Is it possible to combine the Sobolev inequality in two regions into one in the whole region?

Question

Assume that $M$ is a non-compact manifold and $K$ is a compact set of $M$. Assume that Sobolev inequality holds on $M\backslash K$: $$ (\int f^{\frac{2m}{m-2}}~~ d\mu)^\frac{m-2}{m}\le C \int |\nabla f|^2 d\mu, \quad f\in C^\infty_0(M\backslash K) $$ Also on K, Sobolev inequality holds: $$ (\int f^{\frac{2m}{m-2}}~~ d\mu)^\frac{m-2}{m}\le C \int |\nabla f|^2 d\mu, \quad f\in C^\infty_0(K) $$ Is it possible to get Sobolev inequality on whole $M$?