Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces Original Research Article

Let View the MathML sourcebαp(R+1+n) be the space of solutions to the parabolic equation View the MathML source∂tu+(−△)αu=0(α∈(0,1]) having finite View the MathML sourceLp(R+1+n) norm. We characterize nonnegative Radon measures μμ on View the MathML sourceR+1+n having the property View the MathML source‖u‖Lq(R+1+n,μ)≲‖u‖Ẇ1,p(R+1+n), 1≤p≤q<∞1≤p≤q<∞, whenever View the MathML sourceu(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 View the MathML sourceR+1+n satisfying View the MathML source‖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.