Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let bαp(R+1+n) be the space of solutions to the parabolic equation ∂tu+(−△)αu=0(α∈(0,1]) having finite Lp(R+1+n) norm. We characterize nonnegative Radon measures μμ on R+1+n having the property ‖u‖Lq(R+1+n,μ)≲‖u‖Ẇ1,p(R+1+n), 1≤p≤q<∞1≤p≤q<∞, whenever u(t,x)∈bαp(R+1+n)∩Ẇ1.p(R+1+n). Meanwhile, denoting by v(t,x)v(t,x) the solution of the above equation with Cauchy data v0(x)v0(x), we characterize nonnegative Radon measures μμ on R+1+n satisfying ‖v(t2α,x)‖Lq(R+1+n,μ)≲‖v0‖Ẇβ,p(Rn), β∈(0,n)β∈(0,n), p∈[1,n/β]p∈[1,n/β], q∈(0,∞)q∈(0,∞). Moreover, we obtain the decay of v(t,x)v(t,x), an isocapacitary inequality and a trace inequality.