Carleson measure problems for parabolic Bergman spaces and homogeneous Sobolev spaces

Let $b_{\alpha}^{p}(\mathbb{R}^{1+n}_{+})$ be the space of solutions to the parabolic equation $\partial_{t}u+(-\triangle)^{\alpha}u=0$ $(\alpha\in(0, 1])$ having finite $L^{p}(\mathbb{R}^{1+n}_{+})$ norm. We characterize nonnegative Radon measures $\mu$ on $\mathbb{R}^{1+n}_{+}$ having the property $\|u\|_{L^{q}(\mathbb{R}^{1+n}_{+},\mu)}\lesssim \|u\|_{\dot{W}^{1,p}(\mathbb{R}^{1+n}_{+})},$ $1\leq p\leq q<\infty,$ whenever $u(t,x)\in b_{\alpha}^{p}(\mathbb{R}^{1+n}_{+})\cap \dot{W}^{1.p}(\mathbb{R}^{1+n}_{+}).$ Meanwhile, denoting by $v(t,x)$ the solution of the above equation with Cauchy data $v_{0}(x),$ we characterize nonnegative Radon measures $\mu$ on $\mathbb{R}_{+}^{1+n}$ satisfying $\|v(t^{2\alpha},x)\|_{L^{q}(\mathbb{R}_{+}^{1+n}, \mu)}\lesssim\|v_{0}\|_{\dot{W}^{\beta,p}(\mathbb{R}^{n})},$ $\beta\in (0,n),$ $p\in [1, n/\beta],$ $q\in(0, \infty).$ Moreover, we obtain the decay of $v(t,x),$ an iso$-$capacitary inequality and a trace inequality.