We give sufficient conditions for estimates of the form\[∫|u(x)|qdμ(x)⩽C‖u‖s,p1,u∈Hs,p,{\int {\left | {u(x)} \right |} ^q}d\mu (x) \leqslant C\left \| u \right \|_{s,p}^1,\qquad u \in {H^{s,p}},\]to hold, whereμ(x)\mu (x)is a measure and‖u‖s,p{\left \| u \right \|_{s,p}}is the norm of the Sobolev spaceHs,p{H^{s,p}}. Ifdμ=dxd\mu = dx, this reduces to the usual Sobolev inequality. The general form has much wider applications in both linear and nonlinear partial differential equations. An application is given in the last section.