On Sobolev-Poincare-Friedrichs Type Weight Inequalities

In this paper, we prove a Long-Nie type results on Sobolev-Poincare and Friedrichs inequalities (integral(Omega) vertical bar f(x)vertical bar(q)v(x)dx)(1/q) <= C(integral(Omega) vertical bar del f(x)vertical bar(p)omega dx)(1/p), q >= p > 1, where f is a locally Lipschitz function on Omega, the weights v, sigma = omega(-1/p-1) is an element of L-1,L-loc satisfy some cube conditions and Omega is a convex bounded domain in the case of Poincare's inequality. This result generalizes previously known weighted inequalities to more general class of weights.