Two-weight Sobolev-Poincaré inequalities and Harnack inequality for a class of degenerate elliptic operators (Q1335480)

The classical Sobolev-Poincaré inequality \[ \Biggl( -\hskip-.9em\int_ B | f-f_ B|^ q dx\Biggr)^{1/q}\leq Cr \Biggl( -\hskip- .9em\int_ B |\nabla f|^ p dx\Biggr)^{1/p}, \tag{1} \] where \(B\) is a ball in \(\mathbb{R}^ N\) with radius \(r\) and \(-\hskip-.9em\int_ E v dx= {1\over {| E|}} \int_ E v(x)dx\) is the average of \(v\) and \(1\leq p<N\), \(0<q\leq {{pN} \over {N-p}}\), was generalized in various directions, e.g. replacing the Lebesgue averages by averages with respect to measures of the type \(v(x)dx\) and with \(\nabla f\) replaced by expressions containing vector fields. Here, these approaches are unified, at least dealing with a generalized Grushin operator of the form \(\Delta_ \lambda= \Delta_ x+ \lambda^ 2(x) \Delta_ y\) in \(R^ N= \mathbb{R}_ x^ n\times \mathbb{R}_ y^ m\) with \(\lambda\) continuous and nonnegative, and (1) is generalized to \[ \Biggl( -\hskip-.9em\int_{B_ \rho (z_ 0,r)} | g-\mu |^ q u(z)dz \Biggr)^{1/q}\leq Cr \Biggl( -\hskip-.9em\int_{B_ \rho (z_ 0,r)} |\nabla_ \lambda g|^ p v(z)dz \Biggr)^{1/p} \tag{2} \] with appropriate weight functions \(u\), \(v\), \(1\leq p<q <\infty\), with \(\lambda\) a strong \(A_ \infty\) weight satisfying the reverse Hölder inequality and with \(\varphi\) a natural metric associated with \(\Delta_ \lambda\) by means of the so-called sub-unit curves. In (2), \(\mu\) can be chosen as the \(u\)- average of \(g\) over \(B_ \rho (z_ 0,r)\). The result is applied to prove Harnack's inequality for positive weak solutions of some degenerate elliptic equations.