技术，当指数满足跳跃条件并且重量为较小的粉碎机类时，我们将扩展其中的一些结果。 最后，我们将使用这些加权的庞加莱不等式来确定最小化的存在和唯一性，以使Dirichlet Energy积分不可或缺地组成，该问题涉及carnot群体中零边界值的退化$ P（\ cdot）$ laplacian。
In this paper we will establish different weighted Poincaré inequalities with variable exponents on Carnot-Carathéodory spaces or Carnot groups. We will use different techniques to obtain these inequalities. 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 the first order weighted Poincaré inequalities with variable exponents. In the case of Carnot groups we also set up the 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. 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 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.
