Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂t-Δx)u(t,x)=0 on subject to u(0,x)=f(x) on . The main goal of this paper is to characterize such a nonnegative measure μ on that f(x) u(t2,x) induces a bounded embedding from the Sobolev space , p [1,n) into the Lebesgue space , q (0,∞).